3500K.\nFMOS spectra for the 10 new brown dwarfs listed in Table [t1]. The fluxes are on a relative scale with arbitrary offsets. The observed spectrum is shown in black, the best fitting DUSTY model slightly offset, with thin, dashed lines. We masked out the Pa*β* lines for clarity. The identification numbers SONYC-NGC1333-X are abbreviated with S-X. [f20]\nIn Fig. [f4] we compare the FMOS spectra for two of our sources, both originally identified in, with spectra for young brown dwarfs from and old dwarfs with similar spectral types. At late M spectral types, the more rounded peak of the old objects becomes visible (compare the young KPNO4 with the old LHS2924). However, for earlier types the difference between young and old is not significant. The plot also illustrates that sufficient signal-to-noise is required to distinguish between the two types of objects. Our spectra often do not have the necessary quality to make this decision. From these arguments it follows that contamination by late M-dwarfs in the field cannot completely be excluded. Given the compact nature of the NGC1333 cluster (see Sect. [s44]) and the low space densities of such objects, the contamination is considered to be negligible ($\\lesssim 1$).\nSpectral types\n--------------\nSpectral classification for cool dwarfs in the near-infrared relies mostly on the broad H2O absorption features mentioned above. A number of spectral indices has been suggested in the literature. Relevant for us are the indices that are calibrated for young brown dwarfs and thus take into account the fact that the absorption features are sensitive to gravity. Such indices have been proposed by,, and. We find, however, that these schemes are of limited use for our low-resolution and (mostly) low signal-to-noise spectra, mainly because the wavelength offset between the two intervals from which the index is derived is relatively small, e.g. 0.058 *μ**m* for the H2O index or 0.103 *μ**m* for the WH index.\nFor our purposes we require a flux ratio that maximises the contrast between the numerator and the denominator. This is achieved by using the flux ratio between 1.675-1.685 *μ**m* and 1.495-1.505 *μ**m*. The first interval is chosen because it corresponds to the position of the H-band peak for young objects with M6-M9 spectral type. The second interval marks the lowest flux level on the blue side of the H-band peak for such objects.\nlccccc ITG2 & M7.25 & 11.540 & 10.097 & 2.40 & 1.050 \nJ0444 & M7.25 & 12.195 & 10.761 & 2.35 & 1.124 \nCFHT6 & M7.5 & 12.646 & 11.368 & 1.51 & 1.135 \nKPNO2 & M7.5 & 13.925 & 12.753 & 0.93 & 1.136 \nKPNO5 & M7.5 & 12.640 & 11.536 & 0.56 & 1.137 \nCFGH3 & M7.75 & 13.724 & 12.367 & 1.94 & 1.184 \nJ0441 & M7.75 & 13.730 & 12.220 & 2.77 & 1.098 \nCFHT4 & M7.0 & 12.168 & 10.332 & 4.53 & 1.009 \nMHO4 & M7.0 & 11.653 & 10.567 & 0.47 & 1.174 \nKPNO7 & M8.25 & 14.521 & 13.271 & 1.37 & 1.121 \nKPNO1 & M8.5 & 15.101 & 13.772 & 1.78 & 1.203 \nKPNO6 & M8.5 & 14.995 & 13.689 & 1.63 & 1.185 \nKPNO9 & M8.5 & 15.497 & 14.185 & 1.69 & 1.155 \nLRL405 & M8.0 & 15.28 & 13.91 & 2.01 & 1.126 \nJ0457 & M9.25 & 15.771 & 14.484 & 1.56 & 1.240 \nKPNO4 & M9.5 & 14.997 & 13.281 & 3.91 & 1.342 \nKPNO12 & M9.0 & 16.305 & 14.927 & 2.01 & 1.261 \nJ1207 & M8.0 & 12.995 & 11.945 & 0.81 & 1.226 \nJ1139 & M9.0 & 12.686 & 11.503 & 0.99 & 1.336 \nJ1245 & M9.5 & 14.518 & 13.369 & 0.81 & 1.306\nThis index, dubbed HPI for H-peak index, is illustrated in the left and middle panel of Fig. [f4] where we show literature spectra for young and old brown dwarfs in Taurus with spectral types M7 and M9.5 compared with FMOS data for two of the brown dwarfs in NGC1333. The H-band peak clearly is highly sensitive to the spectral type in this regime. We expect that this index increases from mid M to late M spectral types. The figure also demonstrates the advantages of this index at low signal-to-noise ratio.\n \nTo calibrate the HPI, we use literature spectra for 20 young brown dwarfs with spectral types M7-M9.5, for which spectra are publicly available in the SpeX Prism Spectral Libraries. Their spectra have been dereddened using the reddening law *A**λ* = (*λ*/1.235 *μ**m*)− 1.61 ⋅ *A**J* and the relation *A**J* = 0.3088 ⋅ *A**V*. The optical extinction *A**V* is derived from their *J* − *K* colours:\n \n*A**V* = [(*J* − *K*) − (*J* − *K*)0]/0.1844 \nThese relations are based on the extinction law from for *R**V* = 4.0, which is used throughout this paper. We note that *R**V* varies within star forming regions typically from 3 to 5; the adopted value is an average from the values measured by for diffuse and dense regions of the interstellar medium. It is also a reasonable average for our target region NGC1333.\nFor the reasons outlined in we use (*J* − *K*)0 = 1.0. The resulting uncertainty in *A**V* is ∼ 1mag. For example it is possible that we overestimate *A**V* due to the presence of K-band excess from a disk. This induces an uncertainty of up to 0.04 in the HPI. Note that the literature spectral types for the calibrators have been determined in the optical by comparison with templates. The internal accuracy of these optical types is typically ± 0.25 subtypes. The calibrators are listed in Table [t3].\nH-peak index (HPI) calculated as flux ratio between 1.675-1.685 *μ**m* and 1.495-1.505 *μ**m* plotted vs. (optical) spectral type for published spectra from and. The correlation after excluding the outlier MHO4 at M7 is SpT = − 0.84 + 7.66 ⋅ HPI (dashed line). A typical errorbar is overplotted. [f3]\nIn Fig. [f3] we plot the HPI for the 20 calibrators. The plot shows the expected correlation between index and spectral type. The one outlier at spectral type M7 corresponds to the object MHO4 and deviates in its HPI by 3*σ* from a linear fit. Nothing particular can be seen in its spectrum which would explain this discrepancy. Based on the H-band peak, MHO4 appears to have a later spectral type than indicated in the literature. Excluding MHO4, we derive a correlation of\n \nSpT = − 0.84 + 7.66 ⋅ HPI \nwith an 1*σ* scatter of 0.37 subclasses. The 1*σ* scatter in HPI around the correlation is 0.039. As outlined above, this can be attributed to the uncertainty in the extinction. The HPI is properly calibrated for types M7-M9.5, but may also hold for later spectral types as the H-band peak increases in strength in the L-type regime (see below).\nWe note that this correlation does not hold for field dwarfs. As seen in the middle panel of Fig. [f4] these objects have flatter H-band peaks which results in lower values for the HPI for a given spectral type.\nUsing this index we determined spectral types for all 26 objects in the YVLM sample, after dereddening using the corrected extinction derived in Sect. [s33]. 9 of them have published spectral types. Five of them (MBO69, MBO54, MBO77, ASR29, ASR83) are based on an index defined in the K-band. The other four have the Spitzer IDs 131, 104, 118, and 46 in and were classified in the optical. Excluding MBO54, ASR29, Sp 104 and Sp 118 which have a published spectral type of M6 or earlier, for which the HPI is not properly calibrated, the HPI types deviate by -0.4, -0.2, -0.2, -0.1, +0.6 subtypes from the published ones, which provides some reassurance in the usefulness of the HPI. The 10 new objects have spectral types of M6.9 or later, classifying them as likely brown dwarfs. The spectral types for the new and known sources are listed in Table [t1] and [t2] respectively.\nIn the right panel of Fig. [f4] we compare FMOS data for the four latest-type SONYC objects with spectra for young ultracool objects with published spectral types: the M8 companion to HIP78530, the L2 companion to the TW Hya brown dwarf 2M1207, and the L4 companion to 1RXSJ1609-2105B, all three with ages of 5-10Myr. Using the HPI, we get spectral types of M8, L2, and L3.5 for these three comparison objects, consistent with the literature values, which indicates that HPI could be useful for spectral typing of young early L-dwarfs as well. In this plot we approximate the spectral slopes for the two faintest objects in our sample with linear fits on either side of the H-band peak, to facilitate the comparison. The plot demonstrates that objects SONYC-NGC1333-1, 30, 31 are about or later than M8 and clearly earlier than L2, in line with our classification. The object SONYC-NGC1333-36 compares well with the L2 and L4 templates, which makes it the coolest object discovered in NGC1333 so far.\nWe note that the redward slope of the H-band peak appears anomalously steep for SONYC-NGC1333-31 and 36. This is not introduced by the linear fit or the treatment of the spectra, and it is not seen in the other FMOS spectra for NGC1333 or *ρ*Oph (Muzic et al., subm.), which makes a calibration problem unlikely. The effect is difficult to explain, as these two spectra are very noisy, but it could in principle be a real feature, especially since the data is well-matched by the model spectrum (see Sect. [s33]). Better quality spectra are needed to verify the parameters for these objects. Since the HPI is defined for the blueward slope of the H-band peak, this does not affect the spectral typing.\nModel fitting\n-------------\nFor the 26 objects in the YVLM sample the FMOS spectra were compared with AMES-DUSTY models for low gravity (log*g* = 3.5 or, if not available, 4.0) low-mass stars[1](#fn1). Using the extinction law described in Sect. [s32] we calculated a model grid for *T*eff = 1500 to 3900K in steps of 100K and *A**V* = 1 to 20mag in steps of 1mag. The models were binned to 5Å, the same binsize as the FMOS data.\nThe fitting was done in a semi-interactive way. Since extinction and effective temperature cannot be determined separately with low-resolution, low signal-to-noise spectra, we started by adopting the *A**V* determined from the *J* − *K* colour, using Equ. (1). For each of the YVLM objects, we calculated the following test quantity (*O* - observed flux; *M* - model flux; *N* - number of datapoints):\n \n$$\\chi = \\frac{1}{N} \\sum\\_{i=1}^{N} \\frac{(O - M)^2}{M}$$ \nThis was done for the series of models using the ’photometric’ *A**V*; we selected the one with the minimum *χ*, which is typically between 0.005 and 0.05 (with one exception with 0.2). This means that the average deviation between observed and model spectrum in a given wavelength bin of 5Å is in the same range as the noise in the observations.\nUsually a few model spectra (2-4) give indistinguishable *χ*; we adopt the average *T*eff from these best fitting models. A visual inspection of the observed spectra with models for a range of temperatures shows that clear discrepancies are visible for temperatures which are > 200K different from the adopted value, i.e. the uncertainty in the adopted values is ± 200K. We note that effective temperatures are necessarily model dependent; our values should only be interpreted in the context of the AMES-DUSTY models. The best fitting models are plotted as thin, dashed lines in Fig. [f20] for the 10 newly identified objects.\nIn 6/10 cases this initial fit is already convincing. In two more cases, it can be improved by slightly adjusting *A**V* by 1mag, which is within the uncertainty for *A**V* (see Sect. [s32]). The two remaining cases give the best fit when *A**V* is changed by 2mag compared with the initial estimate. In Table [t1] we list the photometric and adjusted value for *A**V* and the best fit value for the effective temperature.\nFor SONYC-NGC1333-36, the coolest object in our sample, we find an effective temperature of 2250K, which is significantly higher than the published values for the two comparison objects shown in Fig. [f4] (right panel). For 1RXSJ1609-2105B find 1800 ± 200K based on DUSTY and Drift-Phoenix models. For 2M1207B, two independent groups determined 1600 ± 100K, again based on comparison with DUSTY spectra, although suggest that the actual value might be as low as 1000K. The DUSTY spectra for < 2000K are clearly not in agreement with our spectrum for SONYC-NGC1333-36. One possible explanation is that model fitting for the comparison objects has been done over the full near-infrared range, whereas in our case *T*eff is mostly fixed by the shape of the H-band peak. We will revisit this issue in Sect. [s34].\nComparison between the spectral types and effective temperatures for the YVLM sample, as determined in this paper (Sect. [s32] and [s33]). A typical errorbar is overplotted. The solid line is a linear fit to the datapoints excluding the 3 outliers marked with squares. We also show the effective temperature scales by and. [f12]\nIn Fig. [f12] we compare the derived effective temperatures with the spectral types determined with the HPI in Sect. [s32]. The expected trend – cooler temperatures correspond to later spectral types – is visible, with only three exceptions. For two of these exceptions (SONYC-NGC1333-17 and -31), the signal-to-noise ratio in the spectrum is very poor ( ∼ 10), i.e. the uncertainties in spectral type and temperature are large. For SONYC-NGC1333-17 there are two published spectral types of M8 and M8.7, which would move the datapoint closer to the general trend. The third outlier (SONYC-NGC1333-33) has an unusual dip in the spectrum around 1.65 *μ**m*, which decreases our spectral type estimate by ∼ 1 subtype, but does not affect the model fitting. Excluding these three outliers, the datapoints are well-approximated by a linear trend (solid line): *T*eff = 4191 − 157 × SpT (where SpT corresponds to the M subtype).\nThe plot also shows the effective temperature scales by and, which are derived using the optical portion of the spectrum. The two scales agree fairly well with each other, although the Mentuch et al. scale is an extrapolation and has not been directly calibrated for *T*eff < 3000K. The trend seen in our data is consistent with these lines. Our datapoints are, however, mostly above the two lines, indicating that we systematically overestimate the temperatures (by ∼ 200K) or the spectral types (by 0.5-1 subtypes).\nWe explored possible reasons for this discrepancy. One option is a systematic error in the extinction. Decreasing *A**V* by 1mag for all objects would shift their spectral types to earlier types, but only by ∼ 0.3 subtypes, not sufficient to explain the effect. Moreover, this would also increase the best estimate for *T*eff by typically 200K and thus cause larger disagreement between datapoints and published effective temperature scales. The inverse is true as well – increasing *A**V* would lead to lower *T*eff, as required, but also to later spectral types. Thus, systematic changes in *A**V* do not resolve the problem. Given the good agreement between our spectral types and literature values (Sect. [s32]), we suspect that the offset is most likely due to a problem with the effective temperatures and could indicate issues with the used model spectra. A more extensive comparison of the effective temperature scales is beyond the scope of this paper.\nSpectral flux distributions\n---------------------------\nTo further test the spectral fitting from Sect. [s33] we compare the full photometric spectral flux distributions (SFD) for some selected sources, as far as available, with the AMES-DUSTY model spectra. We include our own photometry in izJK as well as Spitzer/IRAC data from the C2D ’HREL’ catalogue. In Fig. [f5] we show 3 examples.\n \nObject SONYC-NGC1333-5 is well-matched by a model spectrum for *T*eff of 2800-2900K for *A**V* = 4mag, in line with the parameters derived from the FMOS spectrum (Table [t2]). This object shows colour excess redwards of 5 *μ**m*, presumably due to the presence of a disk (see Sect. [s45]). The best match for object SONYC-NGC1333-30 is obtained for *T*eff of 2500-2700 with *A**V* of 9-10mag, i.e. the object may be slightly cooler than listed in Table [t1], but still within the uncertainty. For the coolest object SONYC-NGC1333-36 a good match is found for temperatures between 1900 and 2100K and *A**V* of 0-2mag. This is again somewhat cooler than the estimate given in Table [t1].\nIn general, when comparing with the full SFD the results are similar to the spectral fits to individual wavelength bands in the near-infrared, maybe except for the regime below 2500K where the SFD comparison yields lower temperatures by ∼ 200K. The comparison also illustrates that the ideal dataset for a characterisation of young brown dwarfs would be a spectrum covering the entire near-infrared domain from 1 to 8 *μ**m*, thus including five broadband features.\nThe brown dwarf population in NGC1333\n=====================================\nThe newly identified very low mass objects in this paper add to the substantial number in the NGC1333 region that have already been confirmed in the literature. In Table [t2] we compile all previously spectroscopically confirmed objects with spectral types of M5 or later or effective temperatures of 3200K or below, from,,, and. Whenever possible, we also re-measured the HPI spectral types for the sources identified in, based on their MOIRCS spectra. In the Table, we list coordinates, photometry, spectral types, effective temperatures, and alternative names. Adding the 10 objects discovered in this paper, the entire sample comprises 51 objects.\nNot all these objects are brown dwarfs; some are very low mass stars. Based on the COND, DUSTY, and BCAH isochrones [2](#fn2), the hydrogen burning limit at 1Myr would be reached at effective temperatures of 2800-2900K, which is found to correspond to an (optical) spectral type of M6-M7. In Table [t2] we made the cut at M5 to include all borderline cases as well. Taking this into account, the number of confirmed brown dwarfs in NGC1333 is about 35 ± 5. This is currently one of the largest and best characterised populations of substellar objects in a single star forming region. In the following we will investigate the mass function, spatial distribution, and disk properties for this sample.\nlllcclll S-1 & 03 28 47.66 & +31 21 54.6 & 17.55 & 15.24 & M9.2*a* & 2600*a*, 2800*b* & MBO139 \nS-2 & 03 28 54.92 & +31 15 29.0 & 15.990 & 14.219 & M7.9*a*, M6.5*b*, M8*d*, M8.6*f* & 2850*a*, 2600*b* & ASR109, Sp 60 \nS-3 & 03 28 55.24 & +31 17 35.4 & 15.090 & 13.433 & M7.9*c*, M8.2*f* & 2900*b* & ASR38 \nS-4 & 03 28 56.50 & +31 16 03.1 & 18.17 & 16.73 & M9.6*f* & 2500*b* & \nS-5 & 03 28 56.94 & +31 20 48.7 & 15.362 & 13.815 & M7.6*a*, M6*b*, M6.8*c* & 2850*a*, 2900*b* & MBO91, Sp 66 \nS-6 & 03 28 57.11 & +31 19 12.0 & 17.24 & 15.34 & M7.3*a*, M8*b*, M8.0*f* & 3250*a*, 2700*b* & MBO148, ASR64, Sp 23 \nS-7 & 03 28 58.42 & +31 22 56.7 & 15.399 & 13.685 & M6.5*b*, M7.1*c*, M7.7*f* & 2800*b* & MBO80, Sp 72 \nS-8 & 03 29 03.39 & +31 18 39.9 & 15.833 & 14.000 & M8.2*a*, M8.5*b*, M7.4*c*, M8.4*f* & 2850*a*, 2600*b* & MBO88, ASR63, Sp 80 \nS-9 & 03 29 05.54 & +31 10 14.2 & 17.072 & 15.667 & M8*b*, M8.6*f* & 2600*b* & \nS-10 & 03 29 05.66 & +31 20 10.7 & 17.113 & 15.485 & & 2500*b* & MBO143, Sp 86 \nS-11 & 03 29 07.17 & +31 23 22.9 & 17.87 & 15.62 & M9.2*f* & (2600)*b* & MBO141 \nS-12 & 03 29 09.33 & +31 21 04.2 & 16.416 & 13.150 & & (2500)*b* & MBO70, Sp 93 \nS-13 & 03 29 10.79 & +31 22 30.1 & 14.896 & 12.928 & M7.5*b*, M7.4*c*, M8.1*f* & 3000*b* & MBO62 \nS-14 & 03 29 14.43 & +31 22 36.2 & 14.606 & 13.035 & M7*b*, M6.6*c*, M7.7*f* & 2900*b* & MBO66 \nS-15 & 03 29 17.76 & +31 19 48.1 & 14.803 & 12.988 & M7.5*b*, M6.5*c*, M7.8*f* & 3000*b* & MBO64, ASR80, Sp 112 \nS-16 & 03 29 28.15 & +31 16 28.5 & 13.054 & 12.091 & M8.5*a*, M7.5*b*, M7.5*e*, M9.1*f* & 2850*a*, 2600*b*, 2761*e* & Sp 164 \nS-17 & 03 29 33.87 & +31 20 36.2 & 16.562 & 15.481 & M7.5*a*, M8*b*, M8.7*f* & 2600*a*, 2500*b* & MBO140 \nS-18 & 03 29 35.71 & +31 21 08.5 & 18.50 & 16.94 & M8.2*f* & 2500*b* & Sp 129 \nS-19 & 03 29 36.36 & +31 17 49.8 & 17.91 & 16.38 & & 2700*b* & \nS-21 & 03 28 47.34 & +31 11 29.8 & 15.484 & 12.702 & & 3100*b* & ASR117 \nASR15 & 03 28 56.94 & +31 15 50.3 & 15.056 & 13.461 & M7.4*c*, M6.0*d* & & \nASR17 & 03 28 57.15 & +31 15 34.5 & 15.405 & 13.186 & M7.4*c*, M6.0*d* & & Sp 68 \nMBO73 & 03 28 58.24 & +31 22 09.3 & 16.004 & 13.367 & M6.4*c* & & Sp 70 \nASR24 & 03 29 11.30 & +31 17 17.5 & 13.977 & 12.915 & M8.2*c*, M8.0*d* & & \nMBO69 & 03 29 24.45 & +31 28 14.9 & 14.041 & 12.686 & M7.0*a*, M7.4*c* & 3200*a* & \nASR29 & 03 29 13.61 & +31 17 43.4 & 16.441 & 13.028 & M5*d* & 3250*a* & \nASR105 & 03 29 04.66 & +31 16 59.1 & 15.550 & 12.665 & M6*d* & & Sp 84 \nASR8 & 03 29 04.06 & +31 17 07.5 & 13.310 & 12.313 & M7*d* & & \nMBO78 & 03 29 00.15 & +31 21 09.2 & 16.466 & 13.349 & M5*d* & & Sp 75 \nSp 45 & 03 28 43.55 & +31 17 36.4 & 12.219 & 10.138 & M5.0*e* & 3125*e* & ASR127 \nSp 46 & 03 28 44.07 & +31 20 52.8 & 14.245 & 12.627 & M7.3*a*, M7.5*e* & 3050*a*, 2829*e* & \nSp 49 & 03 28 47.82 & +31 16 55.2 & 12.940 & 10.909 & M8.0*e* & 2710*e* & ASR111 \nSp 53 & 03 28 52.13 & +31 15 47.1 & 13.161 & 12.029 & M7.0*e* & 2846*e* & ASR45 \nSp 55 & 03 28 52.90 & +31 16 26.4 & 13.616 & 12.476 & M5.0*e* & 3154*e* & ASR46 \nSp 58 & 03 28 54.07 & +31 16 54.3 & 13.027 & 11.599 & M5.0*e* & 3098*e* & ASR42 \nSp 94 & 03 29 09.48 & +31 27 20.9 & 14.154 & 12.692 & M5.0*e* & 3098*e* & MBO60 \nSp 105 & 03 29 13.03 & +31 17 38.3 & 15.231 & 14.158 & M8.0*e* & 2710*e* & ASR28 \nSp 131 & 03 29 37.73 & +31 22 02.4 & 13.987 & 12.958 & M7.6*a*, M7.0*e* & 2850*a*, 2891*e* & MBO65 \nSp 157 & 03 29 12.79 & +31 20 07.7 & 14.676 & 13.294 & M7.6*a*, M7.7*c*, M7.5*e* & 2950*a*, 2812*e* & MBO75, ASR83 \nSp 177 & 03 29 24.83 & +31 24 06.2 & 14.433 & 13.383 & M6.5*a*, M6.7*c*, M6.5*e* & 3000*a*, 2957*e* & MBO77 \nSp 71 & 03 28 58.24 & +31 22 02.1 & 14.912 & 12.406 & M6*e* & 2990*e* & MBO47 \nThe number of brown dwarfs\n--------------------------\nBased on our comprehensive spectroscopy, we can put some constraints on the total number of brown dwarfs in NGC1333 and the mass limits of the current surveys. For this purpose, we use the iz survey presented in. In Fig. [f8] we plot the iz colour-magnitude diagram for the 196 candidates selected in. The confirmed brown dwarfs (Tables [t1] and [t2]), either by us or other groups, are marked with squares; all objects for which we have obtained spectroscopy with crosses. Note that the iz candidates are selected only with a cut in colour and a cut in PSF shape to rule out extended objects. No other selection criteria have been used, i.e. this sample is as unbiased as possible.\nColour-magnitude diagram for the iz candidates (dots), originally identified in. Crosses are objects for which we have obtained spectra in this paper or. Confirmed very low mass objects from this paper or the literature are marked with squares. The horizontal dashed line shows the completeness limit of the survey estimated in. [f8]\nWe have useful spectra for 98/196 candidates; out of these 98, 24 are confirmed by our spectra. In total the iz sample contains 35 confirmed objects with $A\\_V \\lesssim 12$mag. Thus, we have a yield of 24/98 (24%) and would expect to find 24 more objects among the candidates for which we do not have spectra. Since 11 of them (35 minus 24) have already been confirmed by other groups, the expected number of additional very low mass objects from this iz selection is 13.\nThe low-mass end of the diagram deserves additional discussion. The faintest confirmed brown dwarfs in Fig. [f8] are SONYC-NGC1333-1, 30, and 36 at *i* = 23.4 − 23.7. Comparing their effective temperatures with the 1Myr DUSTY and COND isochrones, it seems likely that they have masses of $\\lesssim 0.02\\,M\\_{\\odot}$. If our estimate of *T*eff = 2000K for SONYC-NGC1333-36 is correct (Sect. [s34]), the best mass estimate would be in the range of 0.006 *M*⊙.\nWe have taken spectra for 7 fainter objects, but none of them is a brown dwarf, which might indicate that we have reached the ’bottom of the IMF’, as preliminarily stated in. However we may not be 100% complete in this magnitude range. The formal completeness limits of the iz survey at *i* = 24.8mag, determined from the peak of the histogram of the magnitudes, is shown with a dashed line. This limit has been derived for a field of view of 34ʹ × 27ʹ, but most of the cluster members are located in a smaller region of 10ʹ × 10ʹ which is partially affected by significant background emission from the cloud (see Fig. [f50]. Thus, the completeness limit in the relevant areas might not always reach the value shown in Fig. [f8].\nThus, based on our new data we retract the previous claim by stating a deficit of objects with *M* < 0.02 *M*⊙ in NGC1333, for two reasons: a) The updated brown dwarf census contains a few of objects with masses at or below 0.02 *M*⊙, including one with an estimated mass of 0.006 *M* ⊙ . b) The current survey may not be complete at the lowest masses, i.e. we cannot exclude the presence of a few more objects with *M* < 0.02 *M*⊙.\nThe census for *M* > 0.02 *M*⊙ is more robust. From the 35 confirmed members in the iz diagram we subtract 10 which are probably slightly above the substellar boundary (see discussion in Sect. [s4]). We also subtract the 3 which are likely below 0.02 *M*⊙ and add the 13 which we are still missing. This gives a total number of ∼ 35 brown dwarfs down to 0.02 *M*⊙ and with $A\\_V\\lesssim 12$mag.\nThe star vs. brown dwarf ratio\n------------------------------\nAs a proxy for the shape of the mass function, previous authors have used the ratio of stars to brown dwarfs, where these two groups are defined by a range of masses. These ratios are more robust against uncertainties in the masses than a complete IMF. use a range of 0.08-1.0 *M*⊙ for stars and 0.03-0.08 *M*⊙ for brown dwarfs, hereafter called *R*1. Other authors use 0.08-10 *M*⊙ for stars and 0.02-0.08 *M*⊙ for brown dwarfs, hereafter called *R*2. Since the number of high-mass stars is small, the two ratios *R*1 and *R*2 should be fairly similar.\nThe comparison with NGC1333 is complicated by the fact that no comprehensive spectroscopic census is available for the stars. The best starting point is probably the Spitzer analysis by. They find a total of 137 Class I and II members with disk, from which 94 are in 2MASS. Objects not detected in 2MASS are likely embedded sources with high extinction *A**V* > 10mag and thus not comparable with our brown dwarf sample. We calculated absolute J-band magnitudes for this sample using the dereddening described in Sect. [s32] and assuming a distance of 300pc, which gives a range of *M**J* = 0 − 9mag. Comparing with the BCAH 1Myr isochrone the sample contains 13 objects with *M* > 1.0 *M*⊙, 52 objects with 0.08 < *M* < 1.0 *M*⊙ and 29 with 0.02 < *M* < 0.08 *M*⊙. These are only objects with disks; correcting for a disk fracton of 83% shifts the numbers to 16, 63, 35. The latter number is consistent with the estimate of brown dwarfs in this cluster given in Sect. [s41]. Out of 35 brown dwarfs, the number of objects with masses above 0.03 *M*⊙ would be 28.\nBased on these estimates the ratios for NGC1333 become *R*1 = 63/28 = 2.3 ± 0.5 and *R*2 = 79/35 = 2.3 ± 0.5 (see below for an explanation of the uncertainties). Our value for *R*1 is somewhat larger than our first estimate given in of 1.5 ± 0.3, mainly because we use here the cutoff at 0.03 *M*⊙ to be consistent with.\nThe uncertainties for *R*1 and *R*2 stated above are 1*σ* confidence intervals and have been derived based on the prescription provided by. This prescription is given for population proportions (’success counts’). Therefore, we use the Cameron equation to calculate the confidence intervals *σ**k*1 for the ratio of number of stars to the sum of stars and brown dwarfs (*k*1) and *σ**k*2 for the ratio of number of brown dwarfs to the same sum (*k*2). The confidence intervals for *R*1 and *R*2 are then derived as follows:\n \n$$R\\_{\\mathrm{max}} = \\frac{k1 + \\sigma\\_{k1}}{k2 - \\sigma\\_{k2}}$$ \n \n$$R\\_{\\mathrm{min}} = \\frac{k1 - \\sigma\\_{k1}}{k2 + \\sigma\\_{k2}}$$ \nlll NGC1333 & 2.3 (1.8-2.8) & 2.3 (1.8-2.7) \nONC & 3.3 (2.8-3.9) & 3.8,4.5,5.0 \nUpSco & 3.8 (3.1-4.5) & – \nNGC2024 & 3.8 (2.6-5.2) & – \nChamaeleon & 4.0 (2.3-6.0) & 3.9 (2.9-5.0) \n*ρ*-Oph & 5.1 (3.8-6.4) & 4.8 (3.7-6.0) \nTaurus & 6.0 (4.5-7.7) & 7.3 (5.1-9.6) \nIC348 & 8.3 (6.4-10.5) & 8.0 (6.3-10.0)\nIn Table [t4] we compare the ratios for NGC1333 with the available literature values for *R*1 and *R*2 for other regions. The same numbers are plotted in Fig. [f40] for illustration. To have accurate and consistent confidence intervals, we re-calculated the errors for all literature values as described above. NGC1333 has the lowest ratios measured so far in any star forming region, suggesting that the number of brown dwarfs in NGC1333 is unusually high, which is in line with the conclusion in. In particular, the ratios for NGC1333 deviate by more than 2*σ* from those in IC348. It should be noted that the current census for IC348 is nearly complete down to 0.03 *M*⊙ and covers most of the cluster, which makes the difference to NGC1333 even more striking.\nStar vs. brown dwarf ratios for various star forming regions, from Table [t4]. *R*1 is plotted with squares, *R*2 with circles. The number on the y-axis is identical with the row in Table [t4]. [f40]\nThis finding has to be substantiated with further survey work in diverse regions. In Table [t4] we list the statistical 1*σ* confidence intervals, purely based on the sample sizes. These statistical errors do not take into account additional sources of uncertainty, e.g. unrecognised biases, inconsistencies in sample selection or problems with the mass estimates, i.e. the actual errors may be larger than listed in Table [t4]. In particular, it is important to note that all mass estimates are necessarily model-dependent. For the value in NGC1333 we use the BCAH isochrones, mainly because they cover the brown dwarf regime down to the Deuterium burning limit. The problems and uncertainties of these type of models at very young ages are well-documented.\nThe best way of assessing the overall uncertainties is to compare results from independent surveys. As can be seen in Table [t4], so far the results from independent groups agree within the statistical errorbars, with the possible exception of the ONC.[3](#fn3) Such an independent confirmation is required for NGC1333 as well.\nIf confirmed, the unusually low ratio of stars to brown dwarfs in NGC1333 could point to regional differences in this quantity, possibly indicating environmental differences in the formation of very low mass objects. One option to explain this is turbulence, as very low mass cores which can potentially collapse to brown dwarfs could be assembled by the turbulent flow in a molecular cloud. At first glance this could be a realistic possibility for NGC1333, where the cloud is strongly affected by numerous outflows, although it is not clear if the turbulence in NGC1333 is mainly driven by these outflows. Alternatively, additional brown dwarfs could form by gravitational fragmentation of gas falling into the cluster center. This latter mechanism would benefit from the fact that NGC1333 has a higher stellar density and thus a stronger cluster potential than most other nearby star forming regions.\nSpatial distribution\n--------------------\nIn Fig. [f9] we show the spatial distribution of the sample of very low mass objects listed in Tables [t1] and [t2] (squares). For comparison, the positions of the 137 Class I and Class II sources are overplotted with crosses. The dots indicate the positions of all targets for which we have obtained spectra but which are not confirmed as very low mass objects. Additionally, we show the frequency of objects as a function of distance from the cluster center in Fig. [f30], again for the same three samples, and in addition for all photometric candidates from our iz catalogue (dash-dotted line).\nSpatial distribution of NGC1333 members. Crosses are all 137 objects with Spitzer excess by, squares are all confirmed brown dwarfs (Tables [t1] and [t2]). Objects with spectroscopy for which we can exclude that they are substellar members are shown with dots. [f9]\nHistogram of the distance from the cluster center for confirmed brown dwarfs (solid line, Tables [t1] and [t2]), objects with Spitzer excess by (dashed line), objects for which we can exclude that they are substellar members (dotted line), and all IZ candidates (dash-dotted line). As cluster center we used the average position of the Class I/II sources. [f30]\nIn the two figures, the spatial distribution of brown dwarfs is strongly clustered and indistinguishable from the distribution of the total population of Class I/II sources in NGC1333. For the two samples, the average *α* and *δ* differ only by 0.4’ and 0.3’, respectively, which is < 10% of the cluster radius. Adopting the average position of the Class I/II sources as cluster center, the average distance from the center is similar in the two samples, 5.2’ for the very low mass objects and 5.5’ for the Class I/II sources. The fraction of objects with distance from the cluster center of *d* < 0.1, 0.1 < *d* < 0.2, *d* > 0.2deg is 65, 35, 0% for the very low mass objects and 67, 26, 6% for the Class I/II objects. The scatter in the positions is *σ**α* = 0.071 and *σ**δ* = 0.069deg for the very low mass objects, and *σ**α* = 0.067 and *σ**δ* = 0.085deg for the Class I/II sources. For all these quantities there are no significant differences between the two samples.\nThe figures also show that our spectroscopic follow-up covers an area that is about 1.5-2 as large (in radius) than the cluster itself. We took spectra for 31 candidates with distances of > 0.2deg from the cluster center, but none of them turned out to be a brown dwarf. There are still 43 candidates from the IZ photometry outside 0.2deg (see Sect. [s41]) for which we do not have spectra, but based on our current results, it is unlikely that they contain any very low mass cluster members. Thus, our wide-field follow-up spectroscopy shows that there is no significant population of brown dwarfs at *d* > 0.2deg from the cluster center, corresponding to ∼ 1pc at a distance of 300pc.\nIt has been suggested that gravitational ejection occurs at an early stage in the evolution of substellar objects, either from multiple stellar/substellar systems or from a protoplanetary disk. This ejection is thought to remove the objects from their accretion reservoir and thus sets their masses. In these scenarios one could expect the brown dwarfs to have high spatial velocities in random directions.\nAn ejection velocity of 1kms− 1 would allow the object to travel 1pc in 1Myr, i.e. in the case of NGC1333 this would be sufficient to reach the edge of the cluster. However, the gravitational potential of the cluster will significantly brake the motion of the brown dwarf. Assuming a total cluster mass of 500 *M*⊙ homogenuously distributed in a sphere with 1pc radius, a brown dwarf that gets ejected in the cluster center with 1.5, 2, 3kms− 1 would reach a velocity of approximately 0.5, 1.4, 2.6kms− 1 at a distance of 1pc from the center. All objects with ejection velocities of $\\gtrsim 2$kms− 1 would have moved to distances significantly larger than 1pc over 1Myr. As shown above, the presence of such objects can be excluded from our data.\nThe scenarios by and predict that a substantial fraction of ejected brown dwarfs (more than 50% in some simulations) exceed this velocity threshold of 2kms− 1. These models would require some tuning to reproduce a spatial distribution as observed in NGC1333. However, such simplified scenarios do not take into account that dynamical interactions affect the total cluster population, not exclusively the brown dwarfs. The cluster formation simulations by show that the velocity dispersion in a dense cluster is not expected to increase in the very low mass regime. Although brown dwarfs undergo ejection in the simulations, this does not lead to a velocity offset in comparison to the stars. NGC1333 seems to be consistent with this picture.\nAs a side comment, we note that the parameters in the main simulation in with gas mass of 500 *M*⊙ and cloud radius of 0.4pc are fairly similar to the properties of NGC1333, although the simulation produces a much higher number of stars and brown dwarfs (total stellar mass of 191 *M*⊙ vs. ∼ 50 *M*⊙ in NGC1333).\nDisks\n-----\nIn Fig. [f10] we plot the Spitzer/IRAC colour-colour diagram for the sample listed in Tables [t1] and [t2], again based on the C2D-’HREL’ catalogue. Out of the sample of 51 sources with confirmed spectral type M5 or later, 41 have photometric errors < 40% in all four IRAC bands and are shown in this plot. The figure shows the typical appearance with two groups, one around the origin, the second with significant colour excess in mid-infrared bands due to the presence of circum-(sub)-stellar material. In this sample of 41 objects, 27 show evidence for a disk, i.e. 66 ± 8%. All of them have colours comparable to the Class II sources identified in.\nThe derived disk fraction of 66% is only valid for the sample of 41 objects with reliable Spitzer detection. In the entire sample of 51 very low mass sources in NGC1333 listed in Tables [t1] and [t2], the disk fraction is likely to be smaller, because the ten objects which are not detected by Spitzer are unlikely to have a disk. Correcting for this effect, the disk fraction in the full sample could be as low as 27/51 or 55%. Therefore, we consider the disk fraction of 66% to be an upper limit.\nFor comparison, for the total cluster population derive a disk fraction of 83% from a Spitzer survey. This number has been derived for objects with *K* < 14mag. This magnitude limit was chosen by because it corresponds to *M* = 0.08 *M*⊙ at age of 1Myr and *A**V* = 20mag. Their sample thus includes mostly stars, but also some brown dwarfs (as evident from Table [t2], which contains a number of objects from the sample, marked with ’Sp’.) The Spitzer sample contains a substantial number of objects with *A**V* > 10mag, which are rare among the currently known brown dwarfs. It is possible that some of the heavily embedded brown dwarfs with *A**V* > 10mag have not been found yet. This could explain the discrepancy in the disk fractions.\nOur disk fraction is consistent with the values derived for very low mass members in *σ*Ori, Chamaeleon-I, and IC348 although all three regions are thought to be somewhat older (2-3Myr) than NGC1333.\nA more detailed SED analysis was carried out for objects with an additional datapoint at 24 *μ**m*. 19 of the objects in Fig. [f10] have MIPS fluxes at 24 *μ**m* with errors < 40%. At this wavelength the images are strongly affected by the cloud emission and blending. To make sure that the fluxes are trustworthy, we checked all objects in a 24 *μ**m* image obtained in the Spitzer program #40563 (PI: K. Wood, AOR 23712512), which is deeper than the C2D mosaics. After visual inspection, 3 objects were discarded; the remaining 16 are point sources at 24 *μ**m* and are marked with squares in Fig. [f10].\nSpitzer/IRAC colour-colour diagram for the very low mass sources listed in Tables [t1] and [t2], using the ’HREL’ photometry from the C2D survey. We only plot objects with photometric errors < 40% in all four bands (41 out of 51). The position of the Class II objects from the survey by is indicated by the dashed box. Objects with reliable detection at 24 *μ**m* are marked with squares [f10]\nSpectral energy distributions for 16 very low mass objects with disks in NGC1333 (crosses). The SEDs have been dereddened and scaled to the J-band fluxes. We overplot the typical SED for NGC1333 (solid line), *ρ*Oph (dotted line) Taurus (dashed line), and UpSco (dash-dotted line). We also show the photospheric SED from the 2800K DUSTY-AMES model with small dots. A typical errorbar for the 24 *μ* fluxes is overplotted. [f11]\nIn Fig. [f11] we show their SEDs after dereddening (see Sect. [s32]) and scaling to the J-band flux (crosses). For comparison, the photospheric SED from a model spectrum is overplotted with small dots. To assess the disk evolution in the substellar regime, we derive the typical SED for NGC1333 and three other star forming regions: *ρ*Oph (1Myr), Taurus (2Myr), and UpSco (5Myr). For this purpose we selected the objects which are detected in all four IRAC bands and at 24 *μ**m*. For *ρ*Oph we started with the census in Muzic et al. (subm.) and made use of the C2D data. For Taurus we used published Spitzer data from and. For UpSco the data from was used. When comparing the SEDs from different regions, one has to take into account that the depth of the 24 *μ**m* observations is not the same; thus the median SED is affected by incompleteness at low flux levels. Instead we plot the SED for the object that has the 10th highest flux level at 24 *μ**m* after converting to *λ**F**λ* and scaling to the J-band flux. This represents an estimate for a typical SED unaffected by the depth of the Spitzer observations and the distance to the cluster. Note that all objects used for this exercise are spectroscopically confirmed members of the respective clusters.\nFor wavelengths < 8 *μ**m* the four median SEDs are fairly similar. At 8 *μ**m* the SEDs in the youngest regions (NGC1333, *ρ*Oph) are slightly enhanced. The biggest differences are seen at 24 *μ**m*, particularly when comparing NGC1333 with UpSco. This is mostly due to the fact that NGC1333 harbours a few objects with unusually strong excess emission, which are missing in UpSco, indicating that the objects in NGC1333 are in an early evolutionary stage compared with the other regions. As demonstrated in a large spread in 24 *μ**m* fluxes, as seen in NGC1333, can easily be explained by a range of flaring angles in the disks.\nConclusions\n===========\nAs part of our survey program SONYC, we present a census of very low mass objects in the young cluster NGC1333 based on new follow-up spectroscopy from Subaru/FMOS. To derive reliable spectral types from our data, we define a new spectral index based on the slope of the H-band peak. We find 10 new likely brown dwarfs in this cluster, including one with a spectral type ∼ L3 and two more with spectral type around or later than M9. These objects have estimated masses of 0.006 to 0.02 *M*⊙, the least massive objects identified thus far in this region. This demonstrates that the mass function in this cluster extends down to the Deuterium burning limit and beyond. By combining the findings from our SONYC survey with results published by other groups, we compile a sample of 51 objects with spectral types of M5 or later in this cluster, more than half of them found by SONYC. About 30-40 of them are likely to be substellar. The star vs. brown dwarf ratio in NGC1333 is significantly lower than in other nearby star forming regions, possibly indicating environmental differences in the formation of brown dwarfs. We show that the spatial distribution of brown dwarfs in NGC1333 closely follows the distribution of the stars in the cluster. The disk fraction in the brown dwarf sample is < 66%, lower than for the stellar members, but comparable to the brown dwarf disk fraction in 2-3Myr old regions. The substellar members in NGC1333 show a large fraction of highly flared disks, evidence for the early evolutionary state of the cluster.\nThe authors would like to thank the Subaru staff, especially Dr. Naoyuki Tamura and Dr. Kentaro Aoki, for the assistance during the observations and their preparation. We are grateful to Ms. Yuuki Moritani, Mr. Kiyoto Yabe and Prof. Fumihide Iwamuro (Kyoto University) for their help with the FMOS data reduction. We thank the anonymous referee for a constructive report that helped to improve the paper. This work made use of results from the Spitzer program #40563; we thank Jane Greaves, Chris Poulton, and Kenneth Wood from the University of St. Andrews for their help in the framework of this campaign. We also thank Ewan Cameron for discussing the calculation of confidence intervals and David Lafrenière for making his spectra available to us. AS acknowledges financial support the grant 10/RFP/AST2780 from the Science Foundation Ireland. The research was supported in large part by grants from the Natural Sciences and Engineering Research Council (NSERC) of Canada to RJ. This research has benefitted from the SpeX Prism Spectral Libraries, maintained by Adam Burgasser at http://pono.ucsd.edu/~adam/browndwarfs/spexprism/.\nSpectroscopically excluded objects\n==================================\nIn Tables [t10] and [t11] we provide a full list of objects for which we obtained spectra and which were not classified as young very low mass objects based on the shape of their near-infrared spectrum (see Sect. [s31]). The spectra come from the first campaign with MOIRCS and from the second run with FMOS (this paper). Most of these objects are likely to be either young stellar objects in NGC1333 or background stars with effective temperatures above ∼ 3500K or spectral type earlier than ∼ M3. In Table [t10] we also give the J- and K-band photometry from 2MASS and the identifiers from the photometric surveys by and, if available. Objects without listed identifiers are not known to have a counterpart within 1\".\nllccccl 03 29 18.71 & +31 32 26.4 & 16.722 & 13.839 & IZ & F & \n03 29 52.35 & +31 28 13.7 & 14.919 & 13.829 & IZ & F & \n03 29 48.12 & +31 28 29.4 & 15.550 & 13.663 & IZ & F & \n03 29 37.80 & +31 27 48.4 & 17.274 & 15.249 & IZ & F & \n03 29 34.76 & +31 29 08.1 & 13.647 & 11.532 & IZ & F & MBO 43 \n03 29 11.53 & +31 30 05.6 & 16.740 & 13.786 & IZ & F & [LAL96] 241, 242 \n03 28 55.74 & +31 30 58.0 & 15.175 & 13.390 & IZ & F & [LAL96] 143 \n03 28 52.66 & +31 32 04.3 & 16.563 & 14.730 & IZ & F & \n03 28 46.67 & +31 31 35.4 & 15.647 & 13.933 & IZ & F & \n03 28 37.55 & +31 32 54.5 & 15.107 & 14.232 & IZ & F & \n03 28 23.84 & +31 32 49.3 & 14.714 & 13.655 & IZ & F & \n03 28 44.96 & +31 31 09.9 & 17.057 & 15.211 & IZ & F & \n03 28 46.24 & +31 30 12.1 & 15.089 & 14.069 & IZ & F & [LAL96] 88 \n03 29 45.53 & +31 26 56.5 & 14.870 & 14.023 & IZ & F & \n03 30 05.41 & +31 25 13.1 & 14.921 & 13.680 & IZ & F & \n03 28 49.48 & +31 25 06.6 & 15.957 & 13.658 & IZ & F & MBO 82 \n03 29 10.46 & +31 23 34.8 & 15.633 & 12.762 & JK & F & [LAL96] 231 \n03 28 43.17 & +31 26 06.1 & 15.617 & 13.384 & IZ & F & [LAL96] 78 \n03 28 37.75 & +31 26 32.8 & 16.847 & 14.340 & IZ & F & [LAL96] 60 \n03 27 56.27 & +31 27 00.8 & 16.836 & 13.967 & IZ & F & \n03 28 07.64 & +31 26 42.4 & 15.974 & 13.719 & IZ & F & \n03 28 19.51 & +31 26 39.5 & 15.100 & 13.687 & IZ & F & [LAL96] 13 \n03 28 40.22 & +31 25 49.1 & 15.637 & 12.911 & IZ & F & \n03 28 47.64 & +31 24 06.2 & 14.199 & 11.660 & JK & F & [LAL96] 93 \n03 28 55.22 & +31 25 22.4 & 14.735 & 12.550 & IZ & F & [LAL96] 139 \n03 29 03.32 & +31 23 14.8 & 17.254 & 14.071 & JK & F & [LAL96] 191 \n03 29 03.13 & +31 22 38.2 & 13.724 & 11.323 & JK & F & [LAL96] 189 \n03 29 27.61 & +31 21 10.1 & 14.836 & 13.074 & IZ & F & [LAL96] 324 \n03 29 55.50 & +31 15 30.5 & 15.120 & 14.131 & IZ & F & \n03 29 52.65 & +31 17 22.9 & 16.325 & 14.970 & IZ & F & \n03 29 39.61 & +31 17 43.4 & & & JK & F & \n03 28 35.46 & +31 21 29.9 & 16.052 & 14.197 & IZ & F & [LAL96] 49 \n03 28 48.45 & +31 20 28.4 & 16.842 & 14.283 & IZ & F & [LAL96] 100 \n03 29 10.82 & +31 16 42.7 & 15.652 & 13.039 & JK & F & [LAL96] 235 \n03 29 21.42 & +31 15 55.3 & 15.396 & 14.004 & IZ & F & [LAL96] 305 \n03 29 45.41 & +31 16 23.2 & 15.118 & 13.767 & IZ & F & \n03 29 50.23 & +31 15 47.9 & 16.980 & 15.106 & IZ & F & \n03 30 01.93 & +31 10 50.5 & 14.374 & 12.921 & IZ & F & \n03 29 54.78 & +31 11 41.7 & 16.679 & 15.253 & IZ & F & \n03 29 34.57 & +31 11 23.8 & 16.750 & 14.643 & IZ & F & [LAL96] 351 \n03 29 36.00 & +31 12 49.6 & 15.522 & 14.615 & IZ & F & [LAL96] 353 \n03 29 26.11 & +31 11 36.9 & 14.781 & 12.874 & IZ & F & [LAL96] 320 \n03 28 01.19 & +31 17 36.5 & 15.587 & 14.136 & IZ & F & \n03 27 52.51 & +31 19 38.8 & 14.973 & 13.662 & IZ & F & \n03 28 22.90 & +31 15 21.7 & 15.032 & 13.587 & IZ & F & [LAL96] 20 \n03 29 08.71 & +31 12 01.9 & & & IZ & F & \n03 29 12.24 & +31 12 20.5 & 16.588 & 14.722 & IZ & F & [LAL96] 254 \n03 29 18.45 & +31 11 30.5 & 16.413 & 15.530 & IZ & F & \n03 29 31.00 & +31 11 20.1 & 17.052 & 15.265 & IZ & F & \n03 29 28.99 & +31 10 00.3 & 13.365 & 10.889 & IZ & F & [LAL96] 329 \n03 29 45.40 & +31 10 35.6 & 14.828 & 13.516 & IZ & F & \n03 29 46.48 & +31 08 43.6 & 15.205 & 14.096 & IZ & F & \n03 29 49.16 & +31 09 03.7 & 16.143 & 14.537 & IZ & F & \n03 29 28.95 & +31 07 40.9 & 15.975 & 15.199 & IZ & F & [LAL96] 330 \n03 29 24.73 & +31 07 26.8 & & & IZ & F & \n03 29 20.11 & +31 08 53.7 & 15.923 & 14.813 & IZ & F & \n03 29 09.23 & +31 08 55.4 & 16.330 & 14.912 & IZ & F & \n03 28 57.45 & +31 09 46.5 & 16.623 & 12.738 & IZ & F & [LAL96] 160 \n03 28 06.32 & +31 10 02.0 & 15.731 & 14.579 & IZ & F & \n03 28 22.07 & +31 10 42.9 & 13.867 & 11.859 & IZ & F & [LAL96] 18 \n03 28 39.01 & +31 08 07.6 & 15.143 & 13.389 & IZ & F & [LAL96] 65 \n03 29 03.60 & +31 07 11.9 & 14.995 & 13.351 & IZ & F & [LAL96] 197 \n03 29 05.23 & +31 07 10.6 & 16.776 & 14.766 & IZ & F &\nllccccl 03 28 41.72 & +31 11 15.1 & & & IZ & M & \n03 28 41.97 & +31 12 17.2 & 15.863 & 13.822 & IZ & M & \n03 28 46.21 & +31 12 03.4 & 16.933 & 13.526 & IZ & M & [LAL96] 90 \n03 28 48.99 & +31 12 45.1 & 17.836 & 14.048 & IZ & M & [LAL96] 103 \n03 28 52.10 & +31 16 29.3 & 16.071 & 13.738 & IZ & M & [LAL96] 123 \n03 28 57.25 & +31 07 26.0 & & & IZ & M & [LAL96] 159 \n03 28 58.68 & +31 09 39.2 & 17.052 & 12.945 & IZ & M & [LAL96] 169 \n03 29 00.70 & +31 22 00.8 & 16.236 & 11.764 & IZ & M & [LAL96] 180 \n03 29 17.93 & +31 14 53.5 & 16.625 & 14.039 & IZ & M & [LAL96] 287 \n03 29 18.66 & +31 20 17.8 & 17.510 & 14.608 & IZ & M & MBO 109 \n03 29 19.86 & +31 18 47.7 & 17.205 & 13.321 & IZ & M & [LAL96] 297 \n03 29 28.06 & +31 18 39.0 & 15.307 & 12.846 & IZ & M & [LAL96] 327 \n03 29 32.20 & +31 17 07.3 & 15.503 & 13.784 & IZ & M & [LAL96] 341 \n03 29 37.41 & +31 17 41.6 & 14.886 & 12.761 & IZ & M & [LAL96] 359 \n03 29 08.17 & +31 11 54.6 & 17.247 & 14.980 & IZ & M & [LAL96] 217\n---\n1. downloaded[↩](#fnref1)\n2. downloaded from \nhttp://perso.ens-lyon.fr/isabelle.baraffe/[↩](#fnref2)\n3. A new paper by updates the value of for the ONC to *R*1 = 2.4 ± 0.4, based on an HST survey covering a larger area than previous studies.[↩](#fnref3)\n"},"data_id":{"kind":"string","value":"arxiv_0000350"}}},{"rowIdx":92,"cells":{"content":{"kind":"string","value":" the subsonic and remnant phases are unchanged from those given in and respectively.\nIsothermal evolution of the shocked shell\n-----------------------------------------\nAs first pointed out by, the final state of the shocked gas lying between the lobe surface and the bow shock depends critically on whether it expands adiabatically or isothermally; we assume here that the swept-up gas evolves isothermally. The pressure, $p\\_{\\rm s}(\\theta)$, of the swept-up gas at an angle *θ* from the jet axis is found from the jump conditions at the shock.\nThe cooling rate, and thus intensity of thermal bremsstrahlung radiation, in the shocked shell depends on both the pressure of the shocked gas and its temperature. For low Mach numbers *M*0 the cooling time of the swept-up gas, $t\\_{\\rm s}$, can become short compared with that in the cluster, $t\\_{\\rm x}$, with $t\\_{\\rm s}/t\\_{\\rm x} > 0.05M\\_0$. However, the cooling time of the external gas would have to be at least an order of magnitude less than the source age, *t*, for the shocked gas to suffer significant radiative losses. Such conditions are only possible at the centres of the strongest cooling flow clusters – precisely *not* the locations where powerful, large radio galaxies are typically found. We therefore ignore the cooling of the gas within the shocked shell in our analysis.\n### Mean shocked shell temperature\nThe mean temperature of the shocked gas is found by first calculating the mean pressure and density of the shocked shell. The mean density is taken as the ratio of the mass of ambient gas previously occupying the lobe and shocked shell to the volume of the shocked gas shell:\n \n$$\\begin{split}\n\\bar{\\rho\\_{\\rm s}}(t) = \\frac{\\int\\_0^{\\pi/2} \\int\\_0^{R\\_{\\rm s}(t, \\theta)} \\rho(r) r^2 dr \\sin \\theta d\\theta}{\\int\\_0^{\\pi/2} \\int\\_{R(t, \\theta)}^{R\\_{\\rm s}(t, \\theta)} r^2 dr \\sin \\theta d\\theta},\n\\end{split}$$ \nwhere *ρ*(*r*) is the gas density of the ambient medium. This expression must be solved numerically except for the special case of spherical lobes expanding into a power law environment. The mean temperature of the shocked gas shell is thus given as $T\\_{\\rm s}(t) = \\bar{m} \\bar{p\\_{\\rm s}}(t)/k\\_{\\rm B} \\bar{\\rho\\_{\\rm s}}(t)$ for a shocked gas shell with mean pressure $\\bar{p\\_{\\rm s}}(t)$ and an average particle mass $\\bar{m} \\sim 0.6m\\_{\\rm p}$, where $m\\_{\\rm p}$ is the proton mass. The temperature may of course vary throughout the shocked gas shell but without precise knowledge of any density gradients we simply assume mean values.\n### Bremsstrahlung radiation\nThe X-ray emissivity per unit volume due to thermal bremsstrahlung radiation is \n \n$$\\begin{split}\nJ(\\nu) =& \\frac{Z^2 e^6}{3 \\pi^2 {\\varepsilon\\_0}^3 {m\\_{\\rm e}}^2 c^3} \\left(\\frac{\\pi m\\_{\\rm e}}{6} \\right)^{1/2} \\\\\n&\\quad\\quad (k\\_{\\rm B} \\mathcal{T})^{-5/2} {\\tilde{p}}^{\\!\\:2} e^{-h\\nu/k\\_{\\rm B} \\mathcal{T}} g(\\nu, \\mathcal{T}),\n\\end{split}$$ \nwhere the pressure and temperature (*p̃* and T respectively) may relate to the shocked shell, ambient medium, or any other plasma. Here, $Z \\gtrsim 1$ is the average atomic number of the positively changed particles, *ɛ*0 is the vacuum permittivity, and, at frequencies $h\\nu \\ll k\\_{\\rm B} \\mathcal{T}$, the Gaunt factor has a logarithmic dependence on frequency as\n \n$$\\begin{split}\ng(\\nu, \\mathcal{T}) = \\frac{\\sqrt{3}}{\\pi} \\ln \\left(\\frac{4}{\\zeta} \\frac{k\\_{\\rm B} \\mathcal{T}}{h\\nu} \\right),\n\\end{split}$$ \nwhere here *ζ* = 1.78 is Gauss’ number. In this work, we assume the metallicity of the plasma is 0.3 times the solar value, corresponding to an average atomic number of *Z* ∼ 1.04. The shocked gas pressure used in the bremsstrahlung radiation calculation is that derived from the shock jump conditions on the shell surface at angle *θ* (i.e. $\\tilde{p} = p\\_{\\rm s}(\\theta)$), whilst the mean temperature is derived following the method described in the previous section.\n[fig:apec]\nThe X-ray spectra of smaller clusters at low-redshift are dominated by emission lines at the wavelengths of the *Chandra* and *eRosita* surveys. The APEC plasma code can calculate the X-ray emission spectra from thin-thermal plasma for various temperatures and metallicities. In Figure [fig:apec], we plot the fractional contribution of continuum emission to the total bremsstrahlung X-ray spectrum (including both continuum and line emission) integrated over a typical 0.5-$2\\rm\\, keV$ observing band. The emission lines are found to contribute a significant component of the X-ray flux density at redshifts *z* < 2.5 in clusters with temperatures $kT \\lesssim 1\\rm\\, keV$, corresponding to halo masses $\\lesssim 10^{13.5}\\rm\\, M\\_\\odot$. We use the appropriate APEC model to correct our analytically derived flux densities of the X-ray emission arising from bremsstrahlung radiation.\nX-RAY SURFACE BRIGHTNESS MAPS\n=============================\nWe use the formalism developed in the preceding sections to make predictions for the observed X-ray emission from radio galaxies and their environments. There are three main contributions: lobe inverse-Compton emission (Section [sec:LUMINOSITY MODEL]) and two sources of bremsstrahlung radiation, from the shocked gas shell ahead of the lobes (Section [sec:SHOCKED SHELL BREMSSTRAHLUNG RADIATION]), and from the ambient medium. We calculate the latter in the same way as bremsstrahlung from the shocked gas shell, but adopting density and temperature parameters characteristic of the hot gas in galaxy clusters, as described in.\nX-ray and radio brightness maps\n-------------------------------\nThe lobes and shocked gas shell of each mock radio galaxy are divided into a 512 × 512 × 512 grid of cubic pixels; this grid extends a factor of two beyond the edge of the lobe to enable a comparison with X-ray surface brightness of the ambient medium. Each cell in the cube is classified as either part of the lobe, shocked shell or ambient medium based on the modified RAiSE dynamical model (Section [sec:SHOCKED SHELL BREMSSTRAHLUNG RADIATION]), and the inverse-Compton emissivity or bremsstrahlung radiation calculated as appropriate. The observed emission arising from these extended objects is the integral of all the emissivity along a given line-of-sight. The two-dimensional surface brightness is simply calculated by summing the emissivity from every cell along the depth of the source, assuming the lobe plasma and ambient medium in front of the source is optically thin. The X-ray emission arising from the ambient medium lying outside the grid of cell is calculated analytically for each line-of-sight and added to the total from the numerical grid. For simplicity, we assume that the lobes lie in the plane of the sky.\nThe inverse-Compton and bremsstrahlung emissivity in the (two-dimensional) surface brightness map are derived for the technical characteristics of the *extended Roentgen Survey with an Imaging Telescope Array*. Specifically, at a $1\\rm\\, keV$ observer-frame energy, the half-energy width (on axis) is $15\\rm\\, arcsec$ and the total effective area of the seven mirror systems is $\\sim 1500\\rm\\, cm^2$. The number of $1\\rm\\, keV$ photons falling on these mirrors in a typical 1000$\\rm\\, s$ (1 ks) observing time is thus calculated from the X-ray surface brightness grid. The $151\\rm\\, MHz$ radio frequency emission from the lobes is also calculated following the method of ; this frequency is commonly used by low-frequency Square Kilometre Pathfinder instruments, including the *Low-Frequency Array* (LOFAR) and the *Murchison Widefield Array* (MWA).\nThe X-ray and radio surface brightness distributions are modelled for lobed FR-IIs at a range of redshifts throughout their evolutionary history. Specifically, we simulate AGNs at 41 (log-spaced) source ages between 3 and 300$\\rm\\, Myrs$, redshifts of *z* = 0.1, 0.5 and 1, jet powers of *Q* = 1037, 1038 and $10^{39}\\rm\\, W$, active ages of 30 and 100$\\rm\\, Myrs$ (i.e. when the jet ceases injecting fresh electrons), and host cluster environments with dark halo masses of 1012.5, 1013.5 and $10^{14.5}\\rm\\, M\\_\\odot$. The axis ratio of the lobe is taken as *A* = 4 corresponding to an axis ratio for the shocked shell of $A\\_{\\rm s} = 2$ (see Section [sec:AXISRATIO]). The other parameters in the RAiSE dynamical and synchrotron/inverse-Compton emissivity models take the same values as used by.\nThe surface brightness maps for several informative combinations of these parameters are shown in Figure [fig:surfbright]. Powerful jets expanding into dense environments are found to create dense shells of shocked gas surrounding the lobe. The bremsstrahlung radiation from the shocked gas is much brighter than can be generated by the inverse-Compton radiation process, especially in young sources and at low redshifts (e.g. top panel of Figure [fig:surfbright]). The faint lobes bounded by well-defined lines are seen in *Chandra* X-ray images of Cygnus A and MS0735.6+7421, with cavities also seen in the hosts of weaker FR-Is such as the Perseus cluster. By contrast, in lower mass clusters, the density of swept-up gas is very low causing its temperature to be high to satisfy the shock jump conditions. The increased temperature leads to minimal bremsstrahlung radiation from the shocked gas shell. The inverse-Compton upscattered photons in the lobe are thus the dominant source of X-ray emission, especially at higher redshifts where the cosmic microwave background radiation is stronger (e.g. middle panel of Figure [fig:surfbright]). Finally, the X-ray inverse-Compton emission remains visible well after the radio frequency emission vanishes; in the simulated remnant, the radio emission has retreated towards the freshest electrons at the hotspot, whilst the X-ray emission does a much better job of tracing out the channel evacuated by the radio lobes. A similar result has previously been reported by ; Figure [fig:surfbright] additionally shows that the peak in the X-ray intensity from the lobe is much closer to the core than the full extent of the radio source. Many radio sources appear asymmetric, due to asymmetry in the environments encountered by the two lobes ; this poses a challenge for robustly identifying host galaxies of remnant radio sources. Our results suggest that X-ray observations may provide a useful alternative.\n \n \n[fig:surfbright]\nAn intriguing surface brightness map is produced when propagating weak jets into dense environments (bottom panel of Figure [fig:surfbright]); dense shells of shocked gas build up around the lobe as before, however, the weakened shocks and magnetic fields allow the contact surface near the core to become unstable to turbulent mixing of the dense shell and the lobe. This results in thermal bremsstrahlung emission extending out from the core along the transverse axis, brighter than X-ray emission from the lobe or the thinner (and thus fainter integrated along the line-of-sight), unmixed portions of the shocked shell further from the core. We note that strong X-ray emission may also be observed *along* the jet axis due to inverse-Compton emission from the jets as well as lobes (for older sources at high redshift); a combination of radio and X-ray imaging may distinguish these two scenarios.\nDominant source of X-ray emissivity\n-----------------------------------\nIn this section, we investigate the relative importance of contributions to the X-ray surface brightness from the lobe (inverse-Compton), shocked gas shell (bremsstrahlung), and the ambient medium (bremsstrahlung), for some representative sources at a range of redshifts. The surface brightness for the three regions is calculated as the maximum value along the jet axis. However, the increased bremsstrahlung radiation from sources pinched by Rayleigh-Taylor mixing may present a source of error in this work as the rate of mixing is a poorly quantified model parameter in RAiSE. Fortunately, in the vast majority of extended AGNs this enhanced emission is only a minor contributor to the integrated X-ray luminosity, and when generating surface brightness maps, either the Rayleigh-Taylor mixed region is not resolved from the bright X-ray core (discussed in Section [sec:X-ray source number densities]) or the remainder of the lobe also sits above the surface brightness sensitivity limit (i.e. the shape of the source is correctly determined). The analyses in this work are repeated with the inner third of the lobe masked to exclude this region of enhanced emission; our results are unaffected by this change.\n\n[fig:timeseries1]\n[fig:timeseries2]\nThe evolutionary tracks of the X-ray surface brightness for these representative sources are shown in Figures [fig:timeseries1] and [fig:timeseries2]. The inverse-Compton emission is dominant over the bremsstrahlung radiation from both the shocked gas shell and the ambient medium in poor clusters ($10^{12.5}\\rm\\, M\\_\\odot$; Figure [fig:timeseries1]) at all source ages $\\gtrsim 1\\rm\\, Myr$. By contrast, the shocked gas shell of sources expanding into denser host cluster environments ($\\gtrsim 10^{13.5}\\rm\\, M\\_\\odot$; Figure [fig:timeseries2]) is brighter than the lobe, albeit not appreciably, for at least 20 to $100\\rm\\, Myrs$ with the weakest FR-II jet powers ($Q = 10^{37}\\rm\\, W$); the shocked shell stays brighter than the lobe for longer with either denser environments or higher jet powers. This change in the importance of the lobe and shocked gas shell to the X-ray surface brightness typically results from rapidly falling pressure in the shocked gas as the source approaches pressure equilibrium with the ambient medium. Further, the stronger cosmic microwave background radiation at higher redshifts leads to increased inverse-Compton emission from the lobe causing this source of X-ray surface brightness to become dominant in younger sources.\nFinally, the X-ray surface brightness from the lobe and shocked gas shell remain detectable above that of the ambient medium for a considerable time after the jet switches off. The extended source in Figure [fig:timeseries1], with an active lifetime of 30$\\rm\\, Myrs$, remains detectable over the ambient medium (at the 5*σ* level) for a further 155$\\rm\\, Myrs$ at redshift *z* = 0.1, 115$\\rm\\, Myrs$ at *z* = 0.5, 100$\\rm\\, Myrs$ at *z* = 1, and 65$\\rm\\, Myrs$ at *z* = 2. By contrast, *any* level of synchrotron emission is present at radio frequencies for only 170, 70, 25 and 10$\\rm\\, Myrs$ after the jets switch off, respectively, at these redshifts. Similarly, the extended source in Figure [fig:timeseries2] emits radiation from the lobe and shocked shell at X-ray wavelengths long after the radio emission fades, though this is only a factor of a couple brighter than the ambient medium. Based on these and other test cases, we expect a population of remnant extended radio galaxies undetectable at radio frequencies but visible to X-ray telescopes, in line with literature predictions of a large population of inverse-Compton ‘ghosts’, particularly at high-redshift.\nMOCK EXTENDED AGN POPULATION\n============================\nWe now extend our analysis to make predictions for radio galaxy population statistics. Specifically, we combine the theoretical framework developed in the previous section with the halo mass function and literature radio frequency observations to create a mock population of extended radio galaxies. We then use this simulated population to predict the X-ray luminosity function for extended AGN emission, and investigate the number of objects that could be uniquely detected using large sky X-ray surveys.\nConstruction of mock catalogue\n------------------------------\nWe construct a mock catalogue of extended AGNs, including their angular sizes, surface brightnesses, and integrated flux densities, by running simulations for a dense grid of model parameters. The brightnesses are calculated at radio frequencies for a typical *Low Frequency Array* (LOFAR) survey and in the X-ray for an *eRosita* survey. Observations are used to constrain the input halo mass function for AGN hosts, and the distribution of source active lifetimes and jet kinetic powers. For the remaining parameters, we assume values given in Section [sec:Surface brightness].\n### Halo mass function\n \n[fig:luminfunctions]\n[fig:luminfractions]\nThe mass function of extended AGN hosting clusters is assumed to be the convolution of the halo mass function for all groups and clusters, and the probability of finding an AGN in a given mass host. The mass of dark matter haloes (observed as galaxy groups and clusters) can in general be described by a mass function, which gives the number density of clusters as a function of mass. In this work, and following, we take our dark matter halo masses from the low-redshift mass function of who find that a common Schechter function describes both galaxy groups and clusters. We use the semi-analytic galaxy evolution (SAGE) model of to extend their observations to higher redshifts, finding the mass of the break in the Schechter function scales with redshift as approximately (1 + *z*)− 3.\nThe radio-loud fraction of AGNs (i.e. fraction of sources with brightness above an arbitrary cut in radio luminosity) suggests hosts with higher mass black holes will have more frequent (or longer) phases of activity. made a theoretical prediction for the AGN duty cycle (i.e. the fraction of time a given source is active) as a function of black hole mass, $\\delta \\propto \\rm M\\_\\bullet^{1.5}$; this relationship is consistent with the observed radio-loud fraction for all but the most massive galaxies in which the duty cycle peaks at 100%. The black hole mass in the brightest cluster galaxy (BCG), the mass of the stellar bulge, and the dark matter halo mass, are all known to scale approximately linearly with each other. Individual galaxies show moderate scatter of $0.5\\rm\\, dex$ about the dark matter halo–black hole mass relationship, however the underlying relationship remains when considering several orders of magnitude in halo mass. We therefore convolve the halo mass function with the mass–duty cycle relation to derive the radio AGN mass function. This halo mass function is shown in the left panel of Figure [fig:luminfunctions] as a function of redshift for masses between 1012.5 and $10^{14.75}\\rm\\, M\\_\\odot$. The number density is scaled in the plot based on the number of high-luminosity radio AGNs of FR-II morphology observed by.\nThese scaling relationships predict that all plausible host halo masses occur at similar probability, with a slightly decreased likelihood of finding a radio AGN in the most massive clusters. This convolved mass function will provide a better description of the environments typically hosting AGNs than the raw halo mass function.\n### Source lifetimes\nThe radio AGN lifetime function has generated much recent interest. The large observed fractions of compact sources strongly suggests a dominant population of short-lived radio jets. Recently, used a combination of data from the LOFAR LoTSS survey and dynamical models, to show that the majority of radio AGNs are consistent with models in which the source lifetime distribution is log-uniform. applied self-consistent modelling to LOFAR observations of active, remnant and restarted radio sources in the HETDEX field to similarly infer a dominant short-lived population, consistent with feedback-regulated accretion.\nIn line with these results, we similarly adopt here a log-uniform distribution of the active lifetimes between 3 and 300Myrs, and assume that the duty cycle is (on average) independent of the active lifetime. We note that the chosen distribution for these parameters makes only minimal difference to the results in this work.\n### Jet kinetic powers\nThe distribution of the jet kinetic powers fuelling the extended AGN emission is informed by observations of the luminosity function of powerful FR-IIs in the 3CRR, 6CE and 7CRS samples. fit the high-luminosity function for these objects at $151\\rm\\, MHz$ as a function of redshift. The general shape of their distribution is shown in the right-hand subplot of Figure [fig:luminfunctions], with the number density scaled to match the observed distribution at redshifts *z* = 0.1, 0.5, 1, 2 and 4. We generate our jet power distribution by converting their radio luminosity function into kinetic powers, first by using theory driven jet power–luminosity relationships; i.e. *Q* ∝ *L**ν*6/7, noting that environment and source age introduce scatter to this relation. The jet power corresponding to the turnover in their luminosity function is found by simulating mock catalogues (sampled every $0.2\\rm\\, dex$ in jet power, and including only active sources) for a range of possible turnovers and selecting the best match to the observed radio luminosity distribution. The resulting probability distribution for the kinetic jet power is given by\n \n$$n(Q) = n\\_0 \\left( \\frac{Q}{Q\\_\\star} \\right)^{-1.17\\alpha\\_\\star} \\exp\\bigg[-\\left( \\frac{Q\\_\\star}{Q} \\right)^{1.17} \\bigg],$$ \nwhere *n*0 is a normalisation constant, the turnover in the luminosity function occurs at $Q\\_\\star = 10^{38.1}\\rm\\, W$, and the slope of the power law component of the luminosity function is *α*⋆ = 2.27. The shape of the probability distribution is independent of redshift, however the number density of extended AGNs increases towards higher redshifts. The radio luminosity functions generated from our mock catalogues (using the selected jet power distribution) are shown in the right-hand panel of Figure [fig:luminfunctions] for a range of redshifts; these are in good agreement with the observed luminosity functions.\nX-ray luminosity function\n-------------------------\nThe mock extended radio AGN catalogue, which has been calibrated to successfully describe the observed radio properties of powerful FR-IIs, can now be used to predict their brightness at X-ray wavelengths. The integrated luminosity is calculated for active sources by summing the synchrotron emissivity from the lobe, and the bremsstrahlung radiation from both the shocked gas shell and the ambient medium. The X-ray emission from the ambient medium is included in these calculations since in practice it may be hard to disentangle the emission from the AGN and environment (except in well resolved objects), however we also derive the X-ray luminosity function including only AGN related emission for comparison with previous studies. We expect the X-ray luminosity of most extended radio galaxies to be strongly correlated with the properties of the ambient medium since the brightness of the shocked gas shell is directly related to the mass of gas swept up as the lobe expands. The level of inverse-Compton emission from the lobe, meanwhile, has a more complicated relationship with the density profile of the ambient medium.\nThe accretion disk of AGNs is also often bright at X-ray wavelengths; the spectrum of the accretion disk peaks at ultraviolet wavelengths but a sizeable number of thermal photons from the disk are inverse-Compton upscattered to X-ray energies by the hot corona surrounding the disk. Typical X-ray luminosities from accretion-related nuclear emission are of order 1043 to $10^{45}\\rm\\, erg \\,s^{-1}$. Meanwhile, the highest energy shock-accelerated electrons emit synchrotron radiation at X-ray wavelengths both along the jet and at the terminal hotspot in active sources; these quickly fade in remnants due to radiative losses. The hotspots of typical FR-IIs can reach 1042 to $10^{43}\\rm\\, erg \\,s^{-1}$ at X-ray wavelengths. The brightness of the core and hotspots is not related to the inverse-Compton upscattering of CMB photons and is thus independent of redshift, however it is hypothesised that a beamed inverse-Compton mechanism may operate in the jets. We therefore choose not to include the (rather uncertain) core emission in our luminosity function whilst analyses in subsequent sections will only investigate the detection of emission spatially resolved from the core. Finally, our modelling does not consider synchrotron self-Compton radiation (i.e. inverse-Compton upscattering of synchrotron photons); in Cygnus A (*z* = 0.0561), this mechanism contributes 70-80% of non-thermal X-ray emission in the lobes, however inverse-Compton upscattered CMB photons become dominant at higher redshifts (*z* > 0.4 for a Cygnus A-like source).\nThe X-ray luminosity function derived for a typical *eRosita* survey frequency of $1\\rm\\, keV$ is shown in the central panel of Figure [fig:luminfunctions]. The brightness of the extended AGNs is comparable to, or at most a factor of a few brighter than (on average 30% brighter at *z* = 0.1), typical inactive X-ray clusters in the range $10^{41} < L\\_{\\rm X} < 10^{44}\\rm\\, erg\\, s^{-1}$. The shape of the X-ray luminosity function also resembles that of the AGN halo mass function when including all contributors of extended X-ray emission, in particular at the lowest redshifts. As we show in Figure [fig:luminfractions], this is due to the dominant source of X-ray emission being bremsstrahlung radiation from the ambient medium and shocked gas shell for *z* ≤ 1. Objects with the highest luminosities are typically the highest jet power sources, observed at ages of a few tens of Myrs when inverse-Compton radiation from the lobe begins to increase significantly, contributing in excess of 90% of the total integrated X-ray luminosity from the AGN–host cluster system (see Figure [fig:luminfractions]). By contrast, the X-ray luminosity function at the highest redshifts (*z* > 1) is dominated by inverse-Compton emission for the majority of luminosities (and thus a large fraction of the extended AGN population). The stronger cosmic microwave background radiation at higher redshifts thus boosts the integrated luminosity well above that generated by the hot cluster gas (on average 155% brighter at *z* = 4), leading to an increased population of X-ray bright extended AGNs.\nX-ray source number densities\n-----------------------------\n[fig:numberdensities]\n[fig:remnantdensities]\nWe now use the mock catalogue to calculate the predicted number density of extended AGNs that would be detected using *LOFAR* and *eRosita*. The *LOFAR Two-metre Sky Survey* (LoTSS) has a median sensitivity of $71\\rm\\, \\mu Jy\\, beam^{-1}$ at an observing frequency of 120-168MHz, with approximately 6arcsec full-width half maximum (FWHM) across the synthesised beam. The expected $1\\rm\\, keV$ sensitivity of all-sky surveys using *eRosita* is $14\\rm\\, nJy\\, beam^{-1}$ for their 15arcsec full-width half maximum beam. We smooth the RAiSE images with a circular gaussian filter, mimicking the approximate shape of the *LOFAR* and *eRosita* beams. Extended emission separated from the core by less than the angular resolution of the survey cannot be distinguished from a bright compact core; these pixels are flagged and removed from our mock images. Simulated extended AGNs with emission in at least one beam (i.e. gaussian filtered pixel) exceeding the surface brightness sensitivity limit are assumed to be detectable by *LOFAR* or *eRosita* as appropriate. These sources are by definition resolved, however we make a further classification that the extended AGNs are well resolved (i.e. we can make out the shape of the source) if they have at least five beams along their length.\nThe number density of X-ray and radio detected active-phase, extended AGNs of FR-II morphology is shown in Figure [fig:numberdensities] for a range of source ages and jet kinetic powers. The overwhelming majority of low-redshift sources have extended emission resolvable from their cores in either X-ray and radio frequencies (blue or green shading); approximately half have sufficient resolution to determine the shape of their extended structures (green shading). Specifically, close to all of the active-phase FR-IIs in our catalogue are detectable using radio frequency observations (at *z* ≤ 1), whilst 41% can be seen at X-ray wavelengths at *z* = 0.1 and 2.6% at *z* = 0.5. At both these redshifts *eRosita* can uniquely detect (i.e. no radio detection) only 0.2% of active sources. This pattern continues out to higher redshifts where most active sources are detectable at radio frequencies whilst a rapidly decreasing fraction can be seen at X-ray wavelengths. Importantly, at *z* = 1, only young and relatively compact objects are detectable using X-rays in the active phase; the source shape cannot be determined in any objects. The inverse-Compton emission arising from the overpressured lobes of these extended AGNs is related to the density of the ambient medium and these fade rapidly as they expand into lower-density environments such as the intergalactic medium in low-mass haloes. Moreover, extended AGN emission is not detectable at redshifts *z* ≥ 2 for the *eRosita* surface brightness sensitivity limit investigated (but see Section [sec:Remnant density at high-redshifts]). The number density of extended AGNs with FR-II morphology detectable with *LOFAR* or *eRosita* are summarised in Table [tab:detections] as a function of redshift. We scale the number densities in our catalogue based on the observed high-luminosity function of.\n[tab:detections]\nccccccc && \n&&*eRosita*&*LOFAR*&dual detection&*eRosita* only&*LOFAR* only \n&0.1&-9.17&-8.78&-9.17&-11.40&-9.00 \n&0.5&-9.46&-7.88&-9.49&-10.61&-7.89 \n&1&-11.62&-7.07&-11.63&-13.12&-7.07 \n&0.1&-9.38&-9.07&-9.56&-9.86&-9.24 \n&0.5&-9.59&-8.13&-9.77&-10.07&-8.14 \n&1&-11.80&-7.54&-11.95&-12.32&-7.54 \n&0.1&-4.06&–&–&-4.06&– \n&0.5&-10.57&–&–&-10.57&– \n&1&-29.70&–&–&-29.70&– \nThe number density of remnant FR-IIs detectable with X-ray and radio frequencies is similarly shown in Figure [fig:remnantdensities]. The maximum age of remnants detectable at radio frequencies reduces quickly with increasing redshift (by more than a factor of two over the range *z* = 0.1-1) due to the increasing inverse-Compton losses at high redshift. Crucially, the inverse-Compton emission remains detectable at X-ray wavelengths long after the synchrotron radiation at radio frequencies ceases. This leads to a large population of remnant extended AGNs which are detectable with *eRosita* but cannot be seen at radio frequencies; these sources are typically aged between $100\\rm\\, Myrs$ and $1\\rm\\, Gyr$ (at the lowest redshifts). Specifically, *eRosita* uniquely detects 14% of remnants at *z* = 0.1 (based on non-core emission), decreasing to 1.1% and 0.002% at redshifts *z* = 0.5 and 1 respectively (i.e. a much greater fraction of remnants can only be detected using X-ray surveys than for active sources). We note in Section [sec:Remnant density at high-redshifts] that these number densities increase markedly for the higher redshifts with just a modest improvement in sensitivity. These exclusively X-ray detected remnants have more than five beams across the length of the lobe in 98% of sources at redshift *z* = 0.1; this reduces to 53% at *z* = 0.5 whilst no high-redshift (*z* ≥ 1) remnants are observed with multiple beams across the source. The remnant source number densities are summarised in Table [tab:detections].\n### False positive detections\nMock extended AGNs in our catalogue are only detected if emission from either the lobe or shocked gas shell is present above the survey surface brightness sensitivity limit. However, bremsstrahlung radiation from the brightest X-ray clusters may also be detectable using *eRosita* (in a single beam) without any density enhancement from AGN activity. At redshift, *z* = 0.1, simulated X-ray clusters more massive than $10^{13.43}\\rm\\, M\\_\\odot$ (derived analytically for RAiSE clusters[3](#fn3)) emit sufficiently high levels of bremsstrahlung radiation for their ambient medium to be detected and resolved from an X-ray nucleus. The critical cluster mass increases to $10^{14.94}\\rm\\, M\\_\\odot$ at *z* = 0.5 whilst no realistic mass haloes are predicted to have their ambient medium detected (in a single beam) at redshifts *z* ≥ 1; a small fraction of observed X-ray clusters may of course be atypically gas rich compared to our mock clusters. The number density of ‘false positive’ detections expected due to the misclassification of cluster gas is derived by integrating the cluster mass function above the critical mass at each redshift (see Table [tab:detections]).\nThe number density of X-ray clusters whose hot gas is detectable using *eRosita* far exceeds the expected density of extended AGNs at low-redshift (*z* ∼ 0.1). The presence of non-core emission at X-ray wavelengths therefore cannot be used to suggest AGN activity at these redshifts; however, 98% of exclusively X-ray detected remnants are well-resolved (i.e. have more than five beams across their lobes) and so can potentially be classified based on the shape of their X-ray brightness distribution. The same behaviour is seen at moderate redshifts though ‘false positives’ and actual AGN detections are expected to occur with equal likelihood. By contrast, at high-redshifts (*z* ≥ 1) the number density of X-ray clusters detectable with *eRosita* for a typical survey sensitivity is over seventeen orders of magnitude lower than the expected detection rate for extended AGN emission. Any non-core X-ray emission detected at high-redshifts can therefore be directly attributed to extended AGNs, and not the ambient medium.\n### Remnant density at high-redshifts\nThe number density of extended AGNs of FR-II morphology detectable at high-redshifts using an *eRosita*-like survey is investigated for a range of improved surface brightness sensitivities. In the previous section, we found none of the mock extended AGNs at redshift *z* = 2 and 4 could be detected using the assumed $14\\rm\\, nJy\\, beam^{-1}$ sensitivity; this is reduced by up to a factor of 100 down to $0.14\\rm\\, nJy\\, beam^{-1}$ for the same resolution ($15\\rm\\, arcsec$ FWHM). The expected number density of remnants detectable (and mostly uniquely detectable) by this theoretical increased sensitivity *eRosita* survey is shown in Figure [fig:remnantdensity] for redshifts *z* = 1, 2 and 4 as a function of the surface brightness sensitivity. Remnants can begin to be detected at *z* = 2 for a modest factor of 1.5 increase in sensitivity and at redshift *z* = 4 for an order of magnitude improvement. The rate of false positive detections from the ambient medium is also included in Figure [fig:remnantdensity]; the likelihood of detecting AGN emission and false positives converges at *z* = 1 for a factor of 20 increase in sensitivity, but at higher redshifts the false positive rate remains small for at least a three orders of magnitude improvement in surface brightness sensitivity.\nThe number density of remnants detected at *z* = 1 meanwhile can increase by over four orders of magnitude (before converging to the false positive rate) reaching a level comparable to that found by using radio frequencies; *eRosita* is therefore highly complementary to *LOFAR*. The number density of radio-detected remnants does not increase with improved sensitivity as any objects emitting appreciable synchrotron emission are already detectable. Importantly, the long lasting inverse-Compton emission probes a different population of remnants that has minimal overlap with the radio-selected sample (Table [tab:detections]). The number density of X-ray detected AGNs similarly coverges to the false positive rate with a comparable or greater number of detections to that obtained using radio frequencies; i.e. factor of five and 1200 more remnants at *z* = 2 and 4 respectively. The detection of non-core X-ray emission at high-redshift therefore presents a viable technique to identify previous episodes of AGN activity with X-ray telescopes presently in service; this would be greatly enhanced with only a modest increase in sensitivity. Our work predicts that X-ray wavelengths are capable of detecting at least a factor of ten more remnants than at radio frequencies for redshifts *z* > 2.2, increasing to a factor of 100 for redshifts *z* > 3.1.\n[fig:remnantdensity]\nSimilar results are obtained for extended AGNs in the active phase, however these are more readily observed at radio frequencies and thus are not considered likely targets for such surveys.\nCONCLUSIONS\n===========\nWe have extended the successful *Radio AGN in Semi-analytic Environments* lobe expansion and evolution model to X-ray wavelengths. Specifically, this improved model considers: (1) inverse-Compton upscattering of cosmic microwave background radiation by the synchrotron-emitting electrons in the lobe; (2) the dynamics of the shocked gas shell and the associated bremsstrahlung emission from this dense gas; and (3) emission from the ambient medium surrounding the extended AGN. We construct X-ray surface brightness maps of mock extended AGNs with type-II morphology to understand the relative importance of these radiative mechanisms; in particular, to determine what fraction of the population may be detectable (and correctly recognised as extended AGNs) at X-ray wavelengths.\nX-ray and radio-frequency surface brightness maps are derived for the technical characteristics of the *extended Roentgen Survey with an Imaging Telescope Array* (eRosita) and the *Low-Frequency Array* (LOFAR) instruments respectively. We consider the temporal evolution of the surface brightness in the lobe (inverse-Compton or synchrotron), shocked gas shell (bremsstrahlung) and ambient medium (bremsstrahlung) along the jet axis for typical sources located at increasing redshift from *z* = 0.1 to 2. The inverse-Compton emission from the lobe is dominant over the bremsstrahlung radiation from both the shocked gas shell and ambient medium in poor clusters ($10^{12.5}\\rm\\, M\\_\\odot$) at all sources ages. By contrast, the shocked gas shell is initially brighter than the lobe (for between 20 and $100\\rm\\, Myrs$) for sources expanding into denser environments ($10^{13.5}\\rm\\, M\\_\\odot$); the shocked gas shell stays brighter than the lobe for longer with either denser environments or higher jet powers. The X-ray surface brightness from the lobe and shocked gas shell remain detectable above the ambient medium (at the 5*σ* level) for another 65-$115\\rm\\, Myrs$ after the jet switches off after $30\\rm\\, Myrs$ at redshifts *z* ≥ 0.5. By contrast, synchrotron emission ceases after only 10-$70\\rm\\, Myrs$ at radio frequencies. We find that although both synchrotron and inverse-Compton emission fade more rapidly at higher redshift, synchrotron radiation is much more sharply curtailed, leading to a sizable high-redshift population of remnant radio galaxies emitting exclusively through the inverse-Compton mechanism.\nWe constructed an integrated X-ray luminosity function for extended AGNs by generating a mock population of FR-IIs with jet powers, active lifetimes and host cluster environments based on observational constraints. The integrated X-ray luminosity of most extended AGNs at low redshifts (*z* ≤ 1) is found to be strongly correlated with the properties of the ambient medium. In other words, the bremsstrahlung radiation from either the ambient medium or the shocked gas shell (whose density is strongly correlated with the host environment) is the dominant source of X-ray emission. At these low redshifts, only a small population of AGNs with the highest jet powers and moderate ages of a few tens of Myrs can have inverse-Compton emission contribute in excess of 90% of the total integrated X-ray luminosity from the AGN–host cluster system. By contrast, the X-ray luminosity function at higher redshifts (*z* > 1) is dominated by inverse-Compton emission for the vast majority of the extended AGN population. The stronger microwave background radiation at these high redshifts boosts the lobe contribution to the integrated luminosity well above that generated by the hot cluster gas, leading to an increased population of X-ray bright extended AGNs.\nWe used our mock extended AGN catalogue to explore how many new objects could be detected using both existing and increased sensitivity X-ray observations. We find that most active FR-II sources at redshifts *z* ≤ 1 can be detected at radio frequencies for the sensitivity of the *LOFAR Two-metre Sky Survey* (LoTSS). However, only a small fraction of active or remnant sources can be seen at X-ray wavelengths for typical *eRosita* sensitivity, when excluding core emission. No active extended AGNs are detectable at X-ray wavelengths but not at radio frequencies. By contrast, *eRosita* will find 14% of remnants at *z* = 0.1 which are not visible to *LOFAR*, decreasing to 1.1% and 0.002% at redshifts *z* = 0.5 and 1 respectively. Meanwhile, the surface brightness of the bremsstrahlung radiation from the ambient medium of any realistic mass haloes is expected to become undetectable beyond redshift *z* ≥ 1; i.e. any non-core X-ray detection can be attributed to extended AGN activity. We consider the effectiveness of radio-frequency and X-ray surveys at detecting remnants at these high-redshifts for greatly enhanced surface brightness sensitivity. The number density of X-ray detected remnants at redshift *z* = 1 becomes comparable to the number of radio-frequency detections. 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A. 2001, MNRAS, 322, 536\nYates, P. M., Shabala, S. S., & Krause, M. G. H. 2018, MNRAS, 480, 5286\n---\n1. Email: turner.rj@icloud.com[↩](#fnref1)\n2. RAiSE implicitly assumes a single magnetic field value throughout the lobes; although more complex configurations almost certainly exist these are not well constrained at present, and our approach is standard for lobe modelling.[↩](#fnref2)\n3. RAiSE simulates cluster density profiles based on the observed profiles of, taking the scaling gas mass and virial radius from the semi-analytic galaxy evolution model of. The gas temperature is calculated using an observed relationship with the halo mass.[↩](#fnref3)\n\nRAiSE X: searching for radio galaxies in X-ray surveys\n======================================================\n### Accepted 2020 March 06. Received 2020 March 05; in original form 2019 December 12\n[firstpage]\nWe model the X-ray surface brightness distribution of emission associated with Fanaroff & Riley type-II radio galaxies. Our approach builds on the RAiSE dynamical model which describes broadband radio-frequency synchrotron evolution of jet-inflated lobes in a wide range of environments. The X-ray version of the model presented here includes: (1) inverse-Compton upscattering of cosmic microwave background radiation; (2) the dynamics of the shocked gas shell and associated bremsstrahlung radiation; and (3) emission from the surrounding ambient medium. We construct X-ray surface brightness maps for a mock catalogue of extended FR-IIs based on the technical characteristics of the *eRosita* telescope. The integrated X-ray luminosity function at low redshifts (*z* ≤ 1) is found to strongly correlate with the density of the ambient medium in all but the most energetic sources, whilst at high-redshift (*z* > 1) the majority of objects are dominated by inverse-Compton lobe emission due to the stronger cosmic microwave background radiation. By inspecting our mock spatial brightness distributions, we conclude that any extended X-ray detection can be attributed to AGN activity at redshifts *z* ≥ 1. We compare the expected detection rates of active and remnant high-redshift radio AGNs for *eRosita* and *LOFAR*, and future more sensitive surveys. We find that a factor of ten more remnants can be detected using X-ray wavelengths over radio frequencies at *z* > 2.2, increasing to a factor of 100 for redshifts *z* > 3.1.\ngalaxies: active – galaxies: jets – radio continuum: galaxies – X-rays: galaxies\nINTRODUCTION\n============\nMost galaxies are known to harbour a supermassive black hole (SMBH) at their centre, with the galaxy and SMBH growth rates strongly intertwined over their evolutionary history. The activity of the black hole is tied to the state of its surroundings, since, as hot cluster gas cools by bremsstrahlung radiation, it forms cooling flows that sink towards and are accreted by the SMBH, switching it on. This active galactic nucleus (AGN) imparts a fraction of its accreted energy back into its host environment through either radiative or kinetic-mode feedback. The radio lobes inflated by the relativistic jets of pair-plasma, which emanate from the accretion disk in the kinetic-mode, are responsible for pushing out vast amounts of gas from the host galaxy and thus suppress star formation, or sometimes trigger regions of enhanced growth along shock fronts. find the growth of black holes closely tracks the star formation history of galaxies across cosmic time. The energy input by AGNs into their surroundings through kinetic feedback (primarily by shocks and pressure-volume work) prevent catastrophic cooling of the hot cluster gas. Kinetic-mode feedback is also invoked in cosmological galaxy formation models to explain the missing stellar mass in the brightest galaxies towards the present epoch.\nThe quantification of the energetics of kinetic-mode AGN feedback requires knowledge of the energy imparted in each outburst and the duty cycle of the black hole activity. Jet kinetic powers can be directly measured at radio-frequencies using techniques including hotspot luminosities and spatial shifts of radio cores in VLBI images, or more readily constrained in large sky surveys using dynamical models for the lobes populated by shock-accelerated synchrotron-emitting electrons. Recent high sensitivity and resolution observations by the *Low Frequency Array* (LOFAR) at low frequencies ($150\\rm\\, MHz$) have reinvigorated studies of the AGN duty cycle. found that up to 100 per cent of all large galaxies host an active nucleus, noting that their accretion rate is modulated such that half the energy output is released in outbursts of increased activity lasting less than two percent of the time. Studies of individual objects and large sky surveys have also found populations of remnant and restarted radio galaxies, setting a lower bound for the duty-cycle. used a radio source dynamical model to provide an upper limit on the duty cycle of *δ* < 0.15 in the remnant B20924+30. tighten the range of plausible duty cycles by comparing the size and luminosity functions of radio sources in the Lockman Hole to simulated functions based on dynamical models for a range of lifetime distributions.\nThe shock-accelerated electrons comprising extended radio AGN lobes not only emit synchrotron radiation at radio frequencies, but also produce X-ray emission at keV energies due to the inverse-Compton upscattering of cosmic microwave background photons by lower energy electrons. The inverse-Compton and synchrotron radiative loss mechanisms more rapidly deplete the higher energy (*γ* ∼ 104) electrons involved in the synchrotron radiation compared to the lower energy (*γ* ∼ 103) electrons responsible for the X-ray inverse-Compton emission. Remnant radio galaxies are therefore expected to appear as inverse-Compton ‘ghosts’ for some period of time after the cessation of jet activity before becoming undetectable at both radio and X-ray wavelengths. In particular, the extended X-ray source HDF130 at *z* = 1.99 has been identified as a remnant radio galaxy despite its double-lobed structure only being visible at X-ray wavelengths. Further inverse-Compton ‘ghosts’ should be readily detectable at high redshift since the energy density of the cosmic microwave background increases as (1 + *z*)4, offsettting the reduced X-ray flux density at greater distance. Conversely, only the youngest radio galaxies at high-redshift will be detectable in radio observations due to the extreme radiative losses resulting from the strong microwave background energy density. investigate two distant quasars, ULAS J112001.48+064124.3 at *z* = 7.1 and SDSS J1030+0524 at *z* = 6.3, finding that powerful jets fuelled by super-Eddington accretion rates could exist but be undetectable with current surveys. However, the benefit of increased brightness at higher redshifts may be offset by the convolution of the extended and core emission, which can be of comparable magnitude to the brightest inverse-Compton lobes.\nIn this work, we extend the *Radio AGNs in Semi-analytic Environments* model for the dynamical evolution and synchrotron emissivity of active, remnant and restarted radio galaxies to X-ray wavelengths; in particular, we seek to quantify which radio galaxies have non-core associated X-ray emission detectable with the surface brightness sensitivity of current surveys (e.g. using *Chandra* and *eRosita*). Previous iterations of RAiSE have found success in: (1) reproducing surface brightness and spectral age maps for canonical FR-I (3C31) and FR-II (3C436) type sources ; (2) deriving jet kinetic powers consistent with X-ray inverse-Compton measurements ; and (3) accurately constraining the Hubble constant using low-redshift AGNs. In this paper, we first extend RAiSE to calculate the integrated X-ray luminosity due to the inverse-Compton upscattering of cosmic microwave background radiation by lobe electrons, then create model X-ray surface brightness maps for lobes of type-II morphology (Section [sec:LUMINOSITY MODEL]). These X-ray brightness maps are completed in Section [sec:SHOCKED SHELL BREMSSTRAHLUNG RADIATION] by including the bremsstrahlung radiation both from the shocked shell of gas swept up between the bow shock and lobe plasma, and from the ambient medium. In Section [sec:X-ray surface brightness maps], we investigate the changing importance of the inverse-Compton and bremsstrahlung radiative mechanisms with redshift and intrinsic properties of the source. Finally, we create a mock catalogue of extended radio galaxies based on an observationally informed set of parameters; this catalogue is used to generate the X-ray luminosity function for FR-IIs over cosmic time, and characterise the effectiveness of both radio-frequency (*SKA*-pathfinders) and X-ray surveys at detecting high-redshift remnants (Section [sec:MOCK EXTENDED AGN POPULATION]).\nThe $\\Lambda \\rm CDM$ concordance cosmology with $\\Omega\\_{\\rm M} = 0.3$, ΩΛ = 0.7 and $H\\_0 = 70 \\rm\\,km \\,s^{-1} \\,Mpc^{-1}$ is assumed throughout the paper.\nLOBE INVERSE-COMPTON EMISSIVITY\n===============================\nThe inverse-Compton emissivity model developed in this work is based on the synchrotron model presented by ; it can therefore similarly be applied to any AGN dynamical model, be it analytical or a hydrodynamical simulation. The model assumes that the radiative losses are not taken into account self-consistently in the evolution of the lobe pressure. confirms that synchrotron radiation comprises less than ten percent of the input power for typical radio galaxies (at *z* = 0), whilst inverse-Compton and bremsstrahlung radiation contribute even less to the energy loss during the source expansion. However, the cosmic microwave background radiation responsible for the inverse-Compton losses increases with redshift as (1 + *z*)4; the inverse-Compton radiation can reach up to 40% of the input power in a $100\\rm\\, Myr$ source at *z* = 1, or all of the input power for a $10\\rm\\, Myr$ source at *z* = 4. The results in this work do not consider objects beyond these extremes in age and redshift. The adiabatic expansion of the lobe is therefore assumed to take up the bulk of the input energy as explicitly considered in numerical and published analytical dynamical models.\n proposed a technique to derive the inverse-Compton emissivity from the hotspots of powerful FR-IIs by integrating over the full electron and photon distribution. and later both derived models for the inverse-Compton emission from radio lobes, with the formalisms consistent to those found in the dynamical and synchrotron emissivity models of and. These two models make use of the electron energy distribution assumed in the synchrotron emissivity model, and will therefore have the same strengths and shortcomings as the calculation of the radio emissivity.\nUnresolved, continuously injected electron model\n------------------------------------------------\nThe synchrotron-emitting electrons in radio sources have significant kinetic energy when compared to the energy of the cosmic microwave background (CMB) photons, and thus the electron energy can be transferred to a CMB photon through inverse-Compton scattering.\nAssuming that the electrons emit synchrotron radiation only at some critical frequency $\\nu\\_{\\rm syn} = \\gamma^2 \\nu\\_{\\rm L}$, where $\\nu\\_{\\rm L}$ is the Larmor frequency, the total lossless radio power (integrated over solid angle) emitted by electrons in a volume element *d**V* is \n \n$$\\begin{split}\ndL\\_{\\nu\\_{\\rm syn}} = \\frac{1}{2} \\sigma\\_{\\rm T} c u\\_{\\rm B} \\frac{\\gamma^3}{\\nu\\_{\\rm syn}} n(\\gamma) dV,\n\\end{split}\n\\label{Lsyn}$$ \nwhere $\\sigma\\_{\\rm T}$ is the electron scattering cross-section, *c* is the speed of light, $u\\_{\\rm B}$ is the energy density of the magnetic field, and *n*(*γ*) is the electron energy distribution. The inverse-Compton power is similarly given by\n \n$$\\begin{split}\ndL\\_\\nu = \\frac{1}{2} \\sigma\\_{\\rm T} c u\\_{\\rm c} \\frac{\\gamma^3}{\\nu} n(\\gamma) dV,\n\\end{split}\n\\label{Lic}$$ \nwhere $u\\_{\\rm c} = u\\_{\\rm c0} (1 + z)^4$ is the energy density of the cosmic microwave background radiation at redshift *z*, *e* the electron charge and $u\\_{\\rm c0} = 0.25e\\times10^6 \\rm\\, J \\, m^{-3}$ is the corresponding energy density at the present epoch. The electron population involved in both emission processes is identical, and thus the Lorentz factor and electron energy distribution can be modelled using the equations of derived for synchrotron radiation. For inverse-Compton emission, the relevant frequency is that of upscattered CMB photons, $\\nu = \\gamma^2 \\nu\\_{\\rm cmb}$, where a photon at the peak CMB frequency $\\nu\\_{\\rm cmb} = 5.879\\times10^{10} {\\rm\\, Hz \\, K^{-1}} \\times 2.73 {\\rm\\, K}(1 + z)$ is boosted by a synchrotron-emitting electron with a Lorentz factor *γ*.\nThe peak frequency of the synchrotron-emitting electrons involved in the inverse-Compton upscattering of CMB photons to frequency *ν*, can thus be derived for rest-frame frequencies as\n \n$$\\begin{split}\n\\nu\\_{\\rm syn} = \\frac{3e \\nu \\sqrt{2 \\mu\\_0 u\\_{\\rm B}}}{2\\pi m\\_{\\rm e} \\nu\\_{\\rm cmb}},\n\\end{split}$$ \nwhere *μ*0 is the permeability of free space and $m\\_{\\rm e}$ the electron mass.\nThe full calculation of the inverse-Compton emissivity from the upscattering photons requires the integration over the total spectrum of CMB photons (i.e. a blackbody distribution). The following correction should be applied to the simpler analytically tractable solution which assumes all CMB photons at a given redshift are at a single frequency :\n \n$$\\begin{split}\nJ\\_{\\rm corr}(s) = \\frac{\\pi^4}{15 \\Gamma (\\tfrac{s + 5}{2}) \\zeta (\\tfrac{s + 5}{2})},\n\\end{split}$$ \nwhere Γ and *ζ* are the Gamma and Zeta functions respectively. For a typical electron energy injection index of *s* = 2.4 (i.e. *N*(*γ*) ∝ *γ*− *s*), the use of a single frequency CMB spectrum in this calculation would lead to an error of approximately 40%.\nFollowing the method of, their Equation 3 is modified to yield the integrated inverse-Compton emissivity of the lobe at time *t*:\n \n$$\\begin{split}\nL(\\nu, t) = &\\frac{K(s) {\\nu\\_{\\rm syn}}^{(1 - s)/2}}{J\\_{\\rm corr}(s)} \\left(\\frac{\\nu\\_{\\rm syn}}{\\nu} \\right) \\frac{q^{(s - 3)/4}}{(q + 1)^{(s + 1)/4}} \\\\\n&\\quad \\times p(t)^{(s + 1)/4} V(t) (\\Gamma\\_{\\rm c} - 1) [u\\_{\\rm c0} (z + 1)^4] \\mathcal{Y}(\\gamma, t),\n\\end{split}\n\\label{luminosityloss}$$ \nwhere *K*(*s*) is the source specific constant defined in Equation 5 of, $\\Gamma\\_{\\rm c}$ is the adiabatic index of the lobe plasma, *p*(*t*) is the present-time lobe pressure, and *q* ≡ *u**B*/*u**e* is the ratio of the energy density in the magnetic field to that in the synchrotron-emitting particles [2](#fn2). The differences between the synchrotron and inverse-Compton (here) forms of this equation can be explained as follows: (1) the power law spectrum is cast in terms of the synchrotron frequency rather than the (now inverse-Compton) emitting frequency, (2) the inverse-Compton frequency *ν* in the denominator of Equation [Lic] replaces the synchrotron frequency in Equation [Lsyn], and (3) the inverse-Compton power in Equation [Lic] replaces a linear dependence on the magnetic field energy density $u\\_{\\rm B} = qp/(\\Gamma\\_{\\rm c} - 1)(q + 1)$ in Equation [Lsyn] with a dependence on the energy density of the CMB radiation $u\\_{\\rm c}$.\nThe loss function Y(*γ*, *t*) is defined in Equation 7 of for active and remnant sources as\n \n$$\\begin{split}\n\\mathcal{Y}(t) = \\int\\_0^t \\frac{a\\_{\\rm v}(t\\_{\\rm i})}{t\\_{\\rm i}} \\frac{Q(t\\_{\\rm i})}{Q\\_0} \\left[\\frac{p(t\\_{\\rm i})}{p(t)} \\right]^{1 - 4/(3\\Gamma\\_{\\rm c)}} \\frac{V(t\\_{\\rm i})}{V(t)} \\left[\\frac{\\gamma\\_{\\rm i}}{\\gamma} \\right]^{2 - s} dt\\_{\\rm i},\n\\end{split}\n\\label{resolvedvolume}$$ \nwhere $Q(t\\_{\\rm i})/Q\\_0$ is the instantaneous jet power at the electron injection time scaled by its active value. The integral is over the particle injection times $t\\_{\\rm i}$, where $p(t\\_{\\rm i})$, $V(t\\_{\\rm i})$ and $\\gamma\\_{\\rm i}$ are the lobe pressure, volume and electron Lorentz factor at the time of injection respectively. The constant $a\\_{\\rm v}$ is the average volume expansion rate of the packet of electrons injected at time $t\\_{\\rm i}$, defined through $V \\propto t^{a\\_{\\rm v}(t\\_{\\rm i})}$. The loss function is defined, as before, in terms of the Lorentz factor *γ* of the synchrotron-emitting electrons which radiate at frequency $\\nu\\_{\\rm syn}$; the Lorentz factor of these electrons is the same as for the inverse-Compton upscattering process.\nThe inverse-Compton emission is modelled over the evolutionary history of a source using the RAiSE dynamical model to derive the pressure and volume evolution of the lobe. Luminosity–source age tracks are thus calculated at $1\\rm\\, keV$ observer-frame X-ray energies for four extended AGNs (Figure [fig:LDtracks]). Our base model (Model A) considers a $Q = 3 \\times 10^{38}\\rm\\, W$ jet with an active lifetime of $t = 100\\rm\\, Myrs$ at redshift *z* = 2. The gas density profile is modelled for a $10^{13.5}\\rm\\, M\\_\\odot$ mass cluster following the method of, the lobe axis ratio is set as *A* = 4, the injection index of the electron energy distribution is *s* = 2.4, whilst the other parameters in the RAiSE dynamical and inverse-Compton emissivity models take the same values as used by. Three variations to the base model are considered with either a lower redshift (Model B), higher jet power (Model C) or shorter active lifetime (Model D). The X-ray inverse-Compton luminosity–source age tracks for our four extended AGNs are qualitatively very similar (for example) to those in Figure 3 of, albeit with noticeably brighter luminosities, largely due to the richer cluster cores in our observationally informed density profiles.\n[fig:LDtracks]\nSpatially resolved lobe losses\n------------------------------\nThe spatially resolved RAiSE dynamical model calculates lobe properties as a function of distance from the site of particle acceleration (hotspots for FR-II, flaring points for FR-I sources). Taking the particle acceleration site as the origin of the coordinate system, the inverse-Compton emissivity from a position **r** in the lobe can similarly be derived following.\n \n$$\\begin{split}\ndL(\\nu&, t, \\textbf{r}) = \\frac{K(s) {\\nu\\_{\\rm syn}}^{(1 - s)/2}}{J\\_{\\rm corr}(s)} \\left(\\frac{\\nu\\_{\\rm syn}}{\\nu} \\right) \\frac{q^{(s + 1)/4}}{(q + 1)^{(s + 5)/4}} \\\\\n&\\times p(t, \\textbf{r})^{(s + 1)/4} dV(t, \\textbf{r}) (\\Gamma\\_{\\rm c} - 1) [u\\_{\\rm c0} (z + 1)^4] \\\\\n&\\times \\left[\\frac{p(t\\_{\\rm i}(\\textbf{r}))}{p(t, \\textbf{r})} \\right]^{1 - 4/(3\\Gamma\\_{\\rm c})} \\left[\\frac{\\gamma\\_{\\rm i}(\\textbf{r})}{\\gamma} \\right]^{2 - s},\n\\end{split}\n\\label{resolvedluminosity}$$ \nwhere the volume occupied at the present time by an electron packet injected between $t\\_{\\rm i}$ and $t\\_{\\rm i} + dt\\_{\\rm i}$ is given by\n \n$$\\begin{split}\ndV(t, \\textbf{r}) &= \\int\\_0^t \\frac{Q(t\\_{\\rm i})}{Q\\_0} \\delta(t\\_{\\rm i}' - t\\_{\\rm i}) \\frac{dV(t\\_{\\rm i}', \\textbf{r})}{dt\\_{\\rm i}'} dt\\_{\\rm i}'.\n\\end{split}\n\\label{resolvedvolume}$$ \nFor lobed sources with a high internal sound speed (typically FR-IIs), the volume element can be simplified to $dV(t, \\textbf{r}) = a\\_{\\rm v}(t\\_{\\rm i}) V(t\\_{\\rm i}) dt\\_{\\rm i}/t\\_{\\rm i}$ and the pressures taken as lobe averages; i.e. *p*(*t*, **r**) and $p(t\\_{\\rm i}(\\textbf{r}))$ are the lobe pressure at the present time and the time of electron injection respectively.\n used hydrodynamic simulations to study the spatial distribution of synchrotron-emitting electrons throughout lobed sources, by injecting tracer fields into the jet at regular intervals and tracking their motion. The distribution of these tracer fields within the lobe at each injection age was used to produce a representative synchrotron-emitting electron population at every location across the source. found that the fluid injected at the jet terminal hotspot initially flows smoothly back towards the core, but electrons of different ages disperse broadly after travelling back slightly less than half of the lobe length. model the average location, **r**, and the 2*σ* spread, *d***r**, of the electron packets as a function of synchrotron age (see their Equations 11a,b). These authors also found the behaviour of the flow to be independent of both physical and temporal scales, enabling a single analytic description of the electron distribution to be assumed for all lobed FR-II sources.\nBelow, we use the results of these simulations to inform our analytic models for both active and remnant radio sources. For remnants, the relative locations of the electron packets within the lobe are assumed to remain fixed upon the jet switching off. This assumption is only valid on timescales shorter than the mixing timescales. However, remnant sources are typically observed soon after the jet switches off and this assumption is expected to be appropriate for most objects.\nSHOCKED SHELL BREMSSTRAHLUNG RADIATION\n======================================\nThe bow shock generated by a powerful lobed radio source (FR-II or FR-I) sweeps up the intracluster medium (ICM) in its path as it propagates outwards from the active nucleus. The thermal evolution of this swept-up gas was considered by. Here, we extend the RAiSE model to explicitly include evolution of the shell of shocked gas surrounding the lobe.\nShocked shell dynamical model\n-----------------------------\n### Axis ratio evolution\nHydrodynamical simulations suggest that the bow shocks of radio sources expand in a self-similar manner despite the lobes slowly elongating over their evolutionary history: for example, in their Figure 1, the axis ratio of the shocked shell remains a constant value of ∼ 2.5 once fully formed, whilst the lobe ratio slowly increases from 6.5 to 8. We therefore model the growth of the shocked shell in RAiSE using the same formalism as for lobe evolution, but excluding the late-stage Rayleigh-Taylor mixing which quickly pinches the lobes. The radius of the shocked shell is thus related to that of the intact lobe at each point on its surface as $R\\_{\\rm s}(\\theta) = \\tau R(\\theta)$, where *τ* ≡ *τ*(*θ*) is a constant of proportionality and *θ* is the angle between the surface location and the jet axis.\nSimulations clearly show that the ratio between shocked shell and lobe radii varies across the surface of the shocked shell. We define the axis ratio (length divided by width) of the shocked shell in terms of the axis ratio of the lobe as $A\\_{\\rm s} = A^\\iota$, for some exponent *ι*. Based on the hydrodynamical simulations of, we assume a value of *ι* = 0.5 in this work; in other words, $A\\_{\\rm s} = \\sqrt{A}$.\nThe radial distance to the lobe surface at an angle *θ* from the jet axis is related to the length of the lobe along the jet axis by the ratio given in Equation 21 of,\n \n$$\\begin{split}\n\\eta(\\theta) = \\frac{1}{\\sqrt{A^2 \\sin^2 \\theta + \\cos^2 \\theta}}\n\\end{split},\n\\label{eta}$$ \nand the distance to the shocked shell at angle *θ* is similarly related to the length of the shocked shell along the jet axis by the ratio\n \n$$\\begin{split}\n\\eta\\_s(\\theta) = \\frac{1}{\\sqrt{A^{2\\iota} \\sin^2 \\theta + \\cos^2 \\theta}}\n\\end{split}.\n\\label{eta\\_s}$$ \nThe radius of the shocked shell can thus be related to that of the lobe as $R\\_s(\\theta) = \\tau \\eta\\_{\\rm s}(\\theta) R(\\theta)/\\eta(\\theta)$, where *τ* ∼ 1.05 is the ratio of the shocked shell to lobe radii along the jet axis.\n### Modified shock geometry\nTo describe the expansion of the shocked shell, we follow the geometric approach of. We calculate the component of the expansion rate normal to the surface of the shocked shell by relating it to the expansion rate of the shocked shell along the jet axis, as in Equation 20 of,\n \n$$\\begin{split}\n\\zeta\\_s(\\theta) = \\left[\\frac{A^{2\\iota} \\sin^2 \\theta + \\cos^2 \\theta}{A^{4\\iota} \\sin^2 \\theta + \\cos^2 \\theta} \\right]^{1/2}\n\\end{split}.\n\\label{zeta\\_s}$$ \nAgain following the method of, we derive a second order differential equation in terms of the lobe radius and expansion rate of the lobe surface at an angle *θ* from the jet axis. This differential equation cannot be solved analytically in general, and so we must adopt a numerical scheme using a fourth order Runge-Kutta method in terms of a system of two first order ODEs. The following system of equations must be solved for each small angular element [*θ* − *d**θ*/2, *θ* + *d**θ*/2) of the lobe and shocked shell:\n \n$$\\begin{split}\n\\dot{R} &= v \\\\\n\\dot{v} &= \\frac{3 (\\Gamma\\_{\\rm x} + 1)(\\Gamma\\_{\\rm c} - 1) Q R^{\\beta - 3} d\\lambda}{8 \\pi v (\\zeta\\_{\\rm s}/\\eta\\_{\\rm s})^2 k \\sin\\theta d\\theta} \\left[\\frac{\\tau \\eta\\_{\\rm s}}{\\eta} \\right]^3 + \\frac{(\\beta - 3\\Gamma\\_{\\rm c}) v^2}{2 R} \\\\\n&\\quad\\quad + \\frac{(\\Gamma\\_{\\rm x} - 1) (3 \\Gamma\\_{\\rm c} - \\beta - \\xi) l}{4 R^{\\xi + 1} (\\zeta\\_{\\rm s}/\\eta\\_{\\rm s})^2},\n\\end{split}\n\\label{supersonic system}$$ \nwhere in the strong-shock supersonic limit (near time zero in the numerical scheme) the *θ* dependent constant *d**λ* is defined through the expression:\n \n$$\\begin{split}\n&\\frac{8\\pi k \\sin\\theta d\\theta}{3 (\\Gamma\\_{\\rm x} + 1)} \\left[(3 \\Gamma\\_{\\rm c} - \\beta)R^{2 - \\beta} \\dot{R}^3 + 2R^{3 - \\beta} \\dot{R} \\ddot{R} \\right]\\_{\\theta = 0} \\\\\n&\\quad\\quad \\times {\\eta\\_{\\rm s}}^{3 - \\beta}(\\theta) {\\zeta\\_{\\rm s}}^2(\\theta) \\left[\\frac{\\eta(\\theta)}{\\tau \\eta\\_{\\rm s}(\\theta)} \\right]^3 = (\\Gamma\\_{\\rm c} - 1) Q\\, d\\lambda(\\theta).\n\\end{split}$$ \nHere, *β* and *k* parametrise the local shape of the ambient density profile (i.e. *ρ* = *k**r*− *β*), *ξ* and *l* describe the local shape of the temperature profile (i.e. *T* = *l**r*− *ξ*), whilst $\\Gamma\\_{\\rm x}$ is the adiabatic index of the ambient medium. The shape of density and temperature profiles are based on cluster observations and modelled as piecewise continuous power laws to retain analytically tractable solutions. The differential equations describing the evolution of the lobe in the subsonic and remnant phases are unchanged from those given in and respectively.\nIsothermal evolution of the shocked shell\n-----------------------------------------\nAs first pointed out by, the final state of the shocked gas lying between the lobe surface and the bow shock depends critically on whether it expands adiabatically or isothermally; we assume here that the swept-up gas evolves isothermally. The pressure, $p\\_{\\rm s}(\\theta)$, of the swept-up gas at an angle *θ* from the jet axis is found from the jump conditions at the shock.\nThe cooling rate, and thus intensity of thermal bremsstrahlung radiation, in the shocked shell depends on both the pressure of the shocked gas and its temperature. For low Mach numbers *M*0 the cooling time of the swept-up gas, $t\\_{\\rm s}$, can become short compared with that in the cluster, $t\\_{\\rm x}$, with $t\\_{\\rm s}/t\\_{\\rm x} > 0.05M\\_0$. However, the cooling time of the external gas would have to be at least an order of magnitude less than the source age, *t*, for the shocked gas to suffer significant radiative losses. Such conditions are only possible at the centres of the strongest cooling flow clusters – precisely *not* the locations where powerful, large radio galaxies are typically found. We therefore ignore the cooling of the gas within the shocked shell in our analysis.\n### Mean shocked shell temperature\nThe mean temperature of the shocked gas is found by first calculating the mean pressure and density of the shocked shell. The mean density is taken as the ratio of the mass of ambient gas previously occupying the lobe and shocked shell to the volume of the shocked gas shell:\n \n$$\\begin{split}\n\\bar{\\rho\\_{\\rm s}}(t) = \\frac{\\int\\_0^{\\pi/2} \\int\\_0^{R\\_{\\rm s}(t, \\theta)} \\rho(r) r^2 dr \\sin \\theta d\\theta}{\\int\\_0^{\\pi/2} \\int\\_{R(t, \\theta)}^{R\\_{\\rm s}(t, \\theta)} r^2 dr \\sin \\theta d\\theta},\n\\end{split}$$ \nwhere *ρ*(*r*) is the gas density of the ambient medium. This expression must be solved numerically except for the special case of spherical lobes expanding into a power law environment. The mean temperature of the shocked gas shell is thus given as $T\\_{\\rm s}(t) = \\bar{m} \\bar{p\\_{\\rm s}}(t)/k\\_{\\rm B} \\bar{\\rho\\_{\\rm s}}(t)$ for a shocked gas shell with mean pressure $\\bar{p\\_{\\rm s}}(t)$ and an average particle mass $\\bar{m} \\sim 0.6m\\_{\\rm p}$, where $m\\_{\\rm p}$ is the proton mass. The temperature may of course vary throughout the shocked gas shell but without precise knowledge of any density gradients we simply assume mean values.\n### Bremsstrahlung radiation\nThe X-ray emissivity per unit volume due to thermal bremsstrahlung radiation is \n \n$$\\begin{split}\nJ(\\nu) =& \\frac{Z^2 e^6}{3 \\pi^2 {\\varepsilon\\_0}^3 {m\\_{\\rm e}}^2 c^3} \\left(\\frac{\\pi m\\_{\\rm e}}{6} \\right)^{1/2} \\\\\n&\\quad\\quad (k\\_{\\rm B} \\mathcal{T})^{-5/2} {\\tilde{p}}^{\\!\\:2} e^{-h\\nu/k\\_{\\rm B} \\mathcal{T}} g(\\nu, \\mathcal{T}),\n\\end{split}$$ \nwhere the pressure and temperature (*p̃* and T respectively) may relate to the shocked shell, ambient medium, or any other plasma. Here, $Z \\gtrsim 1$ is the average atomic number of the positively changed particles, *ɛ*0 is the vacuum permittivity, and, at frequencies $h\\nu \\ll k\\_{\\rm B} \\mathcal{T}$, the Gaunt factor has a logarithmic dependence on frequency as\n \n$$\\begin{split}\ng(\\nu, \\mathcal{T}) = \\frac{\\sqrt{3}}{\\pi} \\ln \\left(\\frac{4}{\\zeta} \\frac{k\\_{\\rm B} \\mathcal{T}}{h\\nu} \\right),\n\\end{split}$$ \nwhere here *ζ* = 1.78 is Gauss’ number. In this work, we assume the metallicity of the plasma is 0.3 times the solar value, corresponding to an average atomic number of *Z* ∼ 1.04. The shocked gas pressure used in the bremsstrahlung radiation calculation is that derived from the shock jump conditions on the shell surface at angle *θ* (i.e. $\\tilde{p} = p\\_{\\rm s}(\\theta)$), whilst the mean temperature is derived following the method described in the previous section.\n[fig:apec]\nThe X-ray spectra of smaller clusters at low-redshift are dominated by emission lines at the wavelengths of the *Chandra* and *eRosita* surveys. The APEC plasma code can calculate the X-ray emission spectra from thin-thermal plasma for various temperatures and metallicities. In Figure [fig:apec], we plot the fractional contribution of continuum emission to the total bremsstrahlung X-ray spectrum (including both continuum and line emission) integrated over a typical 0.5-$2\\rm\\, keV$ observing band. The emission lines are found to contribute a significant component of the X-ray flux density at redshifts *z* < 2.5 in clusters with temperatures $kT \\lesssim 1\\rm\\, keV$, corresponding to halo masses $\\lesssim 10^{13.5}\\rm\\, M\\_\\odot$. We use the appropriate APEC model to correct our analytically derived flux densities of the X-ray emission arising from bremsstrahlung radiation.\nX-RAY SURFACE BRIGHTNESS MAPS\n=============================\nWe use the formalism developed in the preceding sections to make predictions for the observed X-ray emission from radio galaxies and their environments. There are three main contributions: lobe inverse-Compton emission (Section [sec:LUMINOSITY MODEL]) and two sources of bremsstrahlung radiation, from the shocked gas shell ahead of the lobes (Section [sec:SHOCKED SHELL BREMSSTRAHLUNG RADIATION]), and from the ambient medium. We calculate the latter in the same way as bremsstrahlung from the shocked gas shell, but adopting density and temperature parameters characteristic of the hot gas in galaxy clusters, as described in.\nX-ray and radio brightness maps\n-------------------------------\nThe lobes and shocked gas shell of each mock radio galaxy are divided into a 512 × 512 × 512 grid of cubic pixels; this grid extends a factor of two beyond the edge of the lobe to enable a comparison with X-ray surface brightness of the ambient medium. Each cell in the cube is classified as either part of the lobe, shocked shell or ambient medium based on the modified RAiSE dynamical model (Section [sec:SHOCKED SHELL BREMSSTRAHLUNG RADIATION]), and the inverse-Compton emissivity or bremsstrahlung radiation calculated as appropriate. The observed emission arising from these extended objects is the integral of all the emissivity along a given line-of-sight. The two-dimensional surface brightness is simply calculated by summing the emissivity from every cell along the depth of the source, assuming the lobe plasma and ambient medium in front of the source is optically thin. The X-ray emission arising from the ambient medium lying outside the grid of cell is calculated analytically for each line-of-sight and added to the total from the numerical grid. For simplicity, we assume that the lobes lie in the plane of the sky.\nThe inverse-Compton and bremsstrahlung emissivity in the (two-dimensional) surface brightness map are derived for the technical characteristics of the *extended Roentgen Survey with an Imaging Telescope Array*. Specifically, at a $1\\rm\\, keV$ observer-frame energy, the half-energy width (on axis) is $15\\rm\\, arcsec$ and the total effective area of the seven mirror systems is $\\sim 1500\\rm\\, cm^2$. The number of $1\\rm\\, keV$ photons falling on these mirrors in a typical 1000$\\rm\\, s$ (1 ks) observing time is thus calculated from the X-ray surface brightness grid. The $151\\rm\\, MHz$ radio frequency emission from the lobes is also calculated following the method of ; this frequency is commonly used by low-frequency Square Kilometre Pathfinder instruments, including the *Low-Frequency Array* (LOFAR) and the *Murchison Widefield Array* (MWA).\nThe X-ray and radio surface brightness distributions are modelled for lobed FR-IIs at a range of redshifts throughout their evolutionary history. Specifically, we simulate AGNs at 41 (log-spaced) source ages between 3 and 300$\\rm\\, Myrs$, redshifts of *z* = 0.1, 0.5 and 1, jet powers of *Q* = 1037, 1038 and $10^{39}\\rm\\, W$, active ages of 30 and 100$\\rm\\, Myrs$ (i.e. when the jet ceases injecting fresh electrons), and host cluster environments with dark halo masses of 1012.5, 1013.5 and $10^{14.5}\\rm\\, M\\_\\odot$. The axis ratio of the lobe is taken as *A* = 4 corresponding to an axis ratio for the shocked shell of $A\\_{\\rm s} = 2$ (see Section [sec:AXISRATIO]). The other parameters in the RAiSE dynamical and synchrotron/inverse-Compton emissivity models take the same values as used by.\nThe surface brightness maps for several informative combinations of these parameters are shown in Figure [fig:surfbright]. Powerful jets expanding into dense environments are found to create dense shells of shocked gas surrounding the lobe. The bremsstrahlung radiation from the shocked gas is much brighter than can be generated by the inverse-Compton radiation process, especially in young sources and at low redshifts (e.g. top panel of Figure [fig:surfbright]). The faint lobes bounded by well-defined lines are seen in *Chandra* X-ray images of Cygnus A and MS0735.6+7421, with cavities also seen in the hosts of weaker FR-Is such as the Perseus cluster. By contrast, in lower mass clusters, the density of swept-up gas is very low causing its temperature to be high to satisfy the shock jump conditions. The increased temperature leads to minimal bremsstrahlung radiation from the shocked gas shell. The inverse-Compton upscattered photons in the lobe are thus the dominant source of X-ray emission, especially at higher redshifts where the cosmic microwave background radiation is stronger (e.g. middle panel of Figure [fig:surfbright]). Finally, the X-ray inverse-Compton emission remains visible well after the radio frequency emission vanishes; in the simulated remnant, the radio emission has retreated towards the freshest electrons at the hotspot, whilst the X-ray emission does a much better job of tracing out the channel evacuated by the radio lobes. A similar result has previously been reported by ; Figure [fig:surfbright] additionally shows that the peak in the X-ray intensity from the lobe is much closer to the core than the full extent of the radio source. Many radio sources appear asymmetric, due to asymmetry in the environments encountered by the two lobes ; this poses a challenge for robustly identifying host galaxies of remnant radio sources. Our results suggest that X-ray observations may provide a useful alternative.\n \n \n[fig:surfbright]\nAn intriguing surface brightness map is produced when propagating weak jets into dense environments (bottom panel of Figure [fig:surfbright]); dense shells of shocked gas build up around the lobe as before, however, the weakened shocks and magnetic fields allow the contact surface near the core to become unstable to turbulent mixing of the dense shell and the lobe. This results in thermal bremsstrahlung emission extending out from the core along the transverse axis, brighter than X-ray emission from the lobe or the thinner (and thus fainter integrated along the line-of-sight), unmixed portions of the shocked shell further from the core. We note that strong X-ray emission may also be observed *along* the jet axis due to inverse-Compton emission from the jets as well as lobes (for older sources at high redshift); a combination of radio and X-ray imaging may distinguish these two scenarios.\nDominant source of X-ray emissivity\n-----------------------------------\nIn this section, we investigate the relative importance of contributions to the X-ray surface brightness from the lobe (inverse-Compton), shocked gas shell (bremsstrahlung), and the ambient medium (bremsstrahlung), for some representative sources at a range of redshifts. The surface brightness for the three regions is calculated as the maximum value along the jet axis. However, the increased bremsstrahlung radiation from sources pinched by Rayleigh-Taylor mixing may present a source of error in this work as the rate of mixing is a poorly quantified model parameter in RAiSE. Fortunately, in the vast majority of extended AGNs this enhanced emission is only a minor contributor to the integrated X-ray luminosity, and when generating surface brightness maps, either the Rayleigh-Taylor mixed region is not resolved from the bright X-ray core (discussed in Section [sec:X-ray source number densities]) or the remainder of the lobe also sits above the surface brightness sensitivity limit (i.e. the shape of the source is correctly determined). The analyses in this work are repeated with the inner third of the lobe masked to exclude this region of enhanced emission; our results are unaffected by this change.\n\n[fig:timeseries1]\n[fig:timeseries2]\nThe evolutionary tracks of the X-ray surface brightness for these representative sources are shown in Figures [fig:timeseries1] and [fig:timeseries2]. The inverse-Compton emission is dominant over the bremsstrahlung radiation from both the shocked gas shell and the ambient medium in poor clusters ($10^{12.5}\\rm\\, M\\_\\odot$; Figure [fig:timeseries1]) at all source ages $\\gtrsim 1\\rm\\, Myr$. By contrast, the shocked gas shell of sources expanding into denser host cluster environments ($\\gtrsim 10^{13.5}\\rm\\, M\\_\\odot$; Figure [fig:timeseries2]) is brighter than the lobe, albeit not appreciably, for at least 20 to $100\\rm\\, Myrs$ with the weakest FR-II jet powers ($Q = 10^{37}\\rm\\, W$); the shocked shell stays brighter than the lobe for longer with either denser environments or higher jet powers. This change in the importance of the lobe and shocked gas shell to the X-ray surface brightness typically results from rapidly falling pressure in the shocked gas as the source approaches pressure equilibrium with the ambient medium. Further, the stronger cosmic microwave background radiation at higher redshifts leads to increased inverse-Compton emission from the lobe causing this source of X-ray surface brightness to become dominant in younger sources.\nFinally, the X-ray surface brightness from the lobe and shocked gas shell remain detectable above that of the ambient medium for a considerable time after the jet switches off. The extended source in Figure [fig:timeseries1], with an active lifetime of 30$\\rm\\, Myrs$, remains detectable over the ambient medium (at the 5*σ* level) for a further 155$\\rm\\, Myrs$ at redshift *z* = 0.1, 115$\\rm\\, Myrs$ at *z* = 0.5, 100$\\rm\\, Myrs$ at *z* = 1, and 65$\\rm\\, Myrs$ at *z* = 2. By contrast, *any* level of synchrotron emission is present at radio frequencies for only 170, 70, 25 and 10$\\rm\\, Myrs$ after the jets switch off, respectively, at these redshifts. Similarly, the extended source in Figure [fig:timeseries2] emits radiation from the lobe and shocked shell at X-ray wavelengths long after the radio emission fades, though this is only a factor of a couple brighter than the ambient medium. Based on these and other test cases, we expect a population of remnant extended radio galaxies undetectable at radio frequencies but visible to X-ray telescopes, in line with literature predictions of a large population of inverse-Compton ‘ghosts’, particularly at high-redshift.\nMOCK EXTENDED AGN POPULATION\n============================\nWe now extend our analysis to make predictions for radio galaxy population statistics. Specifically, we combine the theoretical framework developed in the previous section with the halo mass function and literature radio frequency observations to create a mock population of extended radio galaxies. We then use this simulated population to predict the X-ray luminosity function for extended AGN emission, and investigate the number of objects that could be uniquely detected using large sky X-ray surveys.\nConstruction of mock catalogue\n------------------------------\nWe construct a mock catalogue of extended AGNs, including their angular sizes, surface brightnesses, and integrated flux densities, by running simulations for a dense grid of model parameters. The brightnesses are calculated at radio frequencies for a typical *Low Frequency Array* (LOFAR) survey and in the X-ray for an *eRosita* survey. Observations are used to constrain the input halo mass function for AGN hosts, and the distribution of source active lifetimes and jet kinetic powers. For the remaining parameters, we assume values given in Section [sec:Surface brightness].\n### Halo mass function\n \n[fig:luminfunctions]\n[fig:luminfractions]\nThe mass function of extended AGN hosting clusters is assumed to be the convolution of the halo mass function for all groups and clusters, and the probability of finding an AGN in a given mass host. The mass of dark matter haloes (observed as galaxy groups and clusters) can in general be described by a mass function, which gives the number density of clusters as a function of mass. In this work, and following, we take our dark matter halo masses from the low-redshift mass function of who find that a common Schechter function describes both galaxy groups and clusters. We use the semi-analytic galaxy evolution (SAGE) model of to extend their observations to higher redshifts, finding the mass of the break in the Schechter function scales with redshift as approximately (1 + *z*)− 3.\nThe radio-loud fraction of AGNs (i.e. fraction of sources with brightness above an arbitrary cut in radio luminosity) suggests hosts with higher mass black holes will have more frequent (or longer) phases of activity. made a theoretical prediction for the AGN duty cycle (i.e. the fraction of time a given source is active) as a function of black hole mass, $\\delta \\propto \\rm M\\_\\bullet^{1.5}$; this relationship is consistent with the observed radio-loud fraction for all but the most massive galaxies in which the duty cycle peaks at 100%. The black hole mass in the brightest cluster galaxy (BCG), the mass of the stellar bulge, and the dark matter halo mass, are all known to scale approximately linearly with each other. Individual galaxies show moderate scatter of $0.5\\rm\\, dex$ about the dark matter halo–black hole mass relationship, however the underlying relationship remains when considering several orders of magnitude in halo mass. We therefore convolve the halo mass function with the mass–duty cycle relation to derive the radio AGN mass function. This halo mass function is shown in the left panel of Figure [fig:luminfunctions] as a function of redshift for masses between 1012.5 and $10^{14.75}\\rm\\, M\\_\\odot$. The number density is scaled in the plot based on the number of high-luminosity radio AGNs of FR-II morphology observed by.\nThese scaling relationships predict that all plausible host halo masses occur at similar probability, with a slightly decreased likelihood of finding a radio AGN in the most massive clusters. This convolved mass function will provide a better description of the environments typically hosting AGNs than the raw halo mass function.\n### Source lifetimes\nThe radio AGN lifetime function has generated much recent interest. The large observed fractions of compact sources strongly suggests a dominant population of short-lived radio jets. Recently, used a combination of data from the LOFAR LoTSS survey and dynamical models, to show that the majority of radio AGNs are consistent with models in which the source lifetime distribution is log-uniform. applied self-consistent modelling to LOFAR observations of active, remnant and restarted radio sources in the HETDEX field to similarly infer a dominant short-lived population, consistent with feedback-regulated accretion.\nIn line with these results, we similarly adopt here a log-uniform distribution of the active lifetimes between 3 and 300Myrs, and assume that the duty cycle is (on average) independent of the active lifetime. We note that the chosen distribution for these parameters makes only minimal difference to the results in this work.\n### Jet kinetic powers\nThe distribution of the jet kinetic powers fuelling the extended AGN emission is informed by observations of the luminosity function of powerful FR-IIs in the 3CRR, 6CE and 7CRS samples. fit the high-luminosity function for these objects at $151\\rm\\, MHz$ as a function of redshift. The general shape of their distribution is shown in the right-hand subplot of Figure [fig:luminfunctions], with the number density scaled to match the observed distribution at redshifts *z* = 0.1, 0.5, 1, 2 and 4. We generate our jet power distribution by converting their radio luminosity function into kinetic powers, first by using theory driven jet power–luminosity relationships; i.e. *Q* ∝ *L**ν*6/7, noting that environment and source age introduce scatter to this relation. The jet power corresponding to the turnover in their luminosity function is found by simulating mock catalogues (sampled every $0.2\\rm\\, dex$ in jet power, and including only active sources) for a range of possible turnovers and selecting the best match to the observed radio luminosity distribution. The resulting probability distribution for the kinetic jet power is given by\n \n$$n(Q) = n\\_0 \\left( \\frac{Q}{Q\\_\\star} \\right)^{-1.17\\alpha\\_\\star} \\exp\\bigg[-\\left( \\frac{Q\\_\\star}{Q} \\right)^{1.17} \\bigg],$$ \nwhere *n*0 is a normalisation constant, the turnover in the luminosity function occurs at $Q\\_\\star = 10^{38.1}\\rm\\, W$, and the slope of the power law component of the luminosity function is *α*⋆ = 2.27. The shape of the probability distribution is independent of redshift, however the number density of extended AGNs increases towards higher redshifts. The radio luminosity functions generated from our mock catalogues (using the selected jet power distribution) are shown in the right-hand panel of Figure [fig:luminfunctions] for a range of redshifts; these are in good agreement with the observed luminosity functions.\nX-ray luminosity function\n-------------------------\nThe mock extended radio AGN catalogue, which has been calibrated to successfully describe the observed radio properties of powerful FR-IIs, can now be used to predict their brightness at X-ray wavelengths. The integrated luminosity is calculated for active sources by summing the synchrotron emissivity from the lobe, and the bremsstrahlung radiation from both the shocked gas shell and the ambient medium. The X-ray emission from the ambient medium is included in these calculations since in practice it may be hard to disentangle the emission from the AGN and environment (except in well resolved objects), however we also derive the X-ray luminosity function including only AGN related emission for comparison with previous studies. We expect the X-ray luminosity of most extended radio galaxies to be strongly correlated with the properties of the ambient medium since the brightness of the shocked gas shell is directly related to the mass of gas swept up as the lobe expands. The level of inverse-Compton emission from the lobe, meanwhile, has a more complicated relationship with the density profile of the ambient medium.\nThe accretion disk of AGNs is also often bright at X-ray wavelengths; the spectrum of the accretion disk peaks at ultraviolet wavelengths but a sizeable number of thermal photons from the disk are inverse-Compton upscattered to X-ray energies by the hot corona surrounding the disk. Typical X-ray luminosities from accretion-related nuclear emission are of order 1043 to $10^{45}\\rm\\, erg \\,s^{-1}$. Meanwhile, the highest energy shock-accelerated electrons emit synchrotron radiation at X-ray wavelengths both along the jet and at the terminal hotspot in active sources; these quickly fade in remnants due to radiative losses. The hotspots of typical FR-IIs can reach 1042 to $10^{43}\\rm\\, erg \\,s^{-1}$ at X-ray wavelengths. The brightness of the core and hotspots is not related to the inverse-Compton upscattering of CMB photons and is thus independent of redshift, however it is hypothesised that a beamed inverse-Compton mechanism may operate in the jets. We therefore choose not to include the (rather uncertain) core emission in our luminosity function whilst analyses in subsequent sections will only investigate the detection of emission spatially resolved from the core. Finally, our modelling does not consider synchrotron self-Compton radiation (i.e. inverse-Compton upscattering of synchrotron photons); in Cygnus A (*z* = 0.0561), this mechanism contributes 70-80% of non-thermal X-ray emission in the lobes, however inverse-Compton upscattered CMB photons become dominant at higher redshifts (*z* > 0.4 for a Cygnus A-like source).\nThe X-ray luminosity function derived for a typical *eRosita* survey frequency of $1\\rm\\, keV$ is shown in the central panel of Figure [fig:luminfunctions]. The brightness of the extended AGNs is comparable to, or at most a factor of a few brighter than (on average 30% brighter at *z* = 0.1), typical inactive X-ray clusters in the range $10^{41} < L\\_{\\rm X} < 10^{44}\\rm\\, erg\\, s^{-1}$. The shape of the X-ray luminosity function also resembles that of the AGN halo mass function when including all contributors of extended X-ray emission, in particular at the lowest redshifts. As we show in Figure [fig:luminfractions], this is due to the dominant source of X-ray emission being bremsstrahlung radiation from the ambient medium and shocked gas shell for *z* ≤ 1. Objects with the highest luminosities are typically the highest jet power sources, observed at ages of a few tens of Myrs when inverse-Compton radiation from the lobe begins to increase significantly, contributing in excess of 90% of the total integrated X-ray luminosity from the AGN–host cluster system (see Figure [fig:luminfractions]). By contrast, the X-ray luminosity function at the highest redshifts (*z* > 1) is dominated by inverse-Compton emission for the majority of luminosities (and thus a large fraction of the extended AGN population). The stronger cosmic microwave background radiation at higher redshifts thus boosts the integrated luminosity well above that generated by the hot cluster gas (on average 155% brighter at *z* = 4), leading to an increased population of X-ray bright extended AGNs.\nX-ray source number densities\n-----------------------------\n[fig:numberdensities]\n[fig:remnantdensities]\nWe now use the mock catalogue to calculate the predicted number density of extended AGNs that would be detected using *LOFAR* and *eRosita*. The *LOFAR Two-metre Sky Survey* (LoTSS) has a median sensitivity of $71\\rm\\, \\mu Jy\\, beam^{-1}$ at an observing frequency of 120-168MHz, with approximately 6arcsec full-width half maximum (FWHM) across the synthesised beam. The expected $1\\rm\\, keV$ sensitivity of all-sky surveys using *eRosita* is $14\\rm\\, nJy\\, beam^{-1}$ for their 15arcsec full-width half maximum beam. We smooth the RAiSE images with a circular gaussian filter, mimicking the approximate shape of the *LOFAR* and *eRosita* beams. Extended emission separated from the core by less than the angular resolution of the survey cannot be distinguished from a bright compact core; these pixels are flagged and removed from our mock images. Simulated extended AGNs with emission in at least one beam (i.e. gaussian filtered pixel) exceeding the surface brightness sensitivity limit are assumed to be detectable by *LOFAR* or *eRosita* as appropriate. These sources are by definition resolved, however we make a further classification that the extended AGNs are well resolved (i.e. we can make out the shape of the source) if they have at least five beams along their length.\nThe number density of X-ray and radio detected active-phase, extended AGNs of FR-II morphology is shown in Figure [fig:numberdensities] for a range of source ages and jet kinetic powers. The overwhelming majority of low-redshift sources have extended emission resolvable from their cores in either X-ray and radio frequencies (blue or green shading); approximately half have sufficient resolution to determine the shape of their extended structures (green shading). Specifically, close to all of the active-phase FR-IIs in our catalogue are detectable using radio frequency observations (at *z* ≤ 1), whilst 41% can be seen at X-ray wavelengths at *z* = 0.1 and 2.6% at *z* = 0.5. At both these redshifts *eRosita* can uniquely detect (i.e. no radio detection) only 0.2% of active sources. This pattern continues out to higher redshifts where most active sources are detectable at radio frequencies whilst a rapidly decreasing fraction can be seen at X-ray wavelengths. Importantly, at *z* = 1, only young and relatively compact objects are detectable using X-rays in the active phase; the source shape cannot be determined in any objects. The inverse-Compton emission arising from the overpressured lobes of these extended AGNs is related to the density of the ambient medium and these fade rapidly as they expand into lower-density environments such as the intergalactic medium in low-mass haloes. Moreover, extended AGN emission is not detectable at redshifts *z* ≥ 2 for the *eRosita* surface brightness sensitivity limit investigated (but see Section [sec:Remnant density at high-redshifts]). The number density of extended AGNs with FR-II morphology detectable with *LOFAR* or *eRosita* are summarised in Table [tab:detections] as a function of redshift. We scale the number densities in our catalogue based on the observed high-luminosity function of.\n[tab:detections]\nccccccc && \n&&*eRosita*&*LOFAR*&dual detection&*eRosita* only&*LOFAR* only \n&0.1&-9.17&-8.78&-9.17&-11.40&-9.00 \n&0.5&-9.46&-7.88&-9.49&-10.61&-7.89 \n&1&-11.62&-7.07&-11.63&-13.12&-7.07 \n&0.1&-9.38&-9.07&-9.56&-9.86&-9.24 \n&0.5&-9.59&-8.13&-9.77&-10.07&-8.14 \n&1&-11.80&-7.54&-11.95&-12.32&-7.54 \n&0.1&-4.06&–&–&-4.06&– \n&0.5&-10.57&–&–&-10.57&– \n&1&-29.70&–&–&-29.70&– \nThe number density of remnant FR-IIs detectable with X-ray and radio frequencies is similarly shown in Figure [fig:remnantdensities]. The maximum age of remnants detectable at radio frequencies reduces quickly with increasing redshift (by more than a factor of two over the range *z* = 0.1-1) due to the increasing inverse-Compton losses at high redshift. Crucially, the inverse-Compton emission remains detectable at X-ray wavelengths long after the synchrotron radiation at radio frequencies ceases. This leads to a large population of remnant extended AGNs which are detectable with *eRosita* but cannot be seen at radio frequencies; these sources are typically aged between $100\\rm\\, Myrs$ and $1\\rm\\, Gyr$ (at the lowest redshifts). Specifically, *eRosita* uniquely detects 14% of remnants at *z* = 0.1 (based on non-core emission), decreasing to 1.1% and 0.002% at redshifts *z* = 0.5 and 1 respectively (i.e. a much greater fraction of remnants can only be detected using X-ray surveys than for active sources). We note in Section [sec:Remnant density at high-redshifts] that these number densities increase markedly for the higher redshifts with just a modest improvement in sensitivity. These exclusively X-ray detected remnants have more than five beams across the length of the lobe in 98% of sources at redshift *z* = 0.1; this reduces to 53% at *z* = 0.5 whilst no high-redshift (*z* ≥ 1) remnants are observed with multiple beams across the source. The remnant source number densities are summarised in Table [tab:detections].\n### False positive detections\nMock extended AGNs in our catalogue are only detected if emission from either the lobe or shocked gas shell is present above the survey surface brightness sensitivity limit. However, bremsstrahlung radiation from the brightest X-ray clusters may also be detectable using *eRosita* (in a single beam) without any density enhancement from AGN activity. At redshift, *z* = 0.1, simulated X-ray clusters more massive than $10^{13.43}\\rm\\, M\\_\\odot$ (derived analytically for RAiSE clusters[3](#fn3)) emit sufficiently high levels of bremsstrahlung radiation for their ambient medium to be detected and resolved from an X-ray nucleus. The critical cluster mass increases to $10^{14.94}\\rm\\, M\\_\\odot$ at *z* = 0.5 whilst no realistic mass haloes are predicted to have their ambient medium detected (in a single beam) at redshifts *z* ≥ 1; a small fraction of observed X-ray clusters may of course be atypically gas rich compared to our mock clusters. The number density of ‘false positive’ detections expected due to the misclassification of cluster gas is derived by integrating the cluster mass function above the critical mass at each redshift (see Table [tab:detections]).\nThe number density of X-ray clusters whose hot gas is detectable using *eRosita* far exceeds the expected density of extended AGNs at low-redshift (*z* ∼ 0.1). The presence of non-core emission at X-ray wavelengths therefore cannot be used to suggest AGN activity at these redshifts; however, 98% of exclusively X-ray detected remnants are well-resolved (i.e. have more than five beams across their lobes) and so can potentially be classified based on the shape of their X-ray brightness distribution. The same behaviour is seen at moderate redshifts though ‘false positives’ and actual AGN detections are expected to occur with equal likelihood. By contrast, at high-redshifts (*z* ≥ 1) the number density of X-ray clusters detectable with *eRosita* for a typical survey sensitivity is over seventeen orders of magnitude lower than the expected detection rate for extended AGN emission. Any non-core X-ray emission detected at high-redshifts can therefore be directly attributed to extended AGNs, and not the ambient medium.\n### Remnant density at high-redshifts\nThe number density of extended AGNs of FR-II morphology detectable at high-redshifts using an *eRosita*-like survey is investigated for a range of improved surface brightness sensitivities. In the previous section, we found none of the mock extended AGNs at redshift *z* = 2 and 4 could be detected using the assumed $14\\rm\\, nJy\\, beam^{-1}$ sensitivity; this is reduced by up to a factor of 100 down to $0.14\\rm\\, nJy\\, beam^{-1}$ for the same resolution ($15\\rm\\, arcsec$ FWHM). The expected number density of remnants detectable (and mostly uniquely detectable) by this theoretical increased sensitivity *eRosita* survey is shown in Figure [fig:remnantdensity] for redshifts *z* = 1, 2 and 4 as a function of the surface brightness sensitivity. Remnants can begin to be detected at *z* = 2 for a modest factor of 1.5 increase in sensitivity and at redshift *z* = 4 for an order of magnitude improvement. The rate of false positive detections from the ambient medium is also included in Figure [fig:remnantdensity]; the likelihood of detecting AGN emission and false positives converges at *z* = 1 for a factor of 20 increase in sensitivity, but at higher redshifts the false positive rate remains small for at least a three orders of magnitude improvement in surface brightness sensitivity.\nThe number density of remnants detected at *z* = 1 meanwhile can increase by over four orders of magnitude (before converging to the false positive rate) reaching a level comparable to that found by using radio frequencies; *eRosita* is therefore highly complementary to *LOFAR*. The number density of radio-detected remnants does not increase with improved sensitivity as any objects emitting appreciable synchrotron emission are already detectable. Importantly, the long lasting inverse-Compton emission probes a different population of remnants that has minimal overlap with the radio-selected sample (Table [tab:detections]). The number density of X-ray detected AGNs similarly coverges to the false positive rate with a comparable or greater number of detections to that obtained using radio frequencies; i.e. factor of five and 1200 more remnants at *z* = 2 and 4 respectively. The detection of non-core X-ray emission at high-redshift therefore presents a viable technique to identify previous episodes of AGN activity with X-ray telescopes presently in service; this would be greatly enhanced with only a modest increase in sensitivity. Our work predicts that X-ray wavelengths are capable of detecting at least a factor of ten more remnants than at radio frequencies for redshifts *z* > 2.2, increasing to a factor of 100 for redshifts *z* > 3.1.\n[fig:remnantdensity]\nSimilar results are obtained for extended AGNs in the active phase, however these are more readily observed at radio frequencies and thus are not considered likely targets for such surveys.\nCONCLUSIONS\n===========\nWe have extended the successful *Radio AGN in Semi-analytic Environments* lobe expansion and evolution model to X-ray wavelengths. Specifically, this improved model considers: (1) inverse-Compton upscattering of cosmic microwave background radiation by the synchrotron-emitting electrons in the lobe; (2) the dynamics of the shocked gas shell and the associated bremsstrahlung emission from this dense gas; and (3) emission from the ambient medium surrounding the extended AGN. We construct X-ray surface brightness maps of mock extended AGNs with type-II morphology to understand the relative importance of these radiative mechanisms; in particular, to determine what fraction of the population may be detectable (and correctly recognised as extended AGNs) at X-ray wavelengths.\nX-ray and radio-frequency surface brightness maps are derived for the technical characteristics of the *extended Roentgen Survey with an Imaging Telescope Array* (eRosita) and the *Low-Frequency Array* (LOFAR) instruments respectively. We consider the temporal evolution of the surface brightness in the lobe (inverse-Compton or synchrotron), shocked gas shell (bremsstrahlung) and ambient medium (bremsstrahlung) along the jet axis for typical sources located at increasing redshift from *z* = 0.1 to 2. The inverse-Compton emission from the lobe is dominant over the bremsstrahlung radiation from both the shocked gas shell and ambient medium in poor clusters ($10^{12.5}\\rm\\, M\\_\\odot$) at all sources ages. By contrast, the shocked gas shell is initially brighter than the lobe (for between 20 and $100\\rm\\, Myrs$) for sources expanding into denser environments ($10^{13.5}\\rm\\, M\\_\\odot$); the shocked gas shell stays brighter than the lobe for longer with either denser environments or higher jet powers. The X-ray surface brightness from the lobe and shocked gas shell remain detectable above the ambient medium (at the 5*σ* level) for another 65-$115\\rm\\, Myrs$ after the jet switches off after $30\\rm\\, Myrs$ at redshifts *z* ≥ 0.5. By contrast, synchrotron emission ceases after only 10-$70\\rm\\, Myrs$ at radio frequencies. We find that although both synchrotron and inverse-Compton emission fade more rapidly at higher redshift, synchrotron radiation is much more sharply curtailed, leading to a sizable high-redshift population of remnant radio galaxies emitting exclusively through the inverse-Compton mechanism.\nWe constructed an integrated X-ray luminosity function for extended AGNs by generating a mock population of FR-IIs with jet powers, active lifetimes and host cluster environments based on observational constraints. The integrated X-ray luminosity of most extended AGNs at low redshifts (*z* ≤ 1) is found to be strongly correlated with the properties of the ambient medium. In other words, the bremsstrahlung radiation from either the ambient medium or the shocked gas shell (whose density is strongly correlated with the host environment) is the dominant source of X-ray emission. At these low redshifts, only a small population of AGNs with the highest jet powers and moderate ages of a few tens of Myrs can have inverse-Compton emission contribute in excess of 90% of the total integrated X-ray luminosity from the AGN–host cluster system. By contrast, the X-ray luminosity function at higher redshifts (*z* > 1) is dominated by inverse-Compton emission for the vast majority of the extended AGN population. The stronger microwave background radiation at these high redshifts boosts the lobe contribution to the integrated luminosity well above that generated by the hot cluster gas, leading to an increased population of X-ray bright extended AGNs.\nWe used our mock extended AGN catalogue to explore how many new objects could be detected using both existing and increased sensitivity X-ray observations. We find that most active FR-II sources at redshifts *z* ≤ 1 can be detected at radio frequencies for the sensitivity of the *LOFAR Two-metre Sky Survey* (LoTSS). However, only a small fraction of active or remnant sources can be seen at X-ray wavelengths for typical *eRosita* sensitivity, when excluding core emission. No active extended AGNs are detectable at X-ray wavelengths but not at radio frequencies. By contrast, *eRosita* will find 14% of remnants at *z* = 0.1 which are not visible to *LOFAR*, decreasing to 1.1% and 0.002% at redshifts *z* = 0.5 and 1 respectively. Meanwhile, the surface brightness of the bremsstrahlung radiation from the ambient medium of any realistic mass haloes is expected to become undetectable beyond redshift *z* ≥ 1; i.e. any non-core X-ray detection can be attributed to extended AGN activity. We consider the effectiveness of radio-frequency and X-ray surveys at detecting remnants at these high-redshifts for greatly enhanced surface brightness sensitivity. The number density of X-ray detected remnants at redshift *z* = 1 becomes comparable to the number of radio-frequency detections. 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A. 2001, MNRAS, 322, 536\nYates, P. M., Shabala, S. S., & Krause, M. G. H. 2018, MNRAS, 480, 5286\n---\n1. Email: turner.rj@icloud.com[↩](#fnref1)\n2. RAiSE implicitly assumes a single magnetic field value throughout the lobes; although more complex configurations almost certainly exist these are not well constrained at present, and our approach is standard for lobe modelling.[↩](#fnref2)\n3. RAiSE simulates cluster density profiles based on the observed profiles of, taking the scaling gas mass and virial radius from the semi-analytic galaxy evolution model of. The gas temperature is calculated using an observed relationship with the halo mass.[↩](#fnref3)\n"},"data_id":{"kind":"string","value":"arxiv_0000355"}}},{"rowIdx":93,"cells":{"content":{"kind":"string","value":" velocity during the period Jan.31-Feb. 6 determined from all measurable absorption lines was -1810 ± 40 km s− 1.\nOutflow velocities of the order of 1500 - 2200 kms− 1 were also observed in previous outbursts. Adams and Joy (1920) reported the presence of ``dark\" components of radial velocity up to -2100 km s− 1 a few days after maximum. Joy (1945) observed expansion velocities of 1700 kms− 1 in forbidden emission lines in spectra taken some months after outburst. Similar values were also found by Herbig (1945) and reported by Payne-Gaposchkin (1957).\nThe ``principal absorption system“ is generally associated with the bulk of the mass ejected during outburst, and for most novae has a velocity close to that of the ``nebular system” as determined from the width of the nebular emission lines (Payne-Gaposchkin 1957, Mc Laughlin 1956, Pottasch 1959). The v*e**x**p* value reported by Joy (1945) referred to observations during the nebular phase, three-four months after outburst. That the emission lines in the nebular spectrum showed velocities of the same order as those deduced from the absorption lines of the principal spectrum is an indication of near constant expansion.\nWe note that in the literature on T Pyx little attention has been paid to the fact that for almost three months after the initial halt T Pyx showed a strong continuum with the presence of emission and absorption lines of similar strength. The persistence for almost three months (about t3) of displaced absorption components in the H (and FeII) lines, which are observable by combining the spectroscopic observations of Catchpole (1969) and Chincarini and Rosino (1969) for the 1966 outburst, indicates an optically thick phase of similar duration.\nIn this respect, we recall that, before the outburst of 1966-1967, Mc Laughlin (1965) noted that T Pyx was exceptional among RNe since its photometric and spectroscopic behavior closely resembled that of a typical nova both close to maximum (where it remained for several weeks to within about a magnitude) and in the nebular stage.\nThe mass of the shell ejected in the optically thick phase\n----------------------------------------------------------\nThe similarity between the spectroscopic and photometric characteristics of the outbursts of T Pyx and those of CNe, which allegedly eject about 10− 4 - 10− 5 M⊙, suggests in itself that during outburst T Pyx expelled a shell of comparable mass.\nClassical novae undergo an optically thick phase during which they resemble each other, a fact that can be explained by the same mechanism (i.e. flux redistribution) producing the spectrophotometric light curve (Shore 1998, 2008). To achieve flux redistribution, the material must reach column densities of the order of 1023-1024 cm− 2, which corresponds to masses of about 10− 4 - 10− 5 M⊙.\nAn optically thick stage also characterized the outbursts of T Pyx, as can be directly inferred from the lengthy period of time during which the optical magnitude was close to its maximum value, with t3 ∼ 90*d*, and from the presence of absorption lines of HI and FeII, which lasted for at least 80 days.\nFrom the duration of the optically thick phase (associated to t2) and the observed v*e**x**p*, using simple assumptions, we can estimate the mass of the shell ejected during outburst. The outer radius of the shell can be estimated from the observed expansion velocity (v*e**x**p* ∼ 1500 km s− 1) and the time elapsed from outburst, assuming continuous ejection. This assumption is justified by the persistence of displaced absorption components with similar equivalent widths. The shell radius is:\n \n*R**e**j* ∼ 7.7 × 1014*c**m* ∼ 1.1 × 104*R*⊙. \nThe corresponding shell volume is V ∼ 1.84 × 1045 cm3. These values are conservative because the terminal velocity was probably higher. To reproduce the optically thick stage recognized from the presence and persistence of the absorption lines, the column density must be of the order of 1023 cm− 2. Therefore, the average density in the shell must be close to 108 - 109 cm− 3, (we note that the density should scale as R− 3). This density value agrees with the spectroscopic behavior and the presence of permitted emission lines only. If we assume that the ejecta are homogeneous and consist of ionized hydrogen, the mass might be estimated as\n \n*M**H* = *N**e**m**H**V* ∼ 3.1 × 1029*g* = 1.5 × 10− 4*M*⊙. \nThis value is probably an upper limit because of the assumptions of continuous ejection and homogeneous shell.\nIn an alternative approach, following Williams (1994), one can estimate the hydrogen column density produced by an expanding shell of mass 1.0 × 10− 4 M⊙ :\n \n*N**H* ⋅ *R* = 3.0 × 1052 ⋅ *R*− 2 [*c**m*− 2]. \nFor T Pyx at day 60 (= t2), we find that R2= 5.8 × 1029 [cm2], and N*H* ⋅ R ∼ 5.2 × 1022 [cm− 2]. A mass of the ejecta higher than 10− 4 is therefore required to produce an optically thick stage until day 60.\nIt is well established that in novae, the mass of the ejected envelope is directly correlated with the optical decay time t2 or t3 (Livio, 1994). Therefore, an independent estimate of the ejected shell mass can be derived from the relation:\n \n*l**o**g**M**e**j* = 0.274 ± 0.197 ⋅ *l**o**g**t*2 − 4.355 ± 0.283 \n(Della Valle et al. 2002). Even after considering the large uncertainties in this relationship, a t2=62*d* implies a mass for the envelope M*e**j* ∼ 10− 4 M⊙, which is similar to the mass ejected by classical novae.\nThe data leading to Eq. [eq:dellavalle] suffer from a large scatter. We suspect that the ejecta expansion velocity plays also an important role and should be included in the relation. Shore (2002, 2008) suggested an approximate scaling relation for the optically thick stage:\n \n$$M\\_{ej} \\sim 6.0\\ \\times 10^{-7} \\ \\epsilon \\ N\\_{H,24} \\ V\\_3^2 \\ t\\_3^2 \\\nM\\_{\\odot},$$ \nwhere V3 is the outflow velocity in 103 km s− 1, and *ε* is the filling factor that, for this stage, can be assumed to be of the order of 0.1. For t3=90, f=0.1, N*H*, 24=0.1-1.0, and V3=1.5, the derived values for M*e**j* are in the range 1.5 × 10− 4 - 1.5 × 10− 3 M⊙.\nFinally, the ejected mass can be estimated using the following scaling law from Cassatella et al. (2005), which depends on the approximate assumption that the filling factor in novae ejecta is the same as in V1668 Cyg:\n \n*M**e**j* ∼ 0.044 ⋅ *M*⊙/*v**e**x**p*, \nwhere *v**e**x**p* is in km s− 1 and the constant is set to the values of the ejected mass and the expansion velocity of V1668 Cyg (Stickland et al. 1981). Using *v**e**x**p*=1500 km s− 1, one finds that for T Pyx, *M**e**j* ∼ 2.93 × 10− 5 *M*⊙.\nWe recall that Kato and Hachisu (1991), from their models of steady-state winds for a nova with M1=1.33 X=0.5 and Z=0.02 and the observed t3, suggested the ejection of a massive envelope with M*e**j* of about 10− 5 M⊙ in a single outburst.\nAll the previous results agree with the considerations of Shore (1998), who pointed out that, for a typical ejection velocity of about -2000 kms− 1, a nova with an optical decline time longer than a week must eject a mass higher than 10− 5 M⊙.\nWe also recall the fact that in classical novae close to maximum, the Balmer lines develop P-Cygni profiles, which is a clear indication that a significant amount of material was ejected during the outburst (Starrfield, 1993).\nTherefore, all quantitative methods and the qualitative consideration of the photometric and spectral behavior of T Pyx during the outbursts indicate the presence of a massive envelope with M*e**j* ∼ 10− 4 -10− 5 M⊙.\nThe discrepancy between the mass of the thick shell and the ignition mass\n=========================================================================\nThe ejection of a massive shell during the early (optically thick) outburst phase contrast significantly with the results of the UV + optical observations during quiescence and the theoretical requirements for M*i**g**n*, which imply a $\\dot{M}\\_{pre-OB}$ ∼ 2.2 × 10− 8 and a total mass for the accreted shell M*a**c**c**r* of about ∼ 5.0 × 10− 7 M⊙ (see Sect. 8). It is also in contrast with the conclusions of Sect. 9 that indicated that the closest agreement between the grid models and the observed properties of the system during outburst and Q corresponds to a model with $\\dot{M}$ ∼ 3.0 × 10− 8 M⊙ yr− 1 and M*i**g**n* ≤ 1.3 × 10− 6 M⊙ (Table 6). During outburst, T Pyx has apparently ejected far more material than it has accreted.\nStudies of classical novae containing a massive WD indicated that these objects eject apparently more material than theoretically predicted (Starrfield et al. 1998a; Starrfield et al. 1998b; Vanlandingham et al. 1996; Shore, 1998). These authors emphasized the significant discrepancy between the observed mass of the ejecta and the predicted critical mass of accreted nova envelopes for massive WDs (M*W**D* ≥ 1.25 M⊙), the mass of the observed shell being one order of magnitude (or more) higher than that predicted by the models.\nFor these CNe one could attribute the discrepancy to some inadequacy in the TNR models or in the methods to determine the nebular mass. In the case of T Pyx, the situation, however, differs because there is a serious mismatch between the shell mass indicated by the optical *observations* during outburst (M*e**j* ∼ 10− 4 10− 5 M⊙) and that determined by the UV and optical *observations* during quiescence (which give $\\dot{M}\\_{pre-OB}$ ∼ 2.2 × 10− 8 M⊙ yr− 1 and therefore M*a**c**c**r* ∼ M*i**g**n* = 5.0 × 10− 7 M⊙). Therefore, the mass ejected during outburst is about a factor of 100 higher than both the theoretical ignition mass M*i**g**n* and the total mass accreted before outburst, M*a**c**c**r*.\nWe note that using the post 1966 value of $\\dot{M}$ instead of that inferred for the pre-outburst interval would produce an even larger discrepancy.\nThe role of the distance\n------------------------\nAlthough M*e**j* does not depend on distance, M*a**c**c**r* does (due to its dependence on both L*d**i**s**k* and $\\dot{M}\\_{pre-OB}$), and therefore the mismatch between M*e**j* and M*a**c**c**r* depends crucially on the assumed distance. The uncertainty in the adopted distance was found in Sect. 2 to be of the order of 10 percent. However, we note, that even the adoption of an unlikely distance of, say, 10,000 pc (at about 20 *σ* from the estimated value) would only partially alleviate this inconsistency, and at the expense of an uncomfortably high mass accretion rate, well within the range for the onset of steady burning. This would imply characteristic temperatures and luminosities that are not observed (see also Sect. 14 for a further discussion). A larger distance would also necessarily imply that T Pyx was super-Eddington at maximum, a circumstance that appears unlikely due to its slow photometric and spectroscopic developments during outburst.\nA lower distance would correspond to a lower values of L*d**i**s**k*, and hence $\\dot{M}\\_{pre-OB}$ and M*a**c**c**r*, to values that are theoretically incompatible with the occurrence of outbursts with an average interval of 22 years. It would also exacerbate the discrepancy between the low value of M*a**c**c**r* obtained from the UV observations (and the models) and the apparently high mass of the ejecta, as inferred from the behavior during outburst. Therefore, there is not much leeway to invoke a different distance to explain the discordance.\n The nebula revisited\n=====================\nThe nebula surrounding T Pyx has been the target of several spectroscopic and imaging observations. Duerbeck und Seitter (1979) first reported the presence of a strong nebulosity around T Pyx, with radius r ∼ 5\", whose origin was tentatively attributed to the 1966 outburst and whose strength was described as unusual. By the assumption of an outburst expansion velocity of -900 kms− 1, a low distance ( ∼ 600 pcs) was derived.\nWilliams (1982) obtained spectral scans of the northern portion of the nebula of T Pyx. The spectrum was similar to that of as a typical PN, probably photoionized by radiation from the hot remnant, and lacked the strong CNO enhancements characteristic of the ejecta of classical novae.\nComparing images acquired in 1979.0 and 1982.9, Seitter (1987) found that the nebula did not increase in size during that time interval.\nShara et al. (1989) from deep narrowband CCD images confirmed the faint, extended H*α* + NII halo (twice as large as the inner nebula), first reported by Duerbeck (1987). A smooth, small [OIII] nebula with r ∼ 2“ was also found. Shara et al. (1989) also compared the relative sizes of the main nebula with r ∼ 5” in 1985 and in 1979 but failed as well to find any detectable expansion during the 6 year interval, confirming the finding of Seitter (1987). High resolution imagery data from HST was obtained by Shara et al. (1997). The nebula was resolved into more than two thousand individual knots, and a comparison between images taken at four epochs indicated that these individual knots retained a similar pattern, without any evidence of expansion. These data confirm the apparent stationarity in the 10\" diameter nebula suggested by previous observations. Shara et al. (1997) found an upper limit of 40 ⋅ (1500/d) (km s− 1) for the expansion velocity of the knots. They also detected nine distinct peaks in the brightness distribution, an indication that a multiple nebula model was required.\nMany studies, disappointingly inconclusive, addressed the problem of the mass of the nebula. In this respect, it should be noted that large uncertainties are generally associated with estimates of the mass of novae ejecta, since the mass estimate depends critically on quantities that are not reliably measured, e.g.: distance, electron density, ionization structure in the nebula, geometry, and filling factor. Therefore, it is unsurprising that a range of masses for the nebula of T Pyx has been proposed in the literature.\nFrom the nebular H*β* intensity measured by Williams (1982), WLTO (1987) obtained a lower limit to M*n**e**b* of 10− 6 M⊙, while Shara (1989), using the H*β* intensity with the requirement *ε* ≤ 1 for the filling factor derived an upper limit of 1.0 × 10− 4 M⊙. From the intensity of the H*α* and [NII] lines, Seitter (1987) found a mass close to 8.0 × 10− 5 M⊙. From the H*α* flux and considerations based on HST imagery, Shara et al. (1997) obtained 1.3 × 10− 6 M⊙ to be the most reliable estimate for the nebula mass (with an assumed distance of 1500 pc), the electron density being, allegedly, the main uncertainty factor.\nWe add one more estimate for the mass of the nebula, based on the H*β* flux obtained by Williams (1982) scaled to the entire nebula, after correction for the new reddening and distance. We obtain :\nF*β* ∼ 4.24 × 10− 14 erg cm− 2 s− 1 and L*β* = 6.22 × 1031 erg s− 1.\nwith a relative error of about 25 percent, arising from the uncertainties in the distance and the flux. By combining the two common relations:\n \n*L**β* = 1.24 × 10− 25 *N**e* ⋅ *N*+ ⋅ (4/3) *π* *R**n**e**b*3 ⋅ *ε* \nand\n \n*M**n**e**b* = *μ* ⋅ *N*+ ⋅ *m**H* ⋅ (4/3) *π* *R**n**e**b*3 ⋅ *ε*, \n(where *μ*=1.4 is the mean atomic weight and *ε* is the filling factor), one finds that the mass of the nebula (independent of *ε* and *R**n**e**b*) is\n \n$$M\\_{neb}= 18.67\\; {L\\_{\\beta} \\over N\\_e} \\; g = {0.584 \\over N\\_e}\n\\;M\\_{\\odot}.$$ \nN*e* is inaccurately determined because, as noted by Williams (1982), the spectrum is too poor in number of lines to constrain the physical parameters of the nebula accurately. However, from the presence of a quite strong emission feature near 3722 Å, attributable to [OII] 3726.1 and 3728.8 one can estimate that the nebula density is less than 3.0 × 103 cm− 3, since these two forbidden lines have critical density log N*e**c**r**i**t* = 3.5 and 2.8, respectively. If we assume that N*e* ∼ 103 cm− 3, a value also adopted by Shara et al. (1997) from other considerations, we find that *M**n**e**b* ∼ 5.84 × 10− 4 M⊙.\nA value of this order for the total mass of the nebula, which has apparently increased from the contribution of several successive outbursts, agrees with the ejection during outburst of a massive shell ( ∼ 10− 5 M⊙) as suggested in Sect. 10.2.\nThe fact that the nebula of T Pyx is still clearly observed despite its large distance seems hardly compatible with a mass of 10− 6 M⊙ derived in previous studies. In the most well studied CNe, which on average are closer in distance than T Pyx, the ejected nebula has, at best, a similar strength and/or is barely evident a few years after outburst. We think that the peculiar strength of the nebula of T Pyx may be explained by the fact that most of the gas ejected in successive outbursts has accumulated into a nearly stationary envelope that is strongly irradiated by a more than average luminous UV central source.\nAlternatively, one can guess that the observed nebula of T Pyx was produced in a peculiar event and is not associated with the recorded and/or previous outbursts. Its apparent stationarity and the lack of changes over a timescale of about fifteen years (Shara et al., 1997) supports this interpretation. Williams (1982) noted that the spectrum of the nebula around T Pyx is almost similar to typical planetary nebula, of approximately solar composition.\nWe emphasize that the optically thick shell discussed in Sects. 10.1 and 10.2 was observed only spectroscopically during the outburst phases, and that there is no definite, direct link with the extended nebula. Simple calculations indicate that, if d=3500 pc, the angular radius of the shell, after ten years of constant expansion at 1500 km s− 1, would be less than 1\".\n The filling factor\n-------------------\nIf the radius of the nebula *R**n**e**b* is known, from the observed angular radius r ( ≤ 10\") and the distance, one can obtain the filling factor *ε* with the help of eq. [eq:ellebeta]. After insertion of the values for L*β* = 6.22 × 1031 erg s− 1, R*n**e**b* ∼ 5.2 × 1017 cm, and N*e* ∼ 1.1 N+, one obtains that\n*ε* ⋅ N*e*2 ∼ 1062.7.\nTherefore, for N*e* ∼ 103 cm− 3, *ε* is close to 10− 3. A filling factor of ∼ 10− 3 can be obtained independently from the relation of Harrison and Stringfellow (1994):\n \n*ε* ∼ *N**c**l**V**c**l*/*V**n**e**b* = 8*N**c**l*− 1/2*γ*3/2, \nwhere *γ* is the fraction of the spherical nebula intercepted by the clumps. From the figures of Shara et al. (1997), we estimate that *γ* ∼ 0.05 and, if the number of clumps N*c**l* is ∼ 2000 (Shara et al. 1997), we find that *ε* ∼ 2 × 10− 3. We note that the filling factors estimated for the ejecta of other novae have a wide range of values, from 4 × 10− 1 (Vanlandingham et al., 1999 for Nova LMC 1990 no.1), to 10− 6 (Saizar and Ferland (1994) for nova QU Vul. Williams et al. (1981) found an intermediate value, *ε* ∼ 2 × 10− 3 for the recurrent nova U Sco. Mason et al. (2005) found *ε* ∼ 10− 4-10− 1 for nova SMC 2001 and nova LMC 2002, and Balman and Oegelman (1999) estimated *ε* ∼ 5 × 10− 3-3 × 10− 1 for the shell of GK Per. Mason et al. (2005) have suggested *ε* ∼ 10− 5-10− 2 for T Pyx.\nThe mass balance and the SNIa connection\n========================================\nIt is accepted that Type Ia supernovae represent the complete thermonuclear disruption of mass-accreting white dwarfs that reach the Chandrasekhar limit by accretion (Nomoto et al. 1984; Woosley and Weaver 1986). Within this general framework, there exist single degenerate models (Whelan and Iben, 1973) in which a WD accretes from a non-degenerate companion, and double degenerate models (e.g. Iben and Tutukov, 1984) that involve the merger between two WDs. Hachisu (2002, 2003) and Hachisu and Kato (2001, 2002) proposed a unified picture of binary evolution to SNe Ia in which recurrent novae could be understood to be part of the evolutionary stages of supersoft X-ray sources and symbiotic channels to SNe Ia.\nThe mass M1 must be close to the Chandrasekhar limit before a recurrent nova can become a SN Ia, and the WD must also increase in the long term, after many cycles of accretion and ejection (Starrfield et al., 1985). However, in the case of T Pyx, even if M1 were close to the limit, which is not clearly established since M1 appears to be close to 1.37 M⊙, the results of the previous sections indicate that the mass balance situation is unclear:\nOn the one hand, the photometric and spectroscopic behavior close to outburst, as mentioned in Sect. 10.2, appear to be consistent with the ejection of a rather massive shell, while the UV data and theoretical models (at the specific M1 and $\\dot{M}$ values) indicate that the ignition mass is low ( ∼ 5.0 × 10− 7 *M*⊙). This indicates that during outburst the WD ejects more material than it accumulates and that a secular decrease in the mass of the white dwarf is expected. Therefore, evolution to become a SNIa appears to be excluded.\nOn the other hand, the ejection of a more-massive-than-accreted shell is apparently in contrast with the observational evidence that the chemical composition of the T Pyx nebula is close to solar (Williams, 1982). This appears to exclude any erosion of the white dwarf and implies that the white dwarf does not lose mass after cycles of accretion and ejection. However, this presupposes that the chemical composition of the observed nebula is representative of that of a single shell ejected during outburst.\nIt is unclear whether these substantial discrepancies originates in flaws in the theoretical assumptions or the interpretation of the observations. They certainly highlight the need for accurate values of the most critical parameters of this recurrent nova, i.e. the mass and chemical composition of the shell ejected during outburst.\nIn any case, the behavior of T Pyx raises several doubts about the common assumption that in recurrent novae far less material is ejected during the outburst than is accreted by the white dwarf, and therefore that its mass increases toward the Chandrasekhar limit (Starrfield, 2002). In assessing whether recurrent novae could be progenitors of Type Ia supernovae, we note that Hachisu and Kato (2001), discovered that, among six recurrent novae, only T Pyx is offset significantly from the region occupied in an a orbital period - donor mass plane (their Fig. 3) by the progenitors of SNe Ia.\nNeither a supersoft X-ray source, nor assisted suicide\n======================================================\nTo explain the alleged extremely blue color of T Pyx in quiescence, WLTO (1987) proposed that nuclear burning continues even during its Q state, consistent with the slow outburst development, which suggests that the accreted envelope was only weakly degenerate at the onset of TNR. Patterson et al. (1998) attributed the luminosity of T Pyx (and V Sagittae) to quasi-steady thermonuclear burning and suggested the object to be included in the class of the supersoft X-ray sources.\nHowever, the color of T Pyx (B-V)*o* = -0.26 used in these studies is based on a significant overcorrection for the reddening, assumed to be E*B* − *V*=0.36, instead of the correct value E*B* − *V*=0.25 (see Paper I and earlier communications, e.g. Gilmozzi et al., 1998). It is unfortunate that both the value (B-V)*o*=-0.26 of Patterson et al. (1998) and the statement about the ``extremely blue color\" of T Pyx was adopted widely in the literature (see for example Anupama 2002, and Parthasaraty et al. 2007). We note, incidentally, that in the same paper Patterson et al. (1998) assumed too high a reddening correction (E*B* − *V*=0.33) for V Sge; the IUE data suggest, instead a value close to 0.23. We recall that for T Pyx the observed (B-V) is about 0.14 ± 0.04 (WLTO 1987; Bruch and Engel 1994; Downes et al. 1997; Schaefer 2005; see also Table 7). This would imply that (B-V)*o* ∼ -0.11, close to the value (B-V)*o*= -0.06 given by Szkody (1994).\nPatterson et al.(1998) assumed that M*v*=1.3 and after a significant bolometric correction (based on the assumption of an extremely hot source being present, a consequence of the overestimate of the reddening), derived a quiescent bolometric luminosity higher than 1036 erg s− 1, which considered to be a true lower limit. This encouraged them to invoke nuclear burning on the surface of the WD as the main power source, considering the disturbingly high $\\dot{M}$ ( ≥ 10− 7 M⊙ yr− 1) required in the case of pure accretion power.\nHowever, the IUE and the optical data do not appear to be reproduced by the model depicted by Patterson et al. (1998), since the following observational evidence contradicts with their conclusions:\n1. One of the main results of Paper I was that the de-reddened UVOIR continuum of T Pyx is reproduced well by a single power-law F*λ* ∝ *λ* *α* with a slope *α* = − 2.33, representative of a steady accretion disk. Alternatively, as shown in Paper 1, the UV continuum can be well fitted by a blackbody of temperature 34000 K. There is no way to reconcile these firm observational results with the presence of a supersoft source of typical T ∼ 3.0 × 105 K, since its expected slope in the UV region (*α* = − 3.80) would be inconsistent with the UV observations.\n2. As shown in Sect. 5, the IUE observations in 1980-1996 and the optical and IR photometric data indicate that L*d**i**s**k*=2.7 × 1035 erg s− 1, which corresponds to emission mainly in the directly observed UV range. Therefore, in the absence of any direct or indirect evidence of a hot source, it is unlikely that L*d**i**s**k* is higher than 1036 erg s− 1.\n3. Under steady-state hydrogen-burning conditions, the accretion rate $\\dot{M}$*s**t**e**a**d**y* can be estimated to be:\n \n$$\\dot{M}\\_{steady} \\sim 3.7\\:\\times\\: 10^{-7} (M\\_1-0.4) M\\_{\\odot}\\:yr^{-1},$$ \n(Hachisu and Kato (2001), which is valid for hydrogen content X=0.7 In the case of T Pyx, one calculates $\\dot{M}$*s**t**e**a**d**y* ∼ 3.2 × 10− 7 M⊙*y**r*− 1. This value is almost a factor of 30 higher than the value of $\\dot{M}$ ∼ 1.1 × 10− 8 M⊙ *y**r*− 1 obtained from the IUE data for the post outburst phase.\n4. After correction for inclination, the apparent absolute magnitude of M*v*=1.79 corresponds to a 4-*π* averaged absolute magnitude of M*v**c**o**r**r*=2.53, to be compared with the average value M*v**m**i**n* ∼ 4.0 for ex-novae (Warner, 1995). This implies that T Pyx is more luminous than other ex-novae, as mentioned in Sect. 4, but is not ``extremely bright\". We can explain why T Pyx is brighter than other novae in terms of repeated nova eruptions and heating of the primary, which triggers irradiation of the secondary and produces a higher than average $\\dot{M}$.\n5. The emission line spectrum of T Pyx is not that of an high excitation object: NV *λ* 1240 is nearly absent, HeII *λ* 1640 is weaker than CIV *λ* 1550 and barely present in several spectra, the OIV lines close to *λ* 1405 are absent. A comparison between the IUE spectra of T Pyx and V Sge (Fig.3) clearly shows remarkable differences and a much lower excitation character in T Pyx. From an inspection of IUE spectra of several CVs, we also found that the spectrum of T Pyx is similar to that of the old novae V533 Her and V603 Aql and to some spectra of the intermediate polar TV Col. The old nova RR Pic definitely shows higher excitation than T Pyx, with far stronger NV and HeII emissions.\nOther findings considerations also exclude the H-burning-bloated-WD hypothesis :\n1. The complexity of the optical photometric behavior in T Pyx (Shaefer et al.,1992; Patterson et al., 1998) is difficult to explain if the bulk of the luminosity originates in a spherically symmetric radiation source associated with H-burning on top of a bloated white dwarf.\n2. If the majority of the gas undergoes steady burning, it would be difficult to understand how the remainder that accumulates in the degenerate envelope could burn explosively every 20 years.\n3. The outburst amplitude of T Pyx, ∼ 8.0 magnitudes, is close to that found for classical novae of similar t3 and similar system inclination (Warner, 1995, 2008). If, as assumed by Patterson et al. (1998), the luminosity during quiescence is greater than 1036 erg s− 1 then, with an outburst amplitude of about 8 magnitudes, the luminosity would reach 1040 erg s− 1, implying that T Pyx is an object intermediate between a nova and a SN Ia.\nTo support the hypothesis of steady nuclear burning Patterson et al. (1998) considered *all* CVs of comparable P*o**r**b* and deduced that the $\\dot{M}$ in T Pyx was a factor of 5000 higher than in other CVs of similar P*o**r**b*. One should compare T Pyx with objects that are similar, that is, with recent novae. After outburst, the nova system remains in an excited state and $\\dot{M}$ increases due to the irradiation of the secondary. Patterson et al. (1998) correctly excluded from their Fig. 14 all old novae within 30 yr of the outburst because of their systematically too high luminosity levels. Adopting the same line of reasoning, T Pyx should also have been excluded. In this respect, we note that the $\\dot{M}$ of T Pyx is only slightly higher than that observed in *recent* ex-novae (e.g. RR Pic, V841Oph, HR Del, etc) (Selvelli, 2004).\nBased on the conclusions of Patterson et al. (1998) of an the extremely high luminosity (L*b**o**l* far higher than 1036 erg s− 1), Knigge et al (2000) investigated in detail the evolution of the T Pyx system and proposed that the system is a wind-driven supersoft X-ray source. In this scenario, a strong, radiation-induced wind is excited from the secondary star, and increases the rate of binary evolution, causing the system to destroy itself either by evaporation of the secondary star or in a Type Ia SN if the WD reaches the Chandrasekhar limit. Knigge et al. (2000) therefore proposed that either the primary, the secondary, or both stars may be committing assisted stellar suicide\nThis scenario is, admittedly, highly speculative, and depends crucially on the unsubstantiated assumption that both the temperature and luminosity of T Pyx are extremely high. The IUE and optical data are instead consistent with a more conventional scenario of accretion power, as in other CVs, and we confidently predict that, fortunately, any form of suicide in the near future is extremely unlikely.\nFinally, we note that Greiner et al. (2003) did not find T Pyx to be a supersoft X-ray source, and that T Pyx does not appear in NASA’s HEASARC tool (a master compilation of EUV and X-ray databases).\nThe XMM observations\n====================\nWhile this work was close to completion, the data for X-ray observations of T Pyx by XMM became publicly available. This prompted us to perform a preliminary analysis of the data to verify the presence or absence of a supersoft source.\nT Pyx was observed by XMM-Newton on November 10 2006. All the three EPIC cameras were operated in Full Frame mode with the Medium filter. The total useful exposure time after filtering for high radiation periods was 22.1 ksec. Optical Monitor data were taken simultaneously with the X-ray observations. The values (see Table 7) are consistent with the values given in Paper I (IUE and optical observations) and confirm the stability of the SED with time. The reduction of the XMM EPIC data was carried out with SAS version 7.1, using standard methods. T Pyx was detected as a faint source that had an observed EPIC-pn count-rate of 8.5 × 10− 3 cts s− 1 and emission over the complete range 0.2-8 keV.\nFigure 4 shows the XMM-Newton EPIC-pn spectrum of T Pyx compared to a blackbody of 2.4 × 105 K and a luminosity of 1.0 × 1037 erg s− 1. The blackbody spectrum was simulated assuming an exposure time of 20 ksec, similar to that of the data, with two assumptions: a distance of 3500 pc and a reddening E*B* − *V*=0.25 (as in this paper), and a distance of 3000 pc and a reddening of 0.4 (as assumed by Knigge et al., 2000). The presence of a supersoft component with a temperature of 2.4 × 105 K and a luminosity of 1.0 × 1037 erg s− 1, whose existence was postulated by both Patterson et al. (1998) and Knigge et al.(2000), can be definitely excluded. Such a bright component would be easily visible at soft energies (i.e. below 0.5 keV), and this is not the case. Any supersoft emission, if present, would be several orders of magnitude fainter than expected.\nThe data are compatible with the presence of a relatively hard source of low luminosity, but their low statistical quality (227 counts in the range 0.2-8. keV) does not allows us to perform a reliable fit. We also note that extrapolation of the observed IUE spectrum (power-law) into the X-ray range would result in an unrealistically high flux, several orders of magnitude higher than observed. The existence of such a high flux was already excluded in Paper I on the basis of the intensity of the HeII 1640 Åline.\nWe therefore ask from which process or in which source the observed X-ray emission originates? In a CV (nova or quiescent nova) X-rays may be produced by at least three different mechanisms or regions (see Kuulkers et al. 2006, and Krautter, 2008):\n1. Thermal emission from the hot white dwarf nova remnant in the late outburst phases. The nova becomes a strong X-Ray emitter with a soft SED.\n2. Emission from the inner accretion disk (BL) or the accretion columns (in magnetic CVs). One expects to observe a typical X-ray emission of CVs in quiescence, with a thermal bremsstrahlung spectrum.\n3. Shocks in the circumstellar medium surrounding the nova system where the expanding nova shell and/or a nova wind interact with each other or with pre-existing CS material. The expected hard X-ray spectrum originates from thermal bremsstrahlung with kTeff temperatures 0.2-15 keV, and with Lx ∼ 1033-1034 erg s− 1 (O’Brien et al. 1994).\n4. The corona of an active dwarf M star companion.\nIn T Pyx, as already mentioned, the XMM data would exclude case 1. corresponding to a strong SSS source. Case 2 can also be ruled out because, for a CV accreting at the quite high rates of T Pyx, the BL is optically thick and one would expect to observe (see Patterson and Raymond 1985) only soft X-ray emission with luminosity comparable to the disk luminosity, L*b**l* ∼ L*d**i**s**k* ∼ 2.7 × 1035 erg s− 1. For case 4, we note that a high level of coronal heating at values of L*x*/L*b**o**l* ∼ 10− 3 persists for the active dMe stars, and the emission was detected in surveys of nearby stars (see Giampapa and Fleming, 2002), although any detection would be impossible at the distance of T Pyx.\nWe suggest that the most likely origin of the observed hard X-ray emission is from shocks within the circumstellar envelope. We note that in GK Per, Balman et al. (2006) detected hard X-ray emission by direct imaging with Chandra. The total X-ray spectrum of the nebula consists of two thermal prominent components of emission. GK Per has a large amount of CS material, which is most likely a residual of a planetary nebula phase, and the shell remnant shows a clumpy structure similar to that observed in T Pyx by Shara et al. (1997) with HST. We recall that the studies by Orio et al. (2001) by Orio (2004) of the X-ray emission from classical and recurrent novae demonstrated that emission from shocked ejecta is expected to last about two years, but may last for up to a century, if, for example, there is pre-existing circumstellar material (as in the case of GK Per). Hernanz and Sala (2007) reported on X-ray observations of V4633 Sgr performed with XMM-Newton between 2.6 and 3.5 yr after outburst. The X-ray spectrum is dominated by thermal plasma emission, which most probably originated in the shock heated ejecta.\nUnfortunately, the limited spatial resolution of the available XMM data do not enable any spatially-resolved study to be completed, because the pixel size is about 4 arcsec compared with the optical radius of the nebula which is about 5 arcsec.\nAlso the XMM observations, excluding the possibilities of continuous burning and the supersoft source scenario (a massive white dwarf accreting at high rates), appear to exclude T Pyx becoming a SN Ia by means of the supersoft X-ray source channel described by Hachisu (2002, 2003).\n| | | | | |\n| --- | --- | --- | --- | --- |\n| Filter | *λ**e**f**f* | Observed flux | IUE flux | magn. |\n| | (Å) | (1 × 10− 14) | (1 × 10− 14) | |\n| UVM2 | 2310 | 0.92 | 0.80 | - |\n| UVW1 | 2910 | 0.86 | 0.92 | - |\n| U | 3440 | 0.70 | - | 14.37 |\n| B | 4500 | 0.38 | - | 15.59 |\n| V | 5430 | 0.24 | - | 15.49 |\nThe recurrence time and the next, long-awaited outburst.\n========================================================\nAs reported in Paper I, we started an observing program in 1986 with IUE to monitor T Pyx prior to (and during) the expected next outburst that was supposed to occur in the late eighties of the last millennium. Unfortunately, the star successfully managed to postpone the long-awaited outburst, and at the present time (2008) has surpassed by eighteen years the longest inter-outburst interval so far recorded (24 yrs).\nAs mentioned in Sect. 7, Schaefer (2005) published the results of a study of the inter-outburst interval in the recurrent novae T Pyx and U Sco. From an analysis of the available data, he found that the two novae are relatively bright during short inter-eruption intervals and dim during long intervals, suggesting that the product of the inter-eruption interval times the average bolometrically corrected flux is a constant. Therefore, in the case of T Pyx, the lack of the post-1967 outburst is explained by a lower luminosity and therefore a lower $\\dot{M}$. From the decline in the observed quiescent B magnitude in the time intervals before and after the 1966 outburst, Schaefer (2005) also predicted that the next outburst of T Pyx will occur around 2052.\nWith the help of considerations in Sect. 8 and the data in Tables 3 and 5, we can further investigate this prediction. The recurrence time can be estimated from the (theoretical) M*i**g**n* and the observed mass accretion rate. M*i**g**n* depends mainly on M1 and $\\dot{M}$, while $\\dot{M}$, for a given L*d**i**s**k*, is a function also of M1 and R1, which, in turn, is a function of M1. Tables 3 and 5 list M*i**g**n*, $\\dot{M}$, and the recurrence time *τ*=M*i**g**n*/$\\dot{M}$ (years) for various M1 values. Table 5 (pre-1967 $\\dot{M}$ values) clearly shows that the observed inter-outburst interval (22 years) corresponds to M1 values ∼ 1.36-1.38 M⊙. Table 3, which contains post 1967-outburst $\\dot{M}$ values indicates that the observed interval, which, so far, is longer than 42 years, corresponds to M1 ≤ 1.38 M⊙. The most likely value of M1 is therefore close to 1.37 M⊙.\nWe note that a reduction by a factor two in the mass accretion rate corresponds to an increase in the expected recurrence time *τ* by a larger factor because, for a given M1, M*i**g**n* increases as $\\dot{M}$ decreases. For the relevant values of T Pyx, the decrease by a factor of 2 in $\\dot{M}$ is accompanied by an increase by a factor of about 50 percent in M*i**g**n*. For the next outburst, therefore one expects an increase in the inter-outburst interval *τ* by a factor of approximately 3.0, to values near 60 years (see Table 3). Our prediction for the next outburst date is therefore around A.D. 2025. With this new date, contrasted with that of Schaefer (A.D. 2052), we (or at least some of us) feel a bit more confident about the chance of personally testing this prediction.\nHowever, given the uncertainties in $\\dot{M}$ and M*i**g**n*, the possibility of a more imminent outburst cannot be ruled out. In this case, X-ray and other observations during the first outburst stages will be of paramount importance in determining the mass ejected in a single event.\nSummary and conclusions\n=======================\nWe have accurately determined, from UV and other observations, the accretion disk luminosity of T Pyx during both the pre- and post-1966 inter-outburst phases. For M1 ∼ 1.37 M⊙, we have found that $\\dot{M}\\_{pre-OB}$ ∼ 2.2 × 10− 8 M⊙ yr− 1. By combining the measured accretion rate with the duration of the inter-outburst phase (Δt = 22 yrs), the total accreted mass is inferred to be M*a**c**c**r*=$\\dot{M}\\_{pre-OB}$ ⋅ Δt ∼ 5.2 × 10− 7 M⊙. This value is in excellent agreement with the theoretical ignition mass (M*i**g**n*) ∼ 5.0 × 10− 7M⊙ expected for a massive white dwarf accreting at the quoted rate. Therefore, both the time interval between the last two outburst and the absence of the awaited post-1967 outburst (due to the lower $\\dot{M}$ in the post-1967 time interval) are explained in a self-consistent way.\nThis is the first reliable determination of the mass accreted prior to a nova outburst, *M**a**c**c**r*, owing to the dominance of the accretion disk luminosity over that of the secondary star at UV, optical and IR wavelengths, as well as good observational coverage during the inter-outburst phases. Unfortunately, *M**a**c**c**r* cannot be confidently determined in other cases, such as classical novae, because of their long inter-outburst interval, nor in other recurrent novae, due to the faintness of the source, the lack of systematic UV observations, or the dominance of light from the giant companion over that from the accretion disk.\nIn T Pyx, the consistency between the observed M*a**c**c**r* and the theoretical M*i**g**n* supports the good quality of the observations and the reliability of the models and represents a new, direct confirmation of the validity of the TNR theory, which associates a massive white dwarf with the recurrent nova phenomenon. A detailed comparison of the observed parameters with the theoretical grids of Yaron et al. (2005) indicates that the closest agreement is obtained with a models of a rather massive white dwarf (M1 ∼ 1.25-1.40 M⊙) that accretes at high $\\dot{M}$ rates ($\\dot{M}$ ∼ 10− 8-10− 7 M⊙ yr− 1). However, no combination of the theoretical parameters can reproduce the observed values reliably, t3 being the most difficult parameter to describe.\nThe literature data of the spectroscopic and photometric evolution during the outbursts of T Pyx clearly indicate the occurrence of an optically thick phase that lasted about three months. This implies an ejected mass of M*e**j* ∼ 10− 5 M⊙ or higher, i.e. much higher than the mass of the accreted shell M*a**c**c**r* ∼ 5.2 × 10− 7 M⊙, inferred from UV and other observations during quiescence. Therefore, T Pyx ejected far more material than it has accreted.\nThere is no way to reconcile this discrepancy given the small uncertainty in the value of $\\dot{M}$; even if allowance is made for an uncertainty of a factor two, one obtains an upper limit to M*a**c**c**r* that is smaller by a factor of at least ten than the theoretical value of M*e**j*. Only for accretion rates higher than 4 × 10− 7 M⊙ yr− 1 would the accreted mass M*a**c**c**r* be comparable with the estimated ejected mass M*e**j*. However, these high rates would correspond to the steady-burning regime, while our detailed discussion of Sect. 14 definitely excluded this possibility. Further confirmation of our considerations can be found in the the very recent results of XMM observations that exclude the presence of a super-soft-source in T Pyx.\nThe important point is that far more material appears to have been ejected during the last outburst of T Pyx than has been accreted by the white dwarf. This raises several doubts about the common assumption that the white dwarf in recurrent novae increases in mass toward the Chandrasekhar limit, and about the possible role of RNe as progenitors of SNIa. 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S.1998,, 301, 767\nStarrfield, S.1993, Astrophysics and Space Science Library, 177, 209\nStarrfield, S.1999,, 311, 371\nStarrfield, S.2002, Classical Nova Explosions, AIP 637, 89\nStarrfield, S., Sparks, W. M., & Truran, J. W. 1985,, 291, 136\nStarrfield, S., Vanlandingham, K., Schwarz, G., Hauschildt, P. H., & Shore, S. N. 1998, Stellar Evolution, Stellar Explosions and Galactic Chemical Evolution, 433\nStarrfield, S., Truran, J. W., Wiescher, M. C., & Sparks, W. M. 1998,, 296, 502\nStarrfield, S., Truran, J. W., Sparks, W. M., Hauschildt, P. H., Shore, S. N., Krautter, J., Vanlandingham, K., & Schwarz, G. 1998, Wild Stars in the Old West, 137, 352\nStickland, D. J., Penn, C. J., Seaton, M. J., Snijders, M. A. J., & Storey, P. J. 1981,, 197, 107\nSzkody, P. 1994,, 108, 639\nSzkody, P., & Feinswog, L 1988,, 334, 422\nTownsley, D. M., & Bildsten, L.2004,, 600, 390\nTruran, J. W. 1998, Stellar Evolution, Stellar Explosions and Galactic Chemical Evolution, 425\nvan den Bergh, S., & Younger, P. F 1987,, 70, 125\nVanlandingham, K. M., Starrfield, S., Wagner, R. M., Shore, S. N., & Sonneborn, G. 1996,, 282, 563\nVanlandingham, K. M., Starrfield, S., Shore, S. N., & Sonneborn, G. 1999,, 308, 577\nVogt, N., Barrera, L. H., Barwig, H., & Mantel, K.-H. 1990, Accretion-Powered Compact Binaries, 391\nWade, R. A., & Hubeny, I.1998,, 509, 350\nWarner, B. 1976, Structure and Evolution of Close Binary Systems, 73, 85\nWarner, B. 1986,, 222, 11\nWarner, B. 1987,, 227, 23\nWarner, B. 1995, Cataclysmic Variable Stars, Cambridge Astrophysics Series, Cambridge University Press\nWarner, B. 2008, in Classical Novae, 2nd Edition. Edited by M.F. Bode and A. Evans. Cambridge Astrophysics Series, No. 43, Cambridge University Press, 2008, 16\nWebbink, R. F. 1990, Accretion-Powered Compact Binaries, 177\nWebbink, R. F., Livio, M., Truran, J. W., & Orio, M. 1987,, 314, 653 (WLTO, 1987)\nWhelan, J., & Iben, I. J.1973,, 186, 1007\nWilliams, R. E., Sparks, W. M., Gallagher, J. S., Ney, E. P., Starrfield, S. G., & Truran, J. W. 1981,, 251, 221\nWilliams, R. E.1982,, 261, 170\nWilliams, R. E.1994,, 426, 279\nWoosley, S. E., & Weaver, T. A.1986,, 24, 205\nYaron, O., Prialnik, D., Shara, M. M., & Kovetz, A. 2005,, 623, 398\nYaron, O., 2007, private communication\n\nThe secrets of T Pyx II.A recurrent nova that will not become a SN Ia\n=====================================================================\n### Received.....; accepted\n We compare the observed and theoretical parameters for the quiescent and outburst phases of the recurring nova T Pyx. IUE data were used to derive the disk luminosity and the mass accretion rate, and to exclude the presence of quasi-steady burning at the WD surface. XMM-NEWTON data were used to verify this conclusion. By various methods, we obtained L*d**i**s**k* ∼ 70 L⊙ and $\\dot{M}$ ∼ 1.1 × 10− 8 M⊙yr− 1. These values were about twice as high in the pre-1966-outburst epoch. This allowed the first direct estimate of the total mass accreted before outburst, M*a**c**c**r*=$\\dot{M}\\_{pre-OB}$ ⋅ Δt, and its comparison with the critical ignition mass M*i**g**n*. We found M*a**c**c**r* and M*i**g**n* to be in perfect agreement (with a value close to 5 × 10− 7M⊙) for M1 ∼ 1.37 M⊙, which provides a confirmation of the thermonuclear runaway theory. The comparison of the observed parameters of the eruption phase, with the corresponding values in the grid of models by Yaron and collaborators, provides satisfactory agreement for values of M1 close to 1.35 M⊙ and log$\\dot{M}$ between -8.0 and -7.0, but the observed value of the decay time t3 is higher than expected. The long duration of the optically thick phase during the recorded outbursts of T Pyx, a spectroscopic behavior typical of classical novae, and the persistence of P Cyg profiles, constrains the ejected mass M*i**g**n* to within 10− 5 - 10− 4 M⊙. Therefore, T Pyx ejects far more material than it has accreted, and the mass of the white dwarf will not increase to the Chandrasekhar limit as generally believed in recurrent novae. A detailed study based on the UV data excludes the possibility that T Pyx belongs to the class of the supersoft X-ray sources, as has been postulated. XMM-NEWTON observations have revealed a weak, hard source and confirmed this interpretation. \nIntroduction\n============\nClassical novae (CNe) are semi-detached binary systems in which a white dwarf (WD) primary star accretes hydrogen-rich matter, by means of an accretion disk, from a companion star which fills its Roche lobe. The ``classical nova\" outburst is a thermonuclear runaway (TNR) process on the surface of a white dwarf that is produced when, due to the gradual accumulation of hydrogen-rich material on its surface, the pressure at the bottom of the accreted layer, which is partially or fully degenerate, becomes sufficiently high (P ≥ 1018-1020 dyn cm− 2 ) for nuclear ignition of hydrogen to begin. The critical mass for ignition M*i**g**n* depends primarily on the mass of the white dwarf as M*i**g**n* ∼ M1− 7/3, although more detailed models show that the ignition mass also depends on both the mass accretion rate $\\dot{M}$ and the core WD temperature (Prialnik and Kovetz 1995; Townsley and Bildsten 2004; Yaron et al. 2005; see also Sect. 8).\nRecurrent novae (RNe) are a subclass of classical novae characterized by outbursts with recurrence time of the order of decades. We refer to Webbink et al. (hereinafter WLTO, 1987) as the seminal paper on the nature of recurrent novae and to Warner (1995), Hachisu and Kato (2001), and Shore (2008) for further considerations of this topic. Solid theoretical considerations indicate that a model of a recurrent outburst requires a high accretion rate $\\dot{M}$ (10− 8-10− 7 M⊙) onto a WD of mass close to the Chandrasekhar mass limit (Starrfield 1985; WLTO 1987; Livio 1994). However, nova models (Prialnik and Kovetz 1995; Yaron et al. 2005; see also Sect. 8) appear to evade these strict requirements and allow the occurrence of recurrent outbursts also in a (slightly) less massive WD. The ejecta of RNe are expected to be less massive than those of CNe. This is because on the surface of the massive WD expected in a RN, the critical conditions for ignition are reached with a far less massive envelope. Studies of the ejecta of RNe indicate a mass between 10− 6-10− 7 M⊙, instead of the 10− 4-10− 5 M⊙ observed in ``classical\" novae ( Gehrz et al. 1998, Hernanz and Jose 1998, Starrfield 1999).\nRNe represent a convenient laboratory to compare the predictions of the TNR theory with the observations. From the observed $\\dot{M}$ and the observed duration of the inter-outburst interval, in principle, one can obtain a direct estimate of the total mass accreted (M*a**c**c**r*) between two successive outbursts. This quantity can be compared with both the (theoretical) critical ignition mass M*i**g**n* and the mass of the ejected shell M*e**j*, which can be estimated by spectroscopic methods (and, in principle, by photometric methods if dP/P can be accurately measured, Livio 1991). A similar comparison cannot be made in the case of CNe because the amount of mass accreted prior to outburst is badly determined due to their long inter-outburst interval.\nObservations of RNe in quiescence and outburst can also be used to determine the secular balance between the total accreted mass M*a**c**c**r* and that ejected in the explosive phase M*e**j*, and therefore to investigate the possible role of RNe as progenitors of SN Ia. The ejecta of RNe have an almost ``solar“ chemical composition and are not enriched in ``heavy” elements, such as carbon, oxygen and neon. This has been taken as an indication that the massive white dwarf in RNe is not eroded, will gain mass after each cycle of accretion and ejection, and will eventually explode as a Type Ia supernova (Hachisu and Kato, 2002).\nIn Gilmozzi and Selvelli (2007) (hereinafter Paper I), we studied the UV spectrum of T Pyx in detail and our main conclusion was that the spectral energy distribution (SED) is dominated by an accretion disk in the UV+opt+IR ranges, with a distribution that, after correction for reddening E*B* − *V*=0.25, is described by a power law F*λ* = 4.28 × 10− 6 *λ*− 2.33 erg cm− 2 s− 1 Å− 1, while the continuum in the UV range can also be well represented by a single blackbody of T ∼ 34000 K. The observed UV continuum distribution of T Pyx has remained remarkably constant in both slope and intensity during 16 years of *IUE* observations.\nIn the present study, we use this basic result from the IUE data and other considerations, to constrain the relevant parameters of the system and to attempt to understand the elusive nature of its outbursts. The five recorded outbursts of T Pyx occurred in 1890, 1902, 1920, 1944, and 1966, with a mean recurrence time of 19 ± 5.3 yrs (WLTO). All outbursts were similar in photometric behavior and characterized by a decay time t3 ∼ 90$^{\\rm d}$, a speed class that is substantially slower than in other RNe. We divide the paper into several self-contained sections.\nIn Sect. 2, we discuss the problem of distance, which we determine to be 3500 ± 500 pc. In Sect. 3, we analyze the mass function of the system. Using observational data and other considerations, we obtain 1.25 < *M*1 < 1.4 M⊙, *M*2 ∼ 0.24 M⊙, or *M*2 ∼ 0.12 M⊙ (depending on the as yet uncertain period), and *i* = 25 ± 5 degrees. In Sect. 4, we derive the absolute magnitude at minimum M$\\_{\\rm\nv}$=2.53 ± 0.23. In Sect. 5, we calculate the bolometric disk luminosity $L\\_{\\rm\ndisk} \\sim\n70$ L⊙.\nIn Sect. 6 we use various methods to determine the mass accretion rate. The value depends on $M\\_{\\rm WD}$, and is $\\dot{M} \\sim 1.3\\ \\times\n10^{-8}$ M⊙ yr− 1 for $M\\_{\\rm WD}=1.33$ M⊙. In Sect. 7, we derive the accretion rate before the 1966 outburst, and find that it is about twice the value measured with IUE. In Sect. 8, we compare the accreted mass before the last outburst as determined from the $\\dot{M}$ derived in Sect. 7, $M\\_{\\rm accr} \\sim 4.5\\ \\times\n10^{-7}$ M⊙, with theoretical calculations of the ignition mass. We find good agreement (including the inter-outburst interval) for $M\\_{\\rm WD} \\sim 1.37$ M⊙. In Sect. 9, we compare the observations of T Pyx with the theoretical models of Yaron et al (2005), finding good agreement for all parameters with the exception of *t*3.\nIn Sect. 10, we reanalyze the historical data of the outbursts and reach the inescapable conclusion that the mass ejected in the explosion is $M\\_{\\rm\nej} \\sim 10^{-4}-10^{-5}$ M⊙, about a factor 100 higher than the accreted mass. This discrepancy is discussed in detail in Sect. 11, and leads to the conclusion that the WD loses mass during the ourbursts. In Sect. 12, we revisit the parameters of the extended nebula around T Pyx, and demonstrate why the nebula cannot be used to derive information about the mass ejected during outburst. Section 13 discusses the mass balance and infers that T Pyx is not destined to become a SN Ia as often postulated for recurrent novae. Far from increasing its mass towards the Chandrasekhar limit, the WD in T Pyx is being eroded.\nIn Sect. 14, we discuss the hypothesis that the WD in T Pyx is undergoing steady nuclear burning on its surface, and show that it is incompatible with both historical and UV observations. In Sect. 15, we report about XMM observations confirming our analysis by directly disproving the hypothesis.\nIn Sect. 16, we discuss the recurrence time, and predict that the long awaited next outburst will occur around A.D. 2025. Section 17 reports our conclusions.\nThe distance\n============\nAn accurate knowledge of distance is fundamental to the determination of the disk luminosity L*d**i**s**k* (and M*v*), the mass accretion rate $\\dot{M}$, and the mass of the observed nebula surrounding the system M*n**e**b* (not to be confused with the mass ejected during outburst, M*e**j*).\nIn this section, we revisit the problem of determining the distance to T Pyx. We adopt a new accurate determination of the reddening, E*B* − *V*=0.25 ± 0.02, which is based on the depth of the *λ* 2175 Å interstellar absorption feature (see Paper I).\nAn estimate of the absolute magnitude at maximum M*V**m**a**x* can be obtained from the optical decay time *t*2 or *t*3 by extending to a recurrent nova such as T Pyx, the applicability of the Maximum Magnitude versus Rate of Decline (MMRD) relations obtained using data of classical novae.\nThis is justified by the fact that it is clearly established that the same kind of physical process for the outburst is involved i.e. a TNR on an accreting WD. The suggestion that recurrent novae in M31 are fainter than predicted by the MMRD curve (Della Valle and Livio, 1998), applies only to the RNe subset that, unlike T Pyx, displays a short *t*3. The long *t*3 of T Pyx, the spectral behavior during outburst and the quantity of mass ejected (see Sect. 10) indicate that the same mechanism is operating as in CNe.\nThe optical decay time of T Pyx is well determined (Mayall 1967; Chincarini and Rosino 1969; Duerbeck 1987; WLTO 1987; Warner 1995) with *t*2 ∼ 62*d* and *t*3 ∼ 92*d*. From these values, we derived four different estimates for M*v**m**a**x* ( − 7.01, − 6.75, − 7.00, − 6.64) using the S-shaped MMRD relation of Della Valle and Livio (1995), and the two linear and the one S-shaped relations of Downes and Duerbeck (2000), respectively.\nAn additional estimate of the distance can be obtained by assuming that T Pyx, as a moderately fast nova, radiated at near-Eddington luminosity at maximum. This provides\n \n*L**E**d**d*/*L*⊙ = 4.6 × 104 (*M*1/*M*⊙ − 0.26) \nFor a massive white dwarf (M1 ∼ 1.35 M⊙), this corresponds to M*b**o**l**m**a**x* ∼ -6.98. Close to the epoch of maximum light the observed color index showed oscillations about an average value of (B-V)=0.35 (Eggen et al. 1967, Eggen, 1968), or, after correction for reddening, (B-V)*o* ∼ 0.10. Van den Bergh and Younger (1987) found that novae at maximum light have (B-V)*o* = +0.23 ± 0.06, while Warner (1976) reported that the intrinsic color of novae close to maximum is (B-V)*o* ∼ 0.0. The intermediate value for T Pyx was (B-V)*o* ∼ 0.10, which is indicative of a temperature lower than 10,000 K and suggests that, close to maximum light, the nova radiated mostly in the optical. Since the bolometric correction is quite small in this case (BC ∼ − 0.20), it follows that M*v**m**a**x* = − 6.78, which is quite close to the values obtained above from the four MMRD relations.\nThe mean value of the five estimates for M*v**m**a**x* is − 6.83, with a median value − 6.79 and a formal standard deviation *σ* = 0.16. The five estimates are intermediate between the value ( − 6.4) reported by Payne-Gaposchkin (1957), Catchpole (1968), and Warner(1995) and the single value ( − 7.47) given by Duerbeck (1981).\nAdopting M*v**m**a**x* = − 6.81, *A**V* = 3.15 × *E*(*B* − *V*) = 0.79 and m*v**m**a**x* ∼ 6.7 (Catchpole 1968, Warner 1995, Chincarini and Rosino 1969), one obtains a distance d = 10(1 + (*m* − *M* − *A*)/5) ∼ 3500 pc. The uncertainties in the individual parameters (0.16 for M*v**m**a**x*, 0.1 for m*v**m**a**x*, 0.06 for A*V*) correspond to an uncertainty of about 330 pc in the distance, i.e. to a relative uncertainty of the order of 10 percent.\nA further estimate of the distance can be obtained by comparing the dereddened flux of T Pyx at 1600 Å with the accretion disk models by Wade and Hubeny (1998). As mentioned in Paper I, the most appropriate descriptions of the shape and depth of the lines (models *e**e* and *j**j* at low inclination angles) correspond to high values of white-dwarf mass and accretion rate, although they provide a continuum that is too steep. In any case, these high M1, high $\\dot{M}$ models, (we have extrapolated $\\dot{M}$ to values as high as log $\\dot{M}$=-7.5) clearly show that the ratio F1600*m**o**d**e**l*/F1600*o**b**s*. is about 1000. Applying a scaling of a factor of ∼ (1000)0.5 = 31.5 to the model distance (100 pc) would reconcile these two values. The derived value of ∼ 3150 pc is in fair agreement with the assumed distance of 3500 pc.\nWe recall that a range of distances to T Pyx can be found in the literature. Catchpole (1969) derived a lower limit of 1050 pc from the equivalent width of the CaII K line (EW ∼ 400 mÅ) and the EW-distance relationship derived by Beals and Oke (1953) for stars close to the Galactic plane. The EW measurement of Catchpole is however quite uncertain because of possible contamination by emission lines and the low quality of the spectra (the CaII K line falls close to the edge of the spectrum and only two spectra were of sufficient signal in that region).\nCatchpole (1969), using the two spectra of the highest signal-to-noise-ratio found that the K line of calcium corresponds to radial velocity of +20 km s− 1. According to Catchpole (1969), this velocity agrees well with that of material travelling with the Galactic rotation in the direction of T Pyx; since this direction is, however, close to a node of the rotation curve, the Beals and Oke (1953) method cannot be used to determine the distance.\nWarner (1976), used the calibration given by Munch (1968) for stars close to a node in the Galactic rotation curve, and indicated that a star with a measurement of 0.4 Å for the K line should be at least at a distance of 2 kpc.\nIn a study of the IS Ca II K line towards O- and B-type stars in the Galactic disk, Hunter et al. (2006), presented a new calibration of the the total column density of Ca II K versus distance. In a plot of log N(K) versus log d, the estimated distance of 3500 pc (log d = 3.545) would correspond to log N (CaII) ∼ 13.0. The EW of the K line measured by Catchpole (1969) corresponds to a column density of 4.9 × 1012 (log N=12.69), which would correspond to a distance of 2200 pc. However, this value of column density is a lower limit, because the medium is not optically thin in the K line, and a yet higher value of distance is expected.\nBy applying T Pyx theoretical predictions for light curves of CVs in outburst, Kato (1990) estimated a distance close to 4000 pc, by comparing the observed magnitude at outburst with the theoretical absolute magnitude prediction at outburst.\nIn the following, we assume a distance of *d* = 3500 ± 350 pc. A discussion of the critical role played by distance in the interpretation of the nature of T Pyx is given in Sect. 11.1.\n The system parameters\n======================\nWe aim to determine both the disk luminosity and the mass accretion rate of T Pyx. This requires prior knowledge of the system inclination angle *i* and the mass of the primary M1. In the disk geometry, the inclination angle is critical to the estimate of the disk luminosity, while M1 and R1 (a function of M1) are key parameters in the correlation between $\\dot{M}$ and L*d**i**s**k*. For most CVs, the determination of these and other parameters (M2, P*o**r**b*, 2K) entering the mass function:\n \n$$\\frac{(M\\_2\\cdot sini)^3}{(M\\_1+M\\_2)^2}=1.037 \\times 10^{-7}\\cdot\nK\\_1^3\\cdot P\n\\label{eq:massfun}$$ \nis a difficult task, and accurate solutions have been obtained only for a few eclipsing systems.\nFor T Pyx, the situation, ``prima facie\", does not appear encouraging because no system parameter (not even the orbital period) have been accurately measured. Therefore, one has to start from a restricted range of values corresponding to the most accurately known parameters in the mass function equation, based on theoretical or semi-empirical considerations, and derive the corresponding range of allowed solutions for the unknown relevant parameters.\nThe orbital period(s) and K1\n----------------------------\nThere have been several attempts to measure the orbital period of T Pyx, although, disappointingly, they have resulted in a wide range of values (see Paper I); this is a clear indication that there is no definite photometric clock associated with the system.\nPatterson et al. (1998) detected a stable photometric wave at P*h*=1*h*.829 ± 0*h*.002 (a value close to that of the ``most probable\" photometric period P*h* = 1*h*.828 of Schaefer et al., 1992), although this interpretation is inconsistent with the presence of another signal at 2*h*.635.\nThe only spectroscopic determination of the period is that by Vogt et al. (1990), who, in a preliminary study based on a large number (101) of spectra of limited resolution ( ∼ 3 Å), reported a spectroscopic period of P*h* ∼ 3*h*.439, without quoting uncertainties. This spectroscopic period is almost twice the photometric value determined by Szkody and Feinswog (1988) in the infrared. In other CVs, this behavior is generally interpreted as evidence of ellipsoidal variations in the companion, in a system observed at a high inclination angle. This is not the case for T Pyx because, as convincingly shown in Paper I, the companion does not contribute to the J light curve, unless the distance is less than 200 pc.\nIn the following, we consider system solutions for both P*h* = 1*h*.829 (photometric, Patterson et al, 1998) and P*h*=3*h*.439 (spectroscopic, Vogt et al. 1990). The inclusion of the spectroscopic period could be criticized since it is the result of a preliminary analysis that has not been confirmed by subsequent studies. However, given the complex photometric period structure and in the absence of a definite physical interpretation of the stable signal with P*h*=1*h*.829 (that could even arise from the rotation of a magnetic white dwarf), we believe that the spectroscopic results should be considered, since, in any case, they have not been challenged by similar studies.\nVogt et al. (1990) found that the projected orbital velocity K1 of the primary, derived from an analysis of radial velocities of the emission lines, is approximately 24 ± 5 kms− 1. Taking into account the limited resolution and the large number of spectra, this value is probably an upper limit. We assume that this value of K1 is representative of the primary star radial velocity (independently of the period being considered), although we are aware of the problems encountered in associating radial velocity changes with the motion of the WD.\nThe mass of the primary\n-----------------------\nBoth theoretical considerations and observational determinations (although limited) indicate that the WD masses in classical nova systems are about 1.0 M⊙, i.e. higher than inferred from the standard field WD mass distribution (Ritter et al., 1991). Theoretical models of recurrent novae require an even more massive white dwarf (WLTO, Shara 1989, Livio 1994). Therefore, in the following, we allow the mass of the primary, M1, to vary between 1.25 and 1.4 M⊙.\nThe mass of the secondary\n-------------------------\nIn the absence of any direct information about the secondary (the accretion disk is the dominant luminosity source from the UV to the infrared, Paper I), we can estimate its mass M2 on the basis of the commonly accepted assumptions that: a) the secondary fills its Roche lobe, b) its radius R2 is determined by the Roche geometry, and c) the secondary star obeys a mass-radius relation valid for low mass main sequence stars. In this standard description, the radius of the secondary is well approximated by the equivalent radius R*L* of the Roche lobe itself, i.e. R2 ∼ R*L* and the spherical volume of the secondary is assumed to equal the volume of its Roche lobe. The equivalent volume radii of the Roche lobe are given in tabular form by several authors (Plavec and Kratochvil 1964; Kopal 1972; Eggleton 1983; Mochnacky 1984). For our purposes, it is convenient however to use the analytical approximation by Paczinsky (1971), which is valid for M2/M1 ≤ 0.8:\n \n*R**L*/*A* = 0.457256(*M*2/(*M*1 + *M*2))1/3 \nwhere we have introduced a new value for the constant (instead of 0.46224) because it yields results that differ by less than 1% from those tabulated by Eggleton (1983 ) and Mochnacky (1984).\nBy combining equation [eq:RL] with Kepler’s third law:\n \n*A* = 0.50557*M**t**o**t*1/3*P**h*2/3 \none obtains an approximate relation between P, M2, and R2 ( ∼ R*L*):\n \n*P**h* = 8.997*R*23/2*M*2− 1/2 \nIn the crude assumption M2 ∼ R2, this provides an approximate relation: M2 = 0.111 P*h* by which a rough estimate of the mass of the secondary can be obtained if the orbital period is known. In general, however, the R-M relation for the low MS stars is described more accurately by a relation of the form:\n \n*R*2 ∼ *β**M*2*α* \nBy combining eq. [eq:PRM] and eq [eq:alpha], we derive a more general relation between M2 and P*h*:\n \n*M*2 = *P**h*(2/(3*α* − 1)) ⋅ 8.997(2/(1 − 3*α*)) ⋅ *β*(3/(1 − 3*α*)) \nValues for the parameters *α* and *β* of the low mass main-sequence stars were derived from observational data and theoretical considerations. In particular, Warner (1995) assumed that R2=M213/15 (which provides an approximate fit to the data set by Webbink, 1990) and derived M=0.065 × P1.25, while Smith and Dhillon (1998) found a constitutive relation R2=0.91 M20.75 and derived M=0.038 × P1.58.\nPatterson et al. (2005) derived an empirical mass-radius sequence for CV secondaries based on masses and radii measured primarily for the superhumping CVs. They found that R2=0.62 × M20.61 for P shorter than 2.3 hours, and R2=0.92 × M20.71 for P longer than 2.5 hours, and derived M2=0.032 × P2.38 (P ≤ 2.3) and M2=0.026 × P1.78 (P ≥ 2.5). We inserted the *α* and *β* values of Patterson et al.(2005) in Eq. [eq:alphabeta] and obtained slightly different results, that is, M2=0.02823 P2.41 (short P) and M2=0.02554 P1.77 (long P).\n| | P*h*= 1.829 | P*h*=3.439 |\n| --- | --- | --- |\n| Patterson et al. | 0.134 | 0.234 |\n| Smith and Dhillon | 0.100 | 0.267 |\n| Knigge | 0.140 | 0.236 |\n| Warner | 0.138 | 0.304 |\n| Equation (7) | 0.121 | 0.227 |\nKnigge (2006) obtained an independent M2-R2 relation by revisiting and updating the mass-radius relationship for CV secondaries determined by Patterson et al. (2005). We have fitted with a power-law the data contained in his Table 3 and derived M2-P relations for short P and long P systems. The relations are\n \n$$\\begin{aligned}\nM\\_2 = 0.0371 \\times P^{2.2022}~~~~~~~~~(P \\le 2.2), \\\\ \\nonumber \nM\\_2 = 0.02024 \\times P^{1.9802}~~~~~~~~(P \\ge 3.2).\\end{aligned}$$ \nTable 1 provides the M2 values obtained by these various methods, for P*h*=1*h*.829 (photometric) and P*h*=3*h*.439 (spectroscopic).\nIn the following, we therefore study the system solutions for M2 = 0.12 ± 0.02 M⊙ and M2 = 0.24 ± 0.02 M⊙, separately. The corresponding spectral types are in the range M3V-M5V (Kirkpatrick et al. 1991, Legget 1992).\nThe system inclination\n----------------------\nThe above derived range of allowed values for the system parameters (M1: 1.25-1.4 M⊙, M2 = 0.12 M⊙ (for P*h*=1*h*.829), M2 = 0.24 M⊙ (for P*h*= 3*h*.439), K1: 24 ± 5 km s− 1) can be inserted into Eq. (2) to find the corresponding range of solutions for the system inclination, which is considered to be a free parameter.\nTable 2 clearly shows that, given the observational and theoretical constrains for P, K1, M1, and M2, the solutions for the inclination *i* are in the range between 20 and 30 degrees, a value close to 25 degrees being the most likely value. In particular, for M1 ∼ 1.35 M⊙, (see also the following) the solutions for *i* do not change significantly, and for P*h*=1*h*.829 (M2=0.12 M⊙) and P*h*=3*h*.439 (M2=0.24 M⊙ ) are close to 30 and 20 degrees, respectively (see also Fig. 1).\nPatterson et al. (1998) suggested that T Pyx is being observed at low system inclination, i.e. *i* ∼ 10*o*-20*o*, due to the low amplitude of the orbital signal. Shabbaz et al. (1997) estimated the binary inclination *i* of T Pyx by measuring the separation of the H*α* emission-line peaks, and obtained a lower limit of ∼ 6*o*. A low system inclination is also consistent with the small value of the radial velocity, the sharpness of the emission lines in the optical (Warner, 1995), and the steepness of the UV and optical continua (Paper I). However, the presence of radial velocity variations (Vogt et al. 1990), and the modulations in the photometric variations preclude an *i* value close to zero.\ncccc\nM1 & K1 & *i* (P=1*h*.83) & *i* (P=3*h*.44) \n(M⊙) &(km s− 1) & \n1.25 & 19 & 22.9 & 14.7 \n1.25 & 24 & 29.4 & 18.7 \n1.25 & 29 & 36.4 & 22.8 \n1.30 & 19 & 23.5 & 15.0 \n1.30 & 24 & 30.2 & 19.1 \n1.30 & 29 & 37.4 & 23.3 \n1.35 & 19 & 24.0 & 15.4 \n1.35 & 24 & 31.0 & 19.6 \n1.35 & 29 & 38.4 & 23.9 \n1.40 & 19 & 24.6 & 15.7 \n1.40 & 24 & 31.7 & 20.0 \n1.40 & 29 & 39.5 & 24.4 \nIt should therefore be noted that, in spite of the large uncertainties in the observational data, the adoption of plausible theoretical assumptions and some semi-empirical constraints has enabled a quite restricted range for the values of the system inclination to be defined. In the following we will assume *i* = 25 ± 5 degrees for the system inclination angle.\nThe absolute magnitude at minimum\n=================================\nDuring the quiescent phase after the 1967 outburst, the average optical magnitude of T Pyx is m*v* = 15.3 ± 0.05 (Warner 1995, Duerbeck 1987, Schaefer 1992); for d = 3500 ± 350 pc and A*V*=0.79 ± 0.06, we obtain an (observed) absolute magnitude at minimum M*v**o**b**s*=1.79 ± 0.21 (Warner 1995 defined this to be the *apparent* absolute magnitude since the observed flux in quiescence depends on the inclination angle).\nAssuming that the visual radiation originates in a non-irradiated disk (Paper I), we correct the apparent absolute magnitude M*v**o**b**s*(*i*) for the inclination angle of the disk (see Warner 1987, 1995) by a term Δ*M**v*(*i*) to obtain the ``standard“ or ``reference” absolute magnitude:\n \n*M**v**c**o**r**r* = *M**v**o**b**s*(*i*) − Δ*M**v*(*i*) \nwhere Δ*M**v*(*i*) = − 2.5log(2cos*i*) according to WLTO (1987), or Δ*M**v*(*i*) = − 2.5log[(1 + 1.5cos*i*)cos*i*] according to Paczynski and Schvarzenberg-Czerny (1980).\nFor *i* ∼ 25*o*, one obtains a correction of Δ*M**v*(*i*) ∼ 0.74 ± 0.06, by averaging these two relations, and *M**v**c**o**r**r*= +2.53 ± 0.22 for the absolute magnitude, averaged over all aspect angles. This value is about 1.4 magnitudes brighter than the mean absolute magnitude of nova remnants in the same speed class (see Fig. 2.20 and Table 4.6 of Warner 1995). It is also about 2.0 magnitudes brighter than the mean absolute magnitude of novae at minimum, as obtained from the values given in Table 6 of Downes and Duerbeck (2000).\nThe T Pyx remnant is therefore one of the brightest nova remnants, its absolute magnitude M*v**c**o**r**r*=+2.53 being close to that of the ex-nova HR Del, whose corrected value, M*v**c**o**r**r*=2.30, implies that it is a quite luminous object (see also Paper I).\nThe disk luminosity from IUE and optical data\n=============================================\nAs shown in Paper I, the integrated UV continuum flux of T Pyx in the wavelength range 1180-3230 Å , after correction for reddening, is 1.94 × 10− 10 erg cm− 2s− 1. For *d* = 3500 ± 350 pc, the corresponding integrated luminosity of the UV continuum is:\n \n*L**U**V* ∼ 2.85 × 1035 *e**r**g* *s*− 1 ∼ 74.2 ± 15.0 *L*⊙. \nwhere the 20 percent is caused by the 10 percent uncertainty in the distance. Most old novae have L*U**V* in the range 1-20 L⊙, with V 841 Oph at ∼ 24 L⊙ (Gilmozzi et al., 1994), and V603 Aql and RR Pic at ∼ 11 L⊙ and ∼ 3 L⊙, respectively (Selvelli, 2004). Therefore, T Pyx is also quite bright in the UV and, again, is challenged only by HR Del that has L*U**V* = 56 L⊙ (Selvelli and Friedjung (2002).\nIn the following, we assume that the observed UV and optical luminosity arise from an accretion disk heated by viscous dissipation of gravitational energy. This agrees with the behavior of similar objects, like old novae and nova-like stars. This assumption is also strongly supported by the fact that the continuum energy distribution in the UV range is well reproduced (Paper I) by a power-law with spectral index *λ* − 2.33, as predicted by theoretical optically thick accretion disk models radiating as a sum of blackbodies.\nThe bolometric disk luminosity *L**d**i**s**k* can be estimated from the observed UV and optical luminosity, where the bulk of the continuum radiation is emitted, after correction for the inclination and the unseen luminosity in both the infrared and at *λ* ≤ 1200 Å. The radiation emitted at wavelengths shorter than Ly*α* is strongly absorbed and the energy is redistributed to longer wavelengths (Nofar, Shaviv and Wehrse 1992, Wade and Hubeny 1998).\nThe integration (from *λ* ∼ 1000 Å to the IR) of the power-law distribution which represents the observed UV and optical continuum of T Pyx, corresponds to a flux of about 3.6 × 10− 10 erg cm− 2 s− 1, which corresponds to a bolometric luminosity of 5.24 × 1035 erg s− 1 ∼ 136.5 L⊙. This value refers to the adopted inclination of about 25 degrees. After correction to the ``standard“ inclination of about 57 degrees and by considering an average limb-darkening factor similar to that in the optical, we obtain an angle (4*π*) averaged bolometric disk luminosity (hereinafter L*d**i**s**k*) of about 70 ± 15 L⊙, where the relative uncertainty (21 percent) derives from the combination of the uncertainties in the distance and the inclination. We consider this to be the reference disk luminosity for the ``recent”, post-1967-outburst epoch. We note that the Wade and Hubeny (1998) models indicate (their Fig. 8) that in the UV region close to 1200 A (where the effect is larger) the limb-darkening correction, from *i* of about 25 degrees to *i* of about 60 degrees, corresponds to a flux reduction of about 30 percent (this is limb darkening alone, and the geometric projection factor cos *i* is not included).\nA rough estimate of the 4*π* averaged bolometric disk luminosity can also be obtained from M*v**c**o**r**r* = +2.53. Adopting an average disk temperature of about 28,000 K (the UV continuum indicates ∼ 34,000 K, although this is too hot to match the optical fluxes), the corresponding bolometric correction is *B*.*C*. ∼ − 2.7, and *L**b**o**l* ∼ 92.9 *L*⊙, in fair agreement with the previous estimate. In this comparison, we neglected the effects of temperature stratification in the accretion disk.\nThe mass accretion rate\n=======================\nOne could in principle estimate the mass accretion rate $\\dot{M}$ by comparing the observed spectral distribution with that of proper accretion disk models. This approach is not viable in our case for the following reasons: a) the spectral index of model disks that consist of blackbodies is hardly sensitive to the mass accretion rate $\\dot{M}$; b) most models providing $\\dot{M}$ as a function of the continuum slope do not include the case of a massive WD with high accretion rates; c) disk models depend on a large number of parameters, so that model fitting does not generally provide unique results.\nAlternatively, if, as in our case, one can make the reasonable assumption that the disk is heated by viscous dissipation of gravitational energy, the mass accretion rate $\\dot{M}$ can be obtained from the relation:\n \n$$\\dot{M} = (2R\\_1 L\\_{disk})/(GM\\_1)$$ \nWith this method, $\\dot{M}$ is not model dependent but the knowledge of *L**d**i**s**k* and *M*1 is required.\nNumerically, $\\dot{M}$ can be represented by:\n \n$$\\dot{M}=5.23 \\times 10^{-10} \\phi L\\_{disk}/L\\_{\\odot} \\quad \\rm{with}$$ \n \n*ϕ* = (*R*1/*M*1)/(*R*1*o*/*M*1*o*) = 0.1235 ⋅ *R*1/*M*1, \nwhere R1 is the WD radius in 10− 3 R⊙, R1*o* = 8.10 × 10− 3 R⊙ is the radius of a WD of mass M1 = 1.0 M⊙, and M1 is in solar masses.\nWe obtained average values for R1 as a function of M1 from various WD radius-mass relations in the literature (Hamada and Salpeter 1961; Nauenberg 1972; Anderson 1988; Politano et al. 1990; Livio 1994). By fitting a quadratic function to these average values, for M1 in the range 1.0 to 1.4 M⊙, we found that:\n \n*R*1/*R*⊙ = − 0.01315*M*12 + 0.01777*M*1 + 0.00347. \nThe upper curve in Fig. 2 is a plot of the standard R1 ∼ M1− 1/3 relation, which is generally valid for M1 ≤ 1.0 M⊙, while the lower curve represents the quadratic fit to the average values for M1 in the range 1.0 to 1.4 M⊙, with a relative uncertainty of *σ**R*1/R1 = 0.07.\nColumns 2, 3, and 4 in Table 3 indicate R1, *ϕ*, and $\\dot{M}$ for *M*1 in the range 1.00 - 1.4 *M*⊙. We note that *ϕ* = 1.00 for M1=1.0 M⊙, and decreases to about 0.23 for M1=1.4 M⊙. The accretion rate $\\dot{M}$ in Table 3 is calculated for the (present) luminosity *L**d**i**s**k* ∼ 70 L⊙, derived from UV data. In the case of a massive WD, the mass accretion rate is close to 1.1 ± 0.3 × 10− 8 M⊙ *y**r*− 1. The uncertainty in $\\dot{M}$ is of the order of 23 percent and derives from the uncertainties in the distance, the correction for the inclination, and the value of R1.\ncccccc\nM1 & R1 & *ϕ* & $\\dot{M}$ & M*i**g**n* & *τ* \n(M⊙) & (10− 3R⊙)& &(10− 8M⊙yr− 1) & (10− 7M⊙) & (yrs) \n \n1.00 & 8.10 & 1.000 & 3.66& 108.78 & 297.2 \n1.05 & 7.63 & 0.897 & 3.28& 92.88 & 283.2 \n1.10 & 7.10 & 0.797 & 2.92& 77.75 & 266.3 \n1.15 & 6.51 & 0.699 & 2.56& 63.32 & 247.3 \n1.20 & 5.85 & 0.602 & 2.20& 49.19 & 223.6 \n1.25 & 5.14 & 0.508 & 1.86& 35.21 & 189.3 \n1.30 & 4.35 & 0.413 & 1.51& 22.22 & 147.1 \n1.33 & 3.85 & 0.357 & 1.31& 14.78 & 112.8 \n1.35 & 3.51 & 0.321 & 1.17& 10.35 & 88.5 \n1.36 & 3.33 & 0.302 & 1.11& 8.26 & 74.4 \n1.37 & 3.15 & 0.284 & 1.04& 6.35 & 61.1 \n1.38 & 2.96 & 0.265 & 0.97& 4.61 & 47.5 \n1.39 & 2.78 & 0.247 & 0.90& 3.06 & 34.0 \n1.40 & 2.60 & 0.229 & 0.84& 1.78 & 21.2 \nAn independent confirmation of these values for the mass accretion rate can be obtained from the mass accretion rate versus boundary layer luminosity ($\\dot{M}$-L*B**L*) relation given in Table 2 of Patterson and Raymond (1985), if one assumes that L*B**L* ∼ L*d**i**s**k*. Their Table 2 indicates that (for M1=1.0) a mass accretion rate $\\dot{M}$ of 1.6 × 10− 8 M⊙ *y**r*− 1, close to that derived above, would correspond to a luminosity of 1.3 × 1035 erg s− 1. With the same $\\dot{M}$, a larger luminosity is expected by extrapolation to M1 values close to 1.3 M⊙, in close agreement with the value found for L*d**i**s**k* (2.70 × 1035 erg s− 1).\nThe L1640-$\\dot{M}$ relation reported in the same Table 2 does not, however, provide a consistent result. The average (de-reddened) flux on earth in the HeII 1640 line is 6.32 × 10− 13 erg cm− 2 s− 1 (Paper I) and the corresponding luminosity for d=3500 pc is 9.3 × 1032 erg s− 1. Table 2 of Patterson and Raymond (1985) indicates instead a L1640 = 1.2 × 1032 erg s− 1 for $\\dot{M}$=1018 g s− 1=1.587 × 10− 8 M⊙ yr− 1, M1=1, and extrapolation to higher M1 values does not appear to be able of reproducing the observed HeII luminosity. In T Pyx there is probably a nebular contribution to L1640 that is added to that produced by the BL. It should also be emphasized that the HeII 1640 emission line exhibits significant changes, while the UV continuum remains constant in intensity and slope (see Paper I). This behavior casts doubts about the suitability of the L1640-$\\dot{M}$ method for the specific case of T Pyx.\nAnother estimate of $\\dot{M}$ can be obtained by direct application of the M*v*-$\\dot{M}$ relations reported in the literature. WLTO (1987) derived a relation between the absolute optical magnitude and the accretion rate for the ``reference\" disk model (Warner, 1987), after correction for the disk inclination:\n \n$$M\\_v^{corr.}=-9.48 -5/3 \\cdot log(M\\_1\\cdot\\dot{M})$$ \nwhere M1 is in solar masses and $\\dot{M}$ is in solar masses per year. In the case of T Pyx, for M1 in the range 1.25-1.4 M⊙, one obtains values of $\\dot{M}$ close to 5.0 × 10− 8 M⊙ yr− 1. It should be noted, however, that in the M*v*-$\\dot{M}$ relation given by WLTO there is no explicit dependence on R1, which strongly depends on M1, especially for massive WDs. If $\\dot{M}$ is corrected for the factor *ϕ*=(R1/M1)/(R1*o*/M1*o*) one obtains that $\\dot{M}$(1.25) = 3.50 × 10− 8 M⊙ yr− 1 and $\\dot{M}$(1.40) = 1.60 × 10− 8 M⊙ yr− 1. If one instead adopts the Paczynski and Schvarzenberg-Czerny (1980) correction for *i*, the corresponding $\\dot{M}$ values are 2.85 × 10− 8 M⊙ yr− 1 and 1.22 × 10− 8 M⊙ yr− 1 respectively.\nLipkin et al. (2001) improved the $\\dot{M}$-M*v* relations presented by Retter and Leibowitz (1998) and Retter and Naylor (2000) and derived the new relation\n \n$$\\dot{M}\\_{17} = M\\_1^{-4/3} \\times 10^{(5.69 - 0.4M\\_v^{corr})}$$ \nwhere $\\dot{M}\\_{17}$ is the mass transfer rate in 1017 g s− 1 and M*v**c**o**r**r* is the inclination-corrected absolute magnitude of the disk. The factor M1− 4/3 comes from the factor R1/M1 in the assumption, generally valid for WDs with M ≤ 1 M⊙, that MR3=const. However, this R1-M1 relation (polytropes) is not applicable if the WD is massive because in this case R1 decreases with a far steeper slope as M1 increases (see Sect. 6 and Fig. 2). The adoption of the standard R1 ∼ M1− 1/3 relation for massive WD has the effect of overestimating R1 and in turn overestimating $\\dot{M}$. This is especially true for M1 ∼ 1.4, for which the difference between the R1 values estimated from the R1 ∼ M1− 1/3 relation and the more general R1 - M1 relations (Hamada and Salpeter 1961; Nauenberg 1972; Anderson 1988; Politano et al. 1990; Livio 1994) reaches its maximum value (Fig. 2).\nThe Lipkin et al. (2001) formula corrected for the proper R1/M1 ratio in massive white dwarfs according to the *ϕ* factors of Table 3, provides values of $\\dot{M}$ = 1.38 × 10− 8 and 0.59 × 10− 8 for white dwarfs with M1= 1.25 M⊙ and M1=1.40 M⊙, respectively. These values are about one half of the corresponding values obtained from the WLTO formula. It should be noted, however, that, in the derivation of Lipkin et al. (2001) formula there is the assumption that L*v*/L*b**o**l* ∼ 0.14. In the case of T Pyx, this L*v*/L*b**o**l* ratio is lower by a factor of about two and therefore the $\\dot{M}$ derived using their formula is underestimated by a similar amount. In conclusion, the various methods provide quite consistent results and we can confidently assume that the mass accretion rate derived from ``recent\" UV and optical data is 1.86 ± 0.4 × 10− 8 M⊙*y**r*− 1 for M1=1.25, and 0.84 ± 0.2 × 10− 8 M⊙*y**r*− 1 for M1=1.40 M⊙. For an intermediate value of M1=1.33 M⊙ we obtain that $\\dot{M}$ ∼ 1.3 × 10− 8 M⊙ *y**r*− 1.\nTable 4 summarizes the adopted values of the basic parameters and their estimated errors.\n| Quantity | value |\n| --- | --- |\n| M*v**m**a**x* | -6.81 ± 0.16 |\n| m*v**m**a**x* | 6.7 ± 0.1 |\n| A*v* | 0.79 ± 0.06 |\n| distance | 3500 ± 350 pc |\n| m*v* | 15.30 ± 0.05 |\n| M*v* | 1.79 ± 0.21 |\n| *i* | 25 ± 5 degrees |\n| Δm*v*(i) | 0.74 ± 0.06 |\n| M*v**c**o**r**r* | 2.53 ± 0.23 |\n| L*U**V* | 74 ± 15 L⊙ |\n| L*d**i**s**k* | 70 ± 15 L⊙ |\n| M1 | ∼ 1.36 M⊙ |\n| $\\dot{M}$ | 1.1 ± 0.25 × 10− 8 M⊙ yr− 1 |\n| $\\dot{M}\\_{pre-OB}$ | 2.2 ± 0.5 × 10− 8 M⊙ yr− 1 |\nThe pre-1966-outburst mass accretion rate\n=========================================\nSchaefer (2005) published a large collection of quiescent B-magnitudes for the recurrent novae T Pyx and U Sco using information from archival plates, from the literature, and from his own collection of CCD magnitudes. The photometric data of T Pyx start in 1890 and therefore cover all of the known inter-outburst intervals.\nAccording to these data, during the inter-outburst phase in the years 1945-1966, T Pyx was a factor of 2 brighter in the B band than during the present quiescent phase after the 1967 outburst. Since no significant changes in the (B-V) color index were found during both of these quiescence phases, one can safely deduce that in 1945-1966 T Pyx was a factor of 2 brighter also in terms of visual and bolometric flux. Using the WLTO and the Lipkin et al. (2001) relations, one finds that the mass accretion rate in epochs pre-1966-outburst, hereinafter $\\dot{M}\\_{pre-OB}$, was about twice the values of $\\dot{M}$ for post 1967, obtained in the previous section.\nTherefore, we estimate that the mass accretion rate $\\dot{M}\\_{pre-OB}$ during the last pre-outburst interval (1945-1966) was between 1.68 ± 0.4 × 10− 8 M⊙ *y**r*− 1 and 3.72 ± 0.8 × 10− 8 M⊙ *y**r*− 1, for M1=1.25 and M1=1.40 respectively.\n**These are the values that are compared in Sects. 8 and 9 with the theoretical models of nova.** The post-1967 accretion values (Table 3) are considered in Sect. 16 in the context of the ``missing\" outburst and the next expected outburst.\nThe theoretical ignition mass and the accreted mass\n===================================================\nA nova outburst occurs when, due to the gradual accumulation of H-rich material on the surface of the white dwarf, the pressure at the bottom of the accreted layer becomes sufficiently high for nuclear ignition of H to begin ( Shara 1981, Fujimoto 1982, MacDonald 1983). Since the radius R1 of the white dwarf varies approximately as M1− 1/3 for M1 ≤ 1.0 M⊙, the critical pressure for ignition [eq:PIGN]\n \n*P**i**g**n* = (*G* ⋅ *M*1 ⋅ *M**i**g**n*)/(4*π**R*14) \ncorresponds to a critical ignition mass M*i**g**n* that decreases approximately as M1− 7/3, while, for more massive WDs M*i**g**n*, decreases with a steeper slope (see Table 3 and Fig. 2). In any case, massive white dwarfs need to accrete a small amount of mass to reach the critical conditions.\nFor a given M1 value, the critical ignition mass has been calculated by various authors and the reported values (for a given M1) differ from each other mostly in the choice of the critical pressure at the base of the accreted envelope, which varies between 2.0 × 1019 and 6.0 × 1019 dynes cm− 2 (see Gehrz et al. 1998; Starrfield et al. 1998; Livio and Truran 1992; Hernanz and Jose’ 1998; Truran 1998). These studies indicate that a lower limit to *M**i**g**n* (for a massive white dwarf with *M*1 close to 1.4 *M*⊙) is in the range 2.0 - 4.0 × 10− 6 *M*⊙. However, as first pointed out by MacDonald (1983) and in more detail by Shara (1989) and Prialnik and Kovetz (1995), the behavior of a CN eruption and in particular the critical mass depends (apart from the WD mass) also on the mass accretion rate (since the WD is heated by accretion) and the temperature of the isothermal white dwarf core.\nTownsley and Bildsten (2004) confirmed that the earlier prescriptions for ignition, based on the simple scaling M*i**g**n* ∝ R4 M1− 1 for a unique P*i**g**n*, are inadequate and that a system of a given M1 mass can have value of M*i**g**n* that varies by a factor of 10 for different $\\dot{M}$. At the high $\\dot{M}$ values typical of most CVs, the critical pressure can decrease to values as low as 3 × 1018 dyn cm− 2, and the critical mass decreases accordingly. Thus, in a massive white dwarf accreting at significantly high rates, one expects a value for M*i**g**n* as low as 2-4 × 10− 7 M⊙.\nThe critical envelope mass *M**i**g**n* as a function of M1 and $\\dot{M}$ can be numerically approximated to be :\n \n$$\\begin{aligned}\nlog M\\_{ign}=-2.862+1.542\\cdot M\\_1^{-1.436}ln(1.429-M\\_1)+\\nonumber\\\\\n-0.19(log\\dot{M}+10)^{1.484}\\end{aligned}$$ \n(Kahabka and van Den Heuvel 2006) where M1 is in M⊙ and $\\dot{M}$ is in M⊙ *y**r*− 1.\nTable 5 indicates (for various M1 and hence R1 values) $\\dot{M}\\_{pre-OB}$, the theoretical *M**i**g**n*, the accreted mass M*a**c**c**r*=Δt ⋅ $\\dot{M}\\_{pre-OB}$, where Δt=22 yrs is the pre-1966 inter-outburst interval, and *τ*=*M**i**g**n*/$\\dot{M}\\_{pre-OB}$, that is, the expected recurrence time in years. The mass accretion rate and *M**i**g**n* were calculated using the quadratic fit for R1 as a function of M1 for massive white dwarfs derived in Sect. 6 (see also Table 3), and not the approximation R1 ∝ M1− 1/3. Table 5 clearly shows that, after allowances for errors in the estimate of $\\dot{M}\\_{pre-OB}$, the expected recurrence time *τ* = M*i**g**n*/$\\dot{M}\\_{pre-OB}$ is close to the observed value (22 yr) for M1 ∼ 1.36 - 1.38 M⊙ corresponding to M*i**g**n* and M*a**c**c**r* in the range 3.0 to 6.0 × 10− 7 *M*⊙.\nThis agrees with the estimate of Kato (1990) that the white dwarf mass for T Pyx is between 1.3 and 1.4 M⊙, while in a subsequent paper Kato and Hachisu (1991) assumed M1=1.33 M⊙.\nWe confirmed the results for M*i**g**n* using the approximate relation for the ignition mass as a function of M1 and $\\dot{M}$ given by Kolb et al. (2001):\n \n$$M\\_{ign}=4.4 \\times 10^{-4}\\cdot R\\_1^4\\cdot\nM\\_1^{-1}\\cdot\\dot{M}^{-1/3}$$ \n(where M*i**g**n* and M1 are in M⊙, R1 is in 109 cm, and $\\dot{M}$ is in 10− 9 M⊙ yr− 1) and obtained M*i**g**n* = 2.72 × 10− 7 M⊙. A similar result is also obtained by graphical interpolation in Fig. 5 of Kahabka and Van Den Heuvel (1997), which for M1 ∼ 1.37 and $\\dot{M}$ ∼ 2 × 10− 8 M⊙ yr− 1 provides an ignition mass close to 3.0 × 10− 7 M⊙ (we note, however, that these parameters fall within a region in which only weak flashes are expected). In addition, we derived P*i**g**n* as a function of $\\dot{M}$ and the WD temperature from Fig. 3 of Yaron et al. (2005), and found that P*i**g**n* ∼ 4.8 × 1018 for $\\dot{M}$ = 2 × 10− 8M⊙ yr− 1 (we note that in this regime there is a weak dependence on the WD temperature). The insertion of this value of P*i**g**n* in equation [eq:PIGN] gives M*i**g**n* = 3.8 × 10− 7 M⊙.\nThe consistency of all these results confirms that in the case of T Pyx the ignition mass was close to 4.5 × 10− 7 M⊙.\n| | | | | |\n| --- | --- | --- | --- | --- |\n| M1 | $\\dot{M}\\_{pre-OB}$ | M*i**g**n* | M*a**c**c**r* | *τ* |\n| (M⊙) | (10− 8M⊙yr− 1) | (10− 7M⊙) | (10− 7M⊙) | (yrs) |\n| 1.00 | 7.32 | 77.2 | 16.10 | 105.5 |\n| 1.05 | 6.56 | 67.6 | 14.43 | 103.0 |\n| 1.10 | 5.84 | 56.2 | 12.84 | 96.2 |\n| 1.15 | 5.12 | 45.7 | 11.26 | 89.2 |\n| 1.20 | 4.40 | 36.3 | 9.68 | 82.5 |\n| 1.25 | 3.72 | 25.7 | 8.18 | 69.1 |\n| 1.30 | 3.02 | 16.2 | 6.64 | 53.6 |\n| 1.33 | 2.62 | 11.0 | 5.76 | 42.0 |\n| 1.35 | 2.34 | 7.69 | 5.15 | 32.8 |\n| 1.36 | 2.22 | 6.14 | 4.88 | 27.7 |\n| 1.37 | 2.08 | 4.73 | 4.58 | 22.7 |\n| 1.38 | 1.94 | 3.44 | 4.27 | 17.7 |\n| 1.39 | 1.80 | 2.29 | 3.96 | 12.7 |\n| 1.40 | 1.68 | 1.33 | 3.69 | 7.9 |\nIn the recurrent nova T Pyx, the theoretical ignition mass and observed accreted mass are in excellent agreement for a massive WD, which provides new, independent support of the TNR theory. Theoretical expectations have received only limited confirmation because studies of the system parameters (in U Sco, T Cr B) have provided contradictory results.\nComparison with the nova models of Yaron et al. (2005)\n======================================================\nThe realization that three basic and independent parameters, the white dwarf mass M1, the temperature of its isothermal core T*c*, and the mass transfer rate $\\dot{M}$, control the behavior of a CN eruption, and the improvements in computer power and codes, has enabled an increase in sophistication in simulating a nova outburst. Prialnik and Kovetz (1995) presented an extended grid of multicycle nova evolution models that have been extensively used by researchers. Each observed nova characteristic (e.g. peak luminosity, recurrence time, duration of the high luminosity phase, outburst amplitude, mass of ejecta, average outflow velocity, etc.) can be reproduced by a particular combinations of values of M1,T*c*, and $\\dot{M}$. Following this earlier study, Yaron et al. (2005) extended and refined the resolution in the grid of models, including a considerable number of new parameter combinations. The full grid covers the entire range of observed characteristics, even those of peculiar objects.\nBy matching the observed characteristics of a particular nova with its theoretical counterpart, it is therefore possible to derive information about the mass and temperature of the white dwarf and its average accretion rate. Therefore, the grids in Tables 2 and 3 of Yaron et al. (2005) can be used to determine a set of fundamental parameters that provides the closest agreement with the observed data.\nIn the case of T Pyx, a comparison of the observed parameters for the eruption phase (the recurrence time *τ*, the outburst amplitude A, the decay rate t3, the expansion velocity v*e**x**p*) with the corresponding values in the grid clearly indicates that the closest agreement corresponds to values that are intermediate in between 1.25 and 1.40 for M1, and -8.0 and -7.0 for log$\\dot{M}$; at these high $\\dot{M}$ values, the effects of the core temperature T*c* are minor. Since a denser grid is not available, we calculated intermediate values by graphical and non-linear interpolation in the published values. A satisfactory agreement with the observed data comes from a model (model A) with M1=1.36 and log$\\dot{M}$=7.5 ($\\dot{M}$=3.2 × 10− 8), which provides: *τ* ∼ 20, v*e**x**p* ∼ -1200, A ∼ 7.6, t3 ∼ 13, and *M**i**g**n* ∼ 5.0 × 10− 7. Following our interpolation of the data, Yaron (2007) kindly provided us with results obtained by execution of the code for other intermediate models, that is, WD masses between 1.30 and 1.33, log$\\dot{M}$ between -7 and -8, and different WD temperatures. The closest agreement with the observed values originates in a model with M1=1.33, T1= 107, log$\\dot{M}$=-7.3 (model B), although the model with M1=1.30, T1= 107, and log$\\dot{M}$=-7.15 (model C) is also acceptable. In Table 6, we also include, as an illustration, a model directly obtained from the published Tables 2 and 3, for M1= 1.4 M⊙ and $\\dot{M}$ = 1.0 × 10− 8 (model D).\nThe"},"data_id":{"kind":"string","value":"arxiv_0000356"}}},{"rowIdx":94,"cells":{"content":{"kind":"string","value":"Schottky Signal Modification as a Diagnostic Tool for Coherent Electron Cooling\n===============================================================================\nCoherent electron cooling is a promising technique to cool high-intensity hadron bunches by imprinting the noise in the hadron beam on a beam of electrons, amplifying the electron density modulations, and using them to apply cooling kicks to the hadrons. The typical size for these perturbations can be on the *μ*m scale, allowing us to extend the reach of classical stochastic cooling by several orders of magnitude. However, it is crucial to ensure that the electron and hadron beams are longitudinally aligned within this same *μ*m scale. In order to provide fast feedback for this process, we discuss the extension of signal suppression to coherent electron cooling, and show in both theory and simulation that certain components of the spectral noise in the hadron beam will be predictably modified at the several percent level, which may be detected by observations of the radiation of the hadron beam.\nIntroduction\n============\nIn high-intensity hadron storage rings, intrabeam scattering (IBS) and beam-beam effects will degrade the beam emittance over the length of the store, limiting machine luminosity. In particular, at the planned Electron-Ion Collider (EIC), the IBS times are expected to be at the timescale of a couple hours, and so some form of strong hadron cooling is necessary to achieve the physics goals.\nThe proposed method in this case is microbunched electron cooling (MBEC), a particular form of coherent electron cooling (CeC). This was first introduced in, and the theory has since been developed extensively in. The premise of MBEC is that the hadron beam to be cooled copropagates with an electron beam in a straight “modulator” section, during which time the hadrons will provide energy kicks to the electrons. The two beams are then separated, and the electrons pass through a series of amplifiers to change this initial energy modulation into a density modulation and amplify it. The hadrons travel through their own chicane before meeting the electrons again in a straight “kicker” section. Here, the amplified density modulations in the electron beam provide energy kicks to the hadrons. By tuning the hadron chicane so that the hadron delay in travelling from the modulator to the kicker is energy-dependent, we may arrange it so that the energy kick that the hadron receives in the kicker tends to correct initial energy offsets. If the chicane also gives the hadrons a delay dependent on their transverse phase-space coordinates and if there is non-zero dispersion in the kicker, then the transverse emittance of the hadron beam can also be cooled. In the current design of an MBEC Cooler for the EIC, the typical scale of the electron density modulations at the top energy will be ∼ 1*μ*m. This corresponds to about 4 orders of magnitude higher bandwidth than can be achieved with microwave stochastic cooling, allowing the cooling of dense hadron bunches, but also making alignment a challenge.\nIt is important that the hadron arrives in the kicker at the same time as the density perturbations which it had induced in the electron beam, or else it will receive entirely uncorrelated energy kicks. Comparing the ∼ 100m distance between modulator and kicker to the ∼ 1*μ*m density perturbations in the electron beam, we see that the transit times of the electrons and hadrons must be maintained at a level of ten parts per billion. In order to commission and operate such a system, it is necessary to have some way to quickly measure the relative alignment of the electron and hadron beams at the sub-micron scale. Directly observing cooling would require waits on the timescale of hours, which would make commissioning painful and prevent any sort of fast feedback during operations. The method proposed here is to make use of “signal modification,” an extension of the well-known signal suppression of microwave stochastic cooling to the case of MBEC cooling. The principle of this method is that after the hadron beam has received its cooling kicks, it will propagate to a “detector” where the power of the hadron beam at particular wavelengths may be measured. If the hadron beam is well-aligned with the electron beam, this will produce a predictable change in the spectral content of the hadron beam on a single-pass basis.\nAs our model, we assume that the cooling section consists of a modulator, where the hadron beam imprints on the electrons; a kicker, where the electrons provide an energy kick back to the hadrons; and a detector, where we will observe the density spectrum in the hadron beam. See Fig. [fig:layout].\n\nIn Section [sec:thry], we provide a theoretical derivation of signal modification. In Section [sec:sim], we discuss simulation tools to model this process, and find good agreement with the theoretical predictions. In Section [sec:detection], we discuss what will be needed to measure such a signal experimentally. We present our outlook in Section [sec:conclude] and conclude.\nNote that, although this paper focuses on MBEC, the general form of these results will hold for coherent electron cooling with other amplification mechanisms.\nTheory\n======\nWe derive here the theory of signal modification by directly propagating the particles themselves from the modulator to the detector with arbitrary 6-dimensional transfer matrices. Alternative derivations using the longitudinal Vlasov equation are presented in Appendix [app:alternative]. In subsection [subsec:thrydecohere], we comment on decoherence of the signal when observing a range of frequencies and in subsection [subsec:multikick], we discuss an extension where the kicks are distributed throughout the length of the kicker.\nWe use phase-space coordinates *x*, *x*ʹ, *y*, *y*ʹ, *z*, and *η*, where the first four are the transverse positions and angles, *z* is the particle’s longitudinal position in the bunch, with positive *z* values corresponding to the head of the bunch, and *η* is the fractional energy offset of the particle.\nIn order to characterize the cooling process, we follow and define a longitudinal wake function in the 1D approximation, such that the energy kick a hadron receives in the kicker is the convolution of this wake function with the longitudinal distribution of hadrons in the modulator. Explicitly, the fractional energy kick Δ*η* received by a hadron at longitudinal position *z* within the bunch at the kicker is given by\n \n$$\\begin{aligned}\n\\label{eqtn:wake\\_def}\n\\Delta\\eta(z) = \\frac{q^2}{E\\_0} \\int\\_{-\\infty}^{\\infty}w(z+\\Delta z-z')n(z') dz'\\end{aligned}$$ \nwhere *q* is the hadron charge, *E*0 is the nominal hadron energy, *n*(*z*) is the longitudinal hadron density in the modulator, and Δ*z* is the difference in modulator-to-kicker longitudinal delay between the hadron and electron beams. We also identify a corresponding impedance\n \n$$\\begin{aligned}\n\\label{eqtn:imped\\_def}\n Z(k) = -\\frac{1}{c}\\int\\_{-\\infty}^{\\infty} w(z) e^{-ikz} dz\\end{aligned}$$ \nIn order for these simple longitudinal wakes to hold, we require coherent oscillation of the beams, and therefore the electric field at our frequency of interest must be large compared to the beam sizes. In particular, we require :\n \n$$\\begin{aligned}\n\\label{eqtn:1D}\n \\Sigma\\_{\\perp} \\lesssim \\frac{\\beta\\gamma\\lambda}{2}\\end{aligned}$$ \nwhere Σ⊥ is the transverse beam size, *β* is the relativistic beta, *γ* is the relativistic gamma, and *λ* is the beam oscillation wavelength. Examining the values in Tab. [tab:param], we see that the largest beam size is that of the protons in the modulator and kicker (0.95mm), and the gamma factor is 293, so our approximation is good for wavelengths above ∼ 6.5*μ*m. As we will see, we will be interested mainly in wavelengths of 6*μ*m, at the limit of applicability of the above. However, we note that the requirement of Eq. [eqtn:1D] is likely not a strict one—as it was shown in, in the case when there is not amplification, the 3D theory gives exactly the same result as the 1D model which we employ here.\nIn order for us to neglect Landau damping in the electron beam amplifiers, one usually requires :\n \n$$\\begin{aligned}\n\\label{eqtn:landau\\_damp}\n k \\ll k\\_D,\\end{aligned}$$ \nwhere *k* is the wavenumber of our signal, *k**D* ≡ *ω**p*/*c**σ**β* is the Debye wavenumber, and *σ**β* is the spread in relativistic beta. This electron beam has *k**D* = 4 × 106m− 1 in the amplifiers, larger than the *k* = 106m− 1 of our microbunching. One also has to keep in mind that the concept of Landau damping, with the condition of Eq. [eqtn:landaudamp], is only valid asymptotically, after many plasma oscillations, which is not the case of the microbunching coherent cooling where the electron beam executes $\\frac{1}{4}$ of the plasma oscillation in the amplifier sections. A more accurate analysis of when one can ignore the Landau damping for this case was carried out in Section IV of.\nIn order to ensure that the microbunching is not washed out by the energy spread and divergence of the beam within any of the straight sections, we require :\n \n$$\\begin{aligned}\n \\lambda \\gg \\frac{2L}{\\beta^3\\gamma^2}\\sigma\\_{\\eta}\\end{aligned}$$ \nand\n \n$$\\begin{aligned}\n \\lambda \\gg \\frac{L}{\\beta^2}\\sigma\\_{x'}^2\\end{aligned}$$ \nwhere *σ**η* is the fractional energy spread of the beam, *σ**x*ʹ is the beam divergence, and *L* is the length of a drift. Comparison with the parameters in Tab. [tab:param] shows that these are met for both the electrons and hadrons in all the straight sections.\nWe assume that there is no transverse correlation in the electron or hadron beam. Further discussion of the effects which this might have is available in.\nWe focus our attention on the peak region of the electron and hadron beams and take the limit of a longitudinally infinite and uniform plasma. Since the typical wake wavelength is on the order of a few microns, and the typical bunch lengths are a few mm or longer, this is a reasonable approximation. Finally, we assume that the hadron beam enters the modulator with no correlated structures on the scale of the wake wavelength. Although such structures will be generated within the kicker, their characteristic size is on the micron scale, much less than the millimeter scale longitudinal motion per turn, washing out any memory of the kick by the time the beam enters the modulator again.\nAs illustrated in Fig. [fig:layout], we take the hadrons to have transfer matrix *M**M**K* between modulator and kicker and *M**K**D* between kicker and detector, with the transfer matrix between modulator and detector given by *M**M**D*.\nWe treat our particles as existing in a region of length *L*, much larger than any length scale associated with the wake function, and assume periodic boundary conditions, so that we may arbitrarily shift the limits of integration in our integrals. In this model, we also consider the full 6-dimensional evolution of the hadron beam and ignore any collective effects during beam transport except for the electron-hadron interactions characterized by the wake function, as discussed above. We write the evolution of a hadron’s position between modulator and detector as\n \n$$\\begin{aligned}\n\\label{eqtn:delays}\nz\\_d^{(i)} = &z\\_m^{(i)} + M^{MD}\\_{5u}\\vec{x}\\_u^{(i)}\\\\\\nonumber\n+ &M^{KD}\\_{56}\\sum\\_j \\frac{q^2}{E\\_0} w(z\\_m^{(i)} + M^{MK}\\_{5u}\\vec{x}\\_u^{(i)} + \\Delta z - z\\_m^{(j)})\\end{aligned}$$ \nwhere *z**m*(*i*) is the longitudinal position of particle *i* in the modulator, *z**d*(*i*) is its position within the detector, and *x⃗*(*i*) are the phase-space coordinates of particles *i* in the modulator. We use the convention that the repeated “*u*” subscript refers to sums over the 5 phase-space coordinates *excluding* the longitudinal position. The summation over *j* is over all particles within the hadron beam. The first two terms of this equation describe the modulator to kicker beam evolution by a simple transfer matrix, while the final term gives the contribution to delay due to the extra energy kick our particle receives in the kicker due to the wakes of all particles in the beam, including itself.\nAt the detector, the longitudinal density of the hadron beam is\n \n$$\\begin{aligned}\n\\label{eqtn:rho\\_d}\nn(z) = \\sum\\_i \\delta(z - z\\_d^{(i)})\\end{aligned}$$ \nwith corresponding density in Fourier space\n \n$$\\begin{aligned}\n\\label{eqtn:rho\\_tilde\\_d}\n\\tilde{n}(k) &= \\int\\_{-\\infty}^{\\infty}\\sum\\_i e^{-ikz} \\delta(z - z\\_d^{(i)}) dz\\\\\\nonumber\n&=\\sum\\_i e^{-ikz\\_d^{(i)}}\\end{aligned}$$ \nThe power in the hadron spectrum at a given wavenumber is then given by\n \n$$\\begin{aligned}\n\\label{eqtn:pwr}\n|\\tilde{n}(k)|^2 = &\\sum\\_{i,a}e^{-ik\\big[z\\_d^{(i)} - z\\_d^{(a)}\\big]}\\\\\\nonumber\n=N + &\\sum\\_{i \\neq a}e^{-ik\\big[z\\_m^{(i)} - z\\_m^{(a)} + M^{MD}\\_{5u}\\big(\\vec{x}\\_u^{(i)} - \\vec{x}\\_u^{(a)}\\big)\\big]}\\\\\\nonumber\n&\\times e^{-ikM^{KD}\\_{56}q^2/E\\_0\\sum\\_j w\\big(z\\_m^{(i)} + M^{MK}\\_{5u}\\vec{x}\\_u^{(i)} + \\Delta z - z\\_m^{(j)}\\big)}\\\\\\nonumber\n&\\times e^{ikM^{KD}\\_{56}q^2/E\\_0\\sum\\_j w\\big(z\\_m^{(a)} + M^{MK}\\_{5u}\\vec{x}\\_u^{(a)} + \\Delta z - z\\_m^{(j)}\\big)}\\end{aligned}$$ \nwhere we have substituted in the expression for *z**d*(*i*) given in Eq. [eqtn:delays] and used the fact that the *i* = *a* terms in the sum are all equal to 1, giving us the *N* out front, where *N* is the number of particles in the length-*L* section of the beam.\nTypically, the kick from the wake is small, and so we may Taylor-expand the final two exponentials above to linear order. We thereby obtain\n \n$$\\begin{aligned}\n\\label{eqtn:pwr\\_taylor}\n|\\tilde{n}(k)|^2 &= N + \\sum\\_{i \\neq a}e^{-ik\\big[z\\_m^{(i)} - z\\_m^{(a)} + M^{MD}\\_{5u}\\big(\\vec{x}\\_u^{(i)} - \\vec{x}\\_u^{(a)}\\big)\\big]}\\\\\\nonumber\n\\times \\big[1 &- ikM^{KD}\\_{56}\\frac{q^2}{E\\_0}\\sum\\_j w\\big(z\\_m^{(i)} + M^{MK}\\_{5u}\\vec{x}\\_u^{(i)} + \\Delta z - z\\_m^{(j)}\\big)\\\\\\nonumber\n&+ ikM^{KD}\\_{56}\\frac{q^2}{E\\_0}\\sum\\_j w\\big(z\\_m^{(a)} + M^{MK}\\_{5u}\\vec{x}\\_u^{(a)} + \\Delta z - z\\_m^{(j)}\\big)\\big]\\end{aligned}$$ \nwith the effect of second-order terms considered in Appendix [app:secondorder].\nWe now wish to take the expectation value of the beam power, requiring integrals over the 12 phase space coordinates of particles *i* and *a* and the longitudinal position of particle *j*. However, note that the dependence on *z**m*(*j*) appears only in the argument of the wake functions. If the total integral of the wake is zero, then, if particle *j* is distinct from both particles *i* and *a*, this integral will evaluate to 0. We then need only consider the terms *j* = *i* and *j* = *a* in those sums. The beam power can then be written as\n \n$$\\begin{aligned}\n\\label{eqtn:pwr\\_taylor2}\n|\\tilde{n}(k)|^2 &= N + \\sum\\_{i \\neq a}e^{-ik\\big[z\\_{ia} + M^{MD}\\_{5u}\\big(\\vec{x}\\_u^{(i)} - \\vec{x}\\_u^{(a)}\\big)\\big]}\\\\\\nonumber\n\\times [1 &- ikM^{KD}\\_{56}\\frac{q^2}{E\\_0} w(M^{MK}\\_{5u}\\vec{x}\\_u^{(i)} + \\Delta z)\\\\\\nonumber\n&+ ikM^{KD}\\_{56} \\frac{q^2}{E\\_0}w(M^{MK}\\_{5u}\\vec{x}\\_u^{(a)} + \\Delta z)\\\\\\nonumber\n&- ikM^{KD}\\_{56} \\frac{q^2}{E\\_0}w(z\\_{ia} + M^{MK}\\_{5u}\\vec{x}\\_u^{(i)} + \\Delta z)\\\\\\nonumber\n&+ ikM^{KD}\\_{56} \\frac{q^2}{E\\_0}w(-z\\_{ia} + M^{MK}\\_{5u}\\vec{x}\\_u^{(a)} + \\Delta z)]\\end{aligned}$$ \nwhere we have made the definition *z**i**a* ≡ *z**m*(*i*) − *z**m*(*a*). Since we assume a homogeneous hadron bunch, *z**m*(*i*) and *z**m*(*a*) themselves are irrelevant and *z**i**a* has the same probability distribution as them.\nWe note in the above formula that the first three terms have their only *z**i**a* dependence in the leading exponential, and so performing an average over all *z**i**a* will be zero. We therefore need only focus on the 4th and 5th terms. Approximating the *N*(*N* − 1) terms in the above sum as *N*2, the relevant integral for the 4th term is\n \n$$\\begin{aligned}\n\\label{eqtn:4th\\_term}\n&-N^2ikM\\_{56}^{KD} \\int\\_{-L/2}^{L/2} dz\\_{ia}/L \\int\\_{-\\infty}^{\\infty} d^5\\vec{x}^{(i)} d^5\\vec{x}^{(a)} \\rho\\big(\\vec{x}^{(i)}\\big) \\rho\\big(\\vec{x}^{(a)}\\big)\\\\\\nonumber\n&\\times \\frac{q^2}{E\\_0}w(z\\_{ia} + M^{MK}\\_{5u}\\vec{x}\\_u^{(i)} + \\Delta z)\ne^{-ik\\big[z\\_{ia} + M^{MD}\\_{5u}\\big(\\vec{x}\\_u^{(i)} - \\vec{x}\\_u^{(a)}\\big)\\big]}\\end{aligned}$$ \nwhere $\\rho\\big(\\vec{x}\\big)$ the hadron phase-space density in the modulator within the 5 phase-space coordinates excluding longitudinal position.\nApproximating the longitudinal integral as extending from − ∞ to ∞, using the impedance of Eq. [eqtn:impeddef], and making an appropriate change of variables to *z*ʹ ≡ *z**i**a* + *M*5*u**M**K**x⃗**u*(*i*) + Δ*z*, the longitudinal integral in Eq. [eqtn:4thterm] may be evaluated, leaving us with\n \n$$\\begin{aligned}\n\\label{eqtn:4th\\_term\\_2}\n&N^2ikM\\_{56}^{KD} \\frac{q^2c}{E\\_0}\\frac{1}{L} Z(k) e^{ik\\Delta z}\\\\\\nonumber\n&\\times\\int\\_{-\\infty}^{\\infty} d^5\\vec{x}^{(i)} d^5\\vec{x}^{(a)} \\rho\\big(\\vec{x}^{(i)}\\big) \\rho\\big(\\vec{x}^{(a)}\\big)\\\\\\nonumber\n&\\times e^{-ik\\big[M^{MD}\\_{5u}\\big(\\vec{x}\\_u^{(i)} - \\vec{x}\\_u^{(a)}\\big) - M^{MK}\\_{5u}\\vec{x}\\_u^{(i)}\\big]}\\end{aligned}$$ \nTo perform the remaining integrals, we write the evolution of the phase-space coordinates explicitly in terms of action-angle variables and Courant-Snyder parameters at the start of the transfer matrix, finding\n \n$$\\begin{aligned}\n\\label{eqtn:action\\_angle}\nM\\_{5u}\\vec{x}\\_u &= M\\_{51}(\\sqrt{2J\\_x\\beta\\_x}\\cos(\\phi\\_x)+D\\_x\\eta)\\\\\\nonumber\n&+ M\\_{52}(-\\sqrt{2J\\_x/\\beta\\_x}[\\sin(\\phi\\_x)+\\alpha\\_x\\cos(\\phi\\_x)]+D'\\_x\\eta)\\\\\\nonumber\n&+ M\\_{53}(\\sqrt{2J\\_y\\beta\\_y}\\cos(\\phi\\_y)+D\\_y\\eta)\\\\\\nonumber\n&+ M\\_{54}(-\\sqrt{2J\\_y/\\beta\\_y}[\\sin(\\phi\\_y)+\\alpha\\_y\\cos(\\phi\\_y)]+D'\\_y\\eta)\\\\\\nonumber\n&+ M\\_{56}\\eta\\\\\\nonumber\n&\\\\\\nonumber\n&= (M\\_{51}-\\frac{\\alpha\\_x}{\\beta\\_x}M\\_{52})\\sqrt{2J\\_x\\beta\\_x}\\cos(\\phi\\_x)\\\\\\nonumber\n&- M\\_{52}\\sqrt{2J\\_x/\\beta\\_x}\\sin(\\phi\\_x)\\\\\\nonumber\n&+ (M\\_{53}-\\frac{\\alpha\\_y}{\\beta\\_y}M\\_{54})\\sqrt{2J\\_y\\beta\\_y}\\cos(\\phi\\_y)\\\\\\nonumber\n&- M\\_{54}\\sqrt{2J\\_y/\\beta\\_y}\\sin(\\phi\\_y)\\\\\\nonumber\n&+ (D\\_xM\\_{51} + D'\\_xM\\_{52} + D\\_yM\\_{53} + D'\\_yM\\_{54} + M\\_{56})\\eta\\\\\\nonumber\n&\\\\\\nonumber\n\\equiv & \\hat{M}\\_{51}\\hat{x} + \\hat{M}\\_{52}\\hat{x}' + \\hat{M}\\_{53}\\hat{y} + \\hat{M}\\_{54}\\hat{y}' + \\hat{M}\\_{56}\\hat{\\eta}\\end{aligned}$$ \nwith\n \n$$\\begin{aligned}\n\\label{eqtn:m\\_tilde}\n&\\hat{M}\\_{51} \\equiv M\\_{51}-\\frac{\\alpha\\_x}{\\beta\\_x}M\\_{52}\\\\\\nonumber\n&\\hat{M}\\_{52} \\equiv M\\_{52}\\\\\\nonumber\n&\\hat{M}\\_{53} \\equiv M\\_{53}-\\frac{\\alpha\\_y}{\\beta\\_y}M\\_{54}\\\\\\nonumber\n&\\hat{M}\\_{54} \\equiv M\\_{54}\\\\\\nonumber\n&\\hat{M}\\_{56} \\equiv D\\_xM\\_{51} + D'\\_xM\\_{52} + D\\_yM\\_{53} + D'\\_yM\\_{54} + M\\_{56}\\end{aligned}$$ \nFor a Gaussian beam, the *x̂*, *x̂*ʹ, *ŷ*, *ŷ*ʹ, and *η̂* are normally distributed with\n \n$$\\begin{aligned}\n\\label{eqtn:sigmas}\n &\\sigma\\_{\\hat{x}} = \\sqrt{\\epsilon\\_x\\beta\\_x}\\\\\\nonumber\n &\\sigma\\_{\\hat{x}'} = \\sqrt{\\epsilon\\_x/\\beta\\_x}\\\\\\nonumber\n &\\sigma\\_{\\hat{y}} = \\sqrt{\\epsilon\\_y\\beta\\_y}\\\\\\nonumber\n &\\sigma\\_{\\hat{y}'} = \\sqrt{\\epsilon\\_y/\\beta\\_y}\\\\\\nonumber\n &\\sigma\\_{\\hat{\\eta}} = \\sigma\\_{\\eta}\\end{aligned}$$ \nwhere the *ε* are the horizontal and vertical emittances and *σ**η* is the fractional energy spread. In this case, the remaining ten integrals in Eq. [eqtn:4thterm2] may be performed, yielding\n \n$$\\begin{aligned}\n\\label{eqtn:4th\\_term\\_final}\n&ikM\\_{56}^{KD}\\frac{q^2c}{E\\_0} \\frac{N^2}{L} Z(k) e^{ik\\Delta z}\\\\\\nonumber\n&\\times e^{-\\frac{k^2}{2} \\sum\\_{u\\neq5} \\sigma^2\\_{\\hat{x}\\_u}\\Big[\\big(\\hat{M}^{MD}\\_{5u}\\big)^2 + \\big(\\hat{M}^{MD}\\_{5u} - \\hat{M}^{MK}\\_{5u}\\big)^2\\Big]}\\end{aligned}$$ \nA similar procedure may be applied to the fifth term of Eq. [eqtn:pwrtaylor2], resulting in\n \n$$\\begin{aligned}\n\\label{eqtn:5th\\_term\\_final}\n&-ikM\\_{56}^{KD} \\frac{q^2c}{E\\_0}\\frac{N^2}{L} Z(-k) e^{-ik\\Delta z}\\\\\\nonumber\n&\\times e^{-\\frac{k^2}{2} \\sum\\_{u\\neq5} \\sigma^2\\_{\\hat{x}\\_u}\\Big[\\big(\\hat{M}^{MD}\\_{5u}\\big)^2 + \\big(\\hat{M}^{MD}\\_{5u} - \\hat{M}^{MK}\\_{5u}\\big)^2\\Big]}\\end{aligned}$$ \nMaking use of the fact that, for real wakes, *Z*( − *k*) = *Z*\\*(*k*), and defining *n*0 ≡ *N*/*L* as the mean linear density of the hadrons, we may sum Eqs. [eqtn:4thtermfinal] and [eqtn:5thtermfinal] and incorporate them back into Eq. [eqtn:pwrtaylor2], obtaining\n \n$$\\begin{aligned}\n\\label{eqtn:pwr\\_final}\n|\\tilde{n}(k)|^2 &= N - 2Nn\\_0\\frac{q^2c}{E\\_0}kM\\_{56}^{KD}\\\\\\nonumber\n&\\times[{{\\rm Re}}(Z(k))\\sin(k\\Delta z) + {{\\rm Im}}(Z(k))\\cos(k\\Delta z)]\\\\\\nonumber\n&\\times e^{-\\frac{k^2}{2} \\sum\\_{u\\neq5} \\sigma^2\\_{\\hat{x}\\_u}\\Big[\\big(\\hat{M}^{MD}\\_{5u}\\big)^2 + \\big(\\hat{M}^{MD}\\_{5u} - \\hat{M}^{MK}\\_{5u}\\big)^2\\Big]}\\end{aligned}$$ \nThen, the fractional change in beam power as a function of electron/hadron misalignment is\n \n$$\\begin{aligned}\n\\label{eqtn:pwr\\_relative}\n\\frac{\\Delta |\\tilde{n}(k)|^2}{|\\tilde{n}(k)|^2} &= -2n\\_0\\frac{q^2c}{E\\_0}kM\\_{56}^{KD}\\\\\\nonumber\n&\\times[{{\\rm Re}}(Z(k))\\sin(k\\Delta z) + {{\\rm Im}}(Z(k))\\cos(k\\Delta z)]\\\\\\nonumber\n&\\times e^{-\\frac{k^2}{2} \\sum\\_{u\\neq5} \\sigma^2\\_{\\hat{x}\\_u}\\Big[\\big(\\hat{M}^{MD}\\_{5u}\\big)^2 + \\big(\\hat{M}^{MD}\\_{5u} - \\hat{M}^{MK}\\_{5u}\\big)^2\\Big]}\\end{aligned}$$ \nNote that if ${{\\rm Re}}(Z(k)) = 0$ and ${{\\rm Im}}(Z(k)) > 0$ (which is the case considered below, see Eq. [eqtn:impedapprox]), and if *M*56*K**D* > 0, then perfect alignment of the hadrons and electrons, ie Δ*z* = 0, results in Δ∣*ñ*(*k*)∣2 < 0, corresponding to noise suppression below the shot noise in the beam. Such noise suppression has been previously studied theoretically in and observed experimentally in. Our result Eq. [eqtn:pwrrelative] is an agreement with the theoretical analysis of.\nDecoherence\n-----------\nThe above posits that the amount of signal modification has a purely sinusoidal dependence on the electron/hadron misalignment, ie,\n \n$$\\begin{aligned}\n\\label{eqtn:decoherence\\_ideal}\n \\frac{\\Delta |\\tilde{n}(k)|^2}{|\\tilde{n}(k)|^2} = A(k) \\cos(k\\Delta z + \\theta\\_0)\\end{aligned}$$ \nwhere *A* is some amplitude and *θ*0 is a phase, equal to 0 for an antisymmetric wake.\nHowever, the above derivation assumes an observation at a pure single frequency. If we take the more realistic case that we sample some range of frequencies with bandwidth Δ*k*, the signal amplitude will be\n \n$$\\begin{aligned}\n\\label{eqtn:decoherence\\_true}\n \\frac{\\Delta |\\tilde{n}(k)|^2}{|\\tilde{n}(k)|^2} &\\approx A(k)\\frac{1}{\\Delta k}\\int\\_{k-\\Delta k/2}^{k+\\Delta k/2} \\cos(k'\\Delta z + \\theta\\_0) dk'\\\\\\nonumber\n &= A(k)\\cos(k\\Delta z + \\theta\\_0) \\frac{\\sin(\\Delta k\\Delta z/2)}{\\Delta k\\Delta z/2}\\end{aligned}$$ \nso that the amplitude of the oscillations will decay over lengths of ∼ 1/Δ*k*.\nIntegrated Kicker\n-----------------\nThe above analysis assumes that the full kick to the hadrons takes place at a single point in the kicker. However, in the more realistic case where the kick is applied over the full length of the kicker, we may change Eq. [eqtn:delays] to\n \n$$\\begin{aligned}\n &z\\_d^{(i)} = z\\_m^{(i)} + M^{MD}\\_{5u}\\vec{x}\\_u^{(i)}\\\\\\nonumber\n&+ \\frac{1}{N\\_s}\\sum\\_{s=1}^{N\\_s}M^{K\\_sD}\\_{56}\\sum\\_j \\frac{q^2}{E\\_0} w(z\\_m^{(i)} + M^{MK\\_s}\\_{5u}\\vec{x}\\_u^{(i)} + \\Delta z - z\\_m^{(j)})\\end{aligned}$$ \nwhere we now split our kick in the kicker into *N**s* fractional kicks at locations *K**s*.\nThe rest of the analysis may be carried through in the same way as before, and we arrive at\n \n$$\\begin{aligned}\n \\frac{\\Delta |\\tilde{n}(k)|^2}{|\\tilde{n}(k)|^2} &= -2n\\_0\\frac{q^2c}{E\\_0}\\frac{1}{N\\_s}\\sum\\_{s=1}^{N\\_s}kM\\_{56}^{K\\_sD}\\\\\\nonumber\n&\\times[{{\\rm Re}}(Z(k))\\sin(k\\Delta z) + {{\\rm Im}}(Z(k))\\cos(k\\Delta z)]\\\\\\nonumber\n&\\times e^{-\\frac{k^2}{2} \\sum\\_{u\\neq5} \\sigma^2\\_{\\hat{x}\\_u}\\Big[\\big(\\hat{M}^{MD}\\_{5u}\\big)^2 + \\big(\\hat{M}^{MD}\\_{5u} - \\hat{M}^{MK\\_s}\\_{5u}\\big)^2\\Big]}\\end{aligned}$$ \nHowever, we have not observed any significant difference numerically between using the above equation and Eq. [eqtn:pwrrelative], with the single kick taken at the kicker center.\nSimulation\n==========\nIn order to check the validity and limits of the above theory, we make use of simulation. We first examine the case of a perfectly linear simulation, where the fractional energy kick to a given hadron in the kicker is simply the convolution of the wake function with the longitudinal hadron distribution in the modulator. We then turn our attention to a more detailed simulation, where the electrons and hadrons are tracked with a particle-in-cell code in order to incorporate saturation effects. In this and future sections, we consider the cooling parameters currently planned for 275 GeV protons in the EIC, listed in Tab. [tab:param]. The electron optics are assumed to be kept roughly constant within the modulator, kicker, and amplifiers through the use of focusing. Due to their much higher energy, the hadrons see the modulator and kicker as drifts, with the optics parameters specified at the center. Electron chicanes are defined from the end of one straight section to the start of the next one, while the proton *M*56 and phase advances are evaluated from modulator center to kicker center.\n[!hbt]\n| | |\n| --- | --- |\n| ***Geometry*** | |\n| Modulator Length (m) | 45 |\n| Kicker Length (m) | 45 |\n| Number of Amplifier Straights | 2 |\n| Amplifier Straight Lengths (m) | 37 |\n| ***Proton Parameters*** | |\n| Energy (GeV) | 275 |\n| Protons per Bunch | 6.9e10 |\n| Average Current (A) | 1 |\n| Proton Bunch Length (cm) | 6 |\n| Proton Fractional Energy Spread | 6.8e-4 |\n| Proton Emittance (x/y) (nm) | 11.3 / 1 |\n| Horizontal/Vertical Proton Betas in Modulator (m) | 39 / 39 |\n| Horizontal/Vertical Proton Dispersion in Modulator (m) | 1 / 0 |\n| Horizontal/Vertical Proton Dispersion Derivative in Modulator | -0.023 / 0 |\n| Horizontal/Vertical Proton Betas in Kicker (m) | 39 / 39 |\n| Horizontal/Vertical Proton Dispersion in Kicker (m) | 1 / 0 |\n| Horizontal/Vertical Proton Dispersion Derivative in Kicker | 0.023 / 0 |\n| Proton Horizontal/Vertical Phase Advance (rad) | 4.79 / 4.79 |\n| Proton M56 between Centers of Modulator and Kicker (mm) | -2.26 |\n| ***Electron Parameters*** | |\n| Energy (MeV) | 150 |\n| Electron Bunch Charge (nC) | 1 |\n| Electron Bunch Length (mm) | 7 |\n| Electron Peak Current (A) | 17 |\n| Electron Fractional Slice Energy Spread | 1e-4 |\n| Electron Normalized Emittance (x/y) (mm-mrad) | 2.8 / 2.8 |\n| Horizontal/Vertical Electron Betas in Modulator (m) | 40 / 20 |\n| Horizontal/Vertical Electron Betas in Kicker (m) | 4 / 4 |\n| Horizontal/ Vertical Electron Betas in Amplifiers (m) | 1 / 1 |\n| M56 in First Electron Chicane (mm) | 5 |\n| M56 in Second Electron Chicane (mm) | 5 |\n| M56 in Third Electron Chicane (mm) | -11 |\n| ***Cooling Times*** | |\n| Horizontal/Vertical/Longitudinal IBS Times (hours) | 2.0 / - / 2.9 |\n| Horizontal/Vertical/Longitudinal Cooling Times (hours) | 1.8 / - / 2.8 |\n[tab:param]\nLinear Simulation\n-----------------\nWe simulate a 50*μ*m length of the hadron beam at peak electron and hadron beam currents. Since the bunch lengths of the hadron and electron beams are a few cm and mm, respectively, we may assume constant longitudinal beam densities in our region of interest, and take periodic boundary conditions. We start by seeding 1 million hadron macroparticles in the modulator, representing 23 million real hadrons. In order to match the noise statistics in the real beam at both the modulator and detector, we perform the seeding with sub-Poisson noise, as follows. We create a 2-dimensional grid in the hadron phase space. One coordinate represents the hadron’s longitudinal position at the center of the modulator, and the other represents its delay in travelling from the modulator center to the detector. Using the notation of Eq. [eqtn:mtilde], this delay may be expressed as \n \n$$\\begin{aligned}\n\\label{eqtn:rms\\_delay}\n\\sigma^2\\_{\\Delta z} = &\\hat{M}\\_{51}^2\\beta\\_x\\epsilon\\_x + \\hat{M}\\_{52}^2\\epsilon\\_x/\\beta\\_x\\\\\\nonumber\n+&\\hat{M}\\_{53}^2\\beta\\_y\\epsilon\\_y + \\hat{M}\\_{54}^2\\epsilon\\_y/\\beta\\_y\\\\\\nonumber\n+ &\\hat{M}\\_{56}^2\\sigma\\_{\\eta}^2\\end{aligned}$$ \nThis grid extends in 32768 uniform steps over the 50*μ*m length of beam to be simulated, and in 100 uniform steps between ± 5*σ*Δ*z* in the hadron delay. We assume that the hadrons are distributed uniformly longitudinally and that they have a Gaussian distribution (mean 0, standard deviation *σ*Δ*z*) with respect to the delay. Multiplying this distribution by the total number of hadrons expected within a 50*μ*m slice of beam, we arrive at the expected number of hadrons in each bin. We then add on appropriate pseudo-random Poisson noise to obtain the number of real hadrons in each bin.\nTo assign the macroparticles, we iterate over all the bins, with the loop over position nested inside the loop over delays. We assign each bin in turn a number of macroparticles equal to the number of real hadrons in that bin divided by the number of real hadrons represented by each macroparticle. Each macroparticle is given a position and delay distributed uniformly within the bin. Underflow and overflow are carried to the next bin. Each macroparticle is assigned a horizontal angle (a pseudo-random number between 0 and 2*π*) and a horizontal action (a pseudo-random number drawn from an exponential distribution of mean *ε**x*). Similar procedures are used to obtain the vertical action and angle. The fractional energy error is chosen to provide the hadron with its assigned delay. The actions, angles, and energy errors, together with the optics at the center of the modulator, allow the construction of the particle phase-space coordinates.\nIn the case where the modulator-to-detector transfer matrix is an identity matrix, we assign the hadron fractional energy offset directly, replacing *σ*Δ*z* in the above with *σ**η*.\nWe obtain the position of each hadron macroparticle in the kicker by using the modulator-to-kicker transfer matrix in the design optics, and save this pre-kicker distribution. We then apply a longitudinal shift to all hadrons equally, corresponding to a longitudinal electron-hadron misalignment. We assign an energy kick to each macroparticle by convolving the ideal wake function with the longitudinal hadron density distribution in the modulator. The ideal wake function is computed using the procedures of for the case of two amplification straights and unmatched Gaussian electron and hadron beams with arbitrary horizontal and vertical beam sizes. A plot of this idealized wake is shown in black in Fig. [fig:wakesboth]. Note that this wake is antisymmetiric, which means that ${{\\rm Re}}(Z(k)) = 0$ and the first term in the square brackets on the right hand side of Eq. [eqtn:pwrrelative] vanishes. We then translate the hadrons to the detector element using a kicker-to-detector transfer matrix equal to the inverse of the modulator-to-kicker transfer matrix. We chose this matrix because we had found empirically that it gives a sizable signal and thereby reduces the impact of numerical noise. This will be further justified in subsection [subsec:optimalobs]. Finally, we perform a fast Fourier transform (FFT) on the hadron linear density distribution to obtain the amplitude of the spectrum. Squaring these values gives us the spectral power, and we compute the fractional change from the initial spectral power of the hadrons. We repeat this process from the saved pre-kicker distribution, but apply different longitudinal misalignments. Finally, we repeat this whole procedure from the beginning for 100 different random noise seeds and take the average fractional change in spectral power for each delay value.\n\nWe plot these simulated results and the theory prediction in Fig. [fig:simlinear], finding excellent agreement. Note that the theoretical wake has a constant vertical shift of ∼ 0.04 due to higher-order terms, as discussed in Appendix [app:secondorder].\n\nNonlinear Simulation\n--------------------\nAs a more realistic model, we explicitly track the hadrons and electrons step-by-step through the modulator, kicker, and amplifiers. Much of the process for the hadrons is the same as in the linear case, described in subsection [subsec:simlin]. The main difference is that, rather than simply convolve the wake with the hadron density distribution, we explicitly include the electron beam and model its interactions with itself and the hadrons in a custom particle-in-cell (PIC) code. This is described in, and a summary of its operation and changes is given below. The electrons are initialized analogously to the hadrons with 10 real electrons per macroparticle, but with only their longitudinal coordinates described. As such, the binning is done as a function of the electron longitudinal position and its fractional energy offset. The hadrons are initialized at the center of the modulator as described in subsection [subsec:simlin] using the modulator optics of Tab. [tab:param], then back-propagated to the start of the modulator so as to have the correct initial distribution. We track the electrons and hadrons through the modulator in 2 and 6 phase-space dimensions, respectively, using a simple kick-drift model. We model the inter-particle interactions using the disc model of, and convolve the associated force function with the hadron and electron longitudinal density distributions to obtain the kicks to each species. We then apply the transfer matrix of a short drift length to update the particle positions. This process is repeated through the length of the modulator. The electron and hadron chicanes are modelled as simple transfer matrices. In bringing the hadrons to the kicker, the modulator-to-kicker transfer matrix is multiplied by the inverse transfer matrices of half the modulator and kicker drifts, so as to have the correct transfer matrix between the element centers, while the electrons are explicitly tracked through the amplification section (3 chicanes and 2 straights), using the same PIC code. In the kicker, both species are again tracked with the PIC code to get accurate kicks to the hadrons. In bringing the hadrons to the detector, we multiply the kicker-to-detector transfer matrix by half the inverse transfer matrix of the kicker drift in order to have the correct transfer matrix from the kicker center to the detector. Using the transfer matrix from the center of the kicker is justified by the fact that this is the average location where the kick will take place.\nIn comparing to theory, we reduce the wake amplitude to account for saturation in the electron beam, as described in, and shown in red in Fig. [fig:wakesboth]. We incorporate the ∼ 0.01 offset of Appendix [app:secondorder] into Eq. [eqtn:pwrrelative] and compare with the simulation results, as shown in Fig. [fig:simnonlinear]. Good agreement is observed.\n\nDetection of the Signal\n=======================\nSo far, we have focused on the derivation and validation of the signal modification theory. However, for this to be useful, we must be able to physically detect it. We therefore look for the optimal kicker-to-detector transfer matrix and wavelength at which to observe the signal, and then examine the possibility of detecting the hadron beam density modulation in the radiation of an EIC dipole.\nOptimal Parameters\n------------------\nIn order to perform the optimization, it is helpful to make use of a simplified, analytic form for the wake function. In, it was proposed to approximate the wake using a simple fit function equal to a sine wave with Gaussian decay. Making slight changes to the parameters, we may write\n \n$$\\begin{aligned}\n\\label{eqtn:wake\\_approx}\nw(z) = A \\sin(\\kappa z)e^{-z^2/2\\lambda^2}\\end{aligned}$$ \nwhere *A*, *κ*, and *λ* are fit parameters roughly corresponding to the wake amplitude, wavenumber, and falloff-distance, respectively. We find that this function fits our wake with saturation fairly well, as shown in Fig. [fig:wakefitsat], and we arrive at values of *A* = 78.4MV/pC, *κ* = 1.031/*μ*m, and *λ* = 2.768*μ*m.\n\nThe corresponding impedance of the simplified wake of Eq. [eqtn:wakeapprox] is given by\n \n$$\\begin{aligned}\n\\label{eqtn:imped\\_approx}\n Z(k) = -\\frac{A\\lambda i}{2c}\\sqrt{2\\pi}\\Big[e^{-\\lambda^2(k+\\kappa)^2/2} - e^{-\\lambda^2(k-\\kappa)^2/2}\\Big]\\end{aligned}$$ \nPutting the above into Eq. [eqtn:pwrrelative], we obtain an expression for signal modification amplitude:\n \n$$\\begin{aligned}\n\\label{eqtn:pwr\\_relative2}\n\\frac{\\Delta |\\tilde{n}^2(k)|^2}{|\\tilde{n}^2(k)|^2} &= \\frac{q^2}{E\\_0}A\\lambda n\\_0kM\\_{56}^{KD}\\sqrt{2\\pi}\\cos(k\\Delta z)\\\\\\nonumber\n&\\times \\Big[e^{-\\lambda^2(k+\\kappa)^2/2} - e^{-\\lambda^2(k-\\kappa)^2/2}\\Big]\\\\\\nonumber\n&\\times e^{-\\frac{k^2}{2} \\sum\\_{u\\neq5} \\sigma^2\\_{\\tilde{x}\\_u}\\bigg[\\Big(\\tilde{M}^{MD}\\_{5u}\\Big)^2 + \\Big(\\tilde{M}^{MK}\\_{5u} - \\tilde{M}^{MD}\\_{5u}\\Big)^2\\bigg]}\\end{aligned}$$ \nExamining the above, we see that once we have fixed the cooling parameters, including the transfer matrix from the modulator to the kicker, the only variables which we can alter which will have any impact on the signal modification are the wavenumber of the density modulation we wish to observe and the *M*5*u* transfer elements from the kicker to the detector. (Note that these parameters alone are also sufficient to specify the *M*5*u* transfer matrix elements from the modulator to the detector.) If we have no vertical dispersion, *M*53*M**K* = *M*54*M**K* = 0, and it is easy to see from Eq. [eqtn:pwrrelative2] that the corresponding elements in the kicker-to-detector transfer matrix should also be equal to 0. The magnitude of the signal amplitude in Eq. [eqtn:pwrrelative2] may be easily maximized with respect to *k*, *M*51*K**D*, *M*52*K**D*, and *M*56*K**D*, obtaining values of *M*51*K**D* = − 8.44 × 10− 4, *M*52*K**D* = − 3.00 × 10− 2m, *M*56*K**D* = 2.36 × 10− 3m, and *k* = 1.04 × 106/m, with Eq. [eqtn:pwrrelative2] taking on the numerical value $\\frac{\\Delta |\\tilde{n}^2(k)|^2}{|\\tilde{n}^2(k)|^2} = -0.22\\cos(k\\Delta z)$. (For comparison, the simulations in Section [sec:sim], setting *M**K**D* to the inverse of *M**M**K* were made with near-optimal parameters *M*51*K**D* = − 7.93 × 10− 4, *M*52*K**D* = − 2.86 × 10− 2m, *M*56*K**D* = 2.26 × 10− 3m, and *k* = 1.06 × 106/m.)\nDrift Plus Dipole\n-----------------\nIn the current design of the cooler, it is useful to have as much space as possible dedicated to the modulator, kicker, and amplifiers, making it difficult to also fit in an optimized transfer line between the kicker and detector to achieve optimal signal modification. We therefore consider the case where we simply have a drift after the kicker plus a main arc dipole (field strength of 3.782T, bending radius of 243m). We assume a hard-edge dipole model and that the beam path is normal to the pole face, so that edge effects may be ignored. We also use the fit wake of Fig. [fig:wakefitsat]. For convenience, we define the amplitude of signal modification so that positive values correspond to a reduction in intensity when the electrons and hadrons are aligned. We take an observation point at the start of the dipole and perform a scan of the amplitude of signal modification as a function of the *M*56 value of a drift transfer matrix between the kicker center and the observation point for several wavelengths of hadron density perturbations, as shown in Fig. [fig:scanwavelengthdiscrete]. We find an optimal wavelength of ∼ 6*μ*m, but the 1.2mm *M*56 value corresponds to over 100m of drift, which is too long for the available space. We therefore focus on more reasonable parameters and scan the signal modification amplitude as a function of moderate kicker-to-dipole drift lengths and observation wavelengths, as shown in Fig. [fig:scanwavelength]. Note that these drifts are defined between the end of the kicker and the start of the dipole, and so an extra *M*56 contribution from half the kicker length is also included in the transfer matrix. We find an optimal observation wavelength of 6*μ*m over a range of drift lengths. We then fix a 6*μ*m observation wavelength and scan the signal modification amplitude as a function of drift length and bend angle within the dipole, as shown in Fig. [fig:scandipole]. We find that it is ideal to observe the radiation near the start of the dipole and that, within the region of interest, increased drift lengths result in increased signal modification, with amplitudes of a few percent.\n\n\n\nNote that the inclusion of the *M*51 and *M*52 terms is vital to the correct understanding of signal modification. Fig. [fig:scandriftnegative] shows the result of scanning the amplitude of signal modification as a function of the *M*56 term between the kicker center and observation point. It would appear that we could use a dipole to generate negative *M*56 and see about half the signal modification we could obtain with a drift. However, the dipole will also generate non-trivial *M*51 and *M*52 terms. Performing a scan with the more realistic dipole generates the plot seen in Fig. [fig:scandipolepure]. Virtually no signal modification is observed, and even then only for the small positive *M*56 values generated near the very start of the dipole.\n\n\nFringe Fields\n-------------\nSince we are focused on the radiation at the start of the dipole, careful consideration of the fringe fields is important. We take a model of a dipole consisting of two magnetized poles of uniform magnetization *B*0/*μ*0 separated by a distance 2*w* and of longitudinal length *L* which extend infinitely far in the *x* direction and are each semi-infinite in *y*. See Fig. [fig:dipolegeometry]. It can be analytically shown that, with the origin defined as the dipole center, the fields visible to the beam are given by:\n\n \n$$\\begin{aligned}\nB\\_y(y,z) = \\frac{B\\_0}{2\\pi}\\Bigg[&\\tan^{-1}\\bigg(\\frac{L/2+z}{w+y}\\bigg)\\\\\\nonumber\n+ &\\tan^{-1}\\bigg(\\frac{L/2-z}{w+y}\\bigg)\\\\\\nonumber\n+&\\tan^{-1}\\bigg(\\frac{L/2+z}{w-y}\\bigg)\\\\\\nonumber\n+&\\tan^{-1}\\bigg(\\frac{L/2-z}{w-y}\\bigg)\\Bigg] \\end{aligned}$$ \n \n$$\\begin{aligned}\n B\\_z(y,z) = \\frac{B\\_0}{4\\pi}\\Bigg[&\\ln\\bigg(\\frac{(w+y)^2 + (L/2+z)^2}{(w-y)^2 + (L/2+z)^2}\\bigg)\\\\\\nonumber\n +&\\ln\\bigg(\\frac{(w-y)^2 + (L/2-z)^2}{(w+y)^2 + (L/2-z)^2}\\bigg)\\Bigg]\\end{aligned}$$ \nWe choose the dimensions of the RHIC dipoles, with *w* = 4cm and *L* = 9.441m. As before, *B*0 = 3.782T. Although this is not an exact fit to any particular dipole, it provides a proof-of-principle study. In order to counteract the fringe fields, so that we arrive at a reasonable transfer matrix inside the main arc dipole where the field is strong enough to produce detectable radiation, we also include a 1m long screen dipole with *w* = 4cm and a weak field opposite to that of the main arc dipole (*B*0 = − 0.40T).\nWe check that this produces reasonable results by altering the simulation used in subsection [subsec:simlin] to bring the hadrons from the kicker center to a point 5m after the end of the kicker through the use of a 27.5m drift transfer matrix, then track them with the realistic fields through a drift of 4m, the 1m screen dipole, and 91.5mm into the main arc dipole. The total integrated field seen by the proton making this journey is 0.13T-m, corresponding to a bend of 0.14 mrad. Due to the longer computation time, only 8 random seeds are used. We compare this to the theoretical prediction of a 32.5m drift after the kicker center (corresponding to 10m past the end of the kicker) plus a 0.14mrad bend in the main arc dipole. The results are shown in Fig. [fig:sigsupwrealdipole]. Good agreement is observed.\n\nRadiation\n---------\nSo far, we have focused entirely on the changes to the hadron density spectrum. However, this cannot be observed directly. Instead, we will monitor the radiation produced by the hadron beam as it moves through a dipole. In the far-field approximation, the electric field from an accelerated charge is given by \n \n$$\\begin{aligned}\n\\label{eqtn:jackson}\n \\vec{E}(\\vec{x},t\\_0) = \\frac{q}{4\\pi\\epsilon\\_0c}\\Bigg[\\frac{1}{|\\vec{r}\\_{0} - \\vec{r}|}\\frac{\\hat{n}\\times\\big[(\\hat{n}-\\vec{\\beta})\\times\\dot{\\vec{\\beta}}\\big]}{\\big(1-\\vec{\\beta}\\cdot\\hat{n}\\big)^3}\\Bigg]\\_t\\end{aligned}$$ \nwhere the observation point is *r⃗*0, the observation time is *t*0, and the hadron position, *r⃗*, relativistic beta, *β⃗*, and its derivative, $\\dot{\\vec{\\beta}}$, are evaluated at the retarded time, *t*, with *t*0 = *t* + ∣*r⃗*0 − *r⃗*∣/*c*.\nSince we had seen in subsection [subsec:dipoleobs] that it is useful to observe the signal modification immediately at the start of the dipole, we consider the edge radiation. We write the frequency-space electric field\n \n$$\\begin{aligned}\n\\label{eqtn:fft\\_E}\n \\vec{E}(\\vec{x},\\omega) &= \\int\\_{-\\infty}^{\\infty} \\vec{E}(\\vec{x},t\\_0) e^{i\\omega t\\_0} dt\\_0\\\\\\nonumber\n &= \\frac{q}{4\\pi\\epsilon\\_0c}\\int\\_{-\\infty}^{\\infty}\\Bigg[\\frac{1}{|\\vec{r}\\_{0} - \\vec{r}|}\\frac{\\hat{n}\\times\\big[(\\hat{n}-\\vec{\\beta})\\times\\dot{\\vec{\\beta}}\\big]}{\\big(1-\\vec{\\beta}\\cdot\\hat{n}\\big)^2}\\Bigg]\\_t e^{i\\omega t\\_0} dt\\end{aligned}$$ \nwhere we have simplified using the fact that *d**t*0/*d**t* = 1 − *β⃗* ⋅ *n̂*.\nThe intensity of the radiation is given by \n \n$$\\begin{aligned}\n\\label{eqtn:intensity}\n S = \\alpha \\frac{\\Delta \\omega}{\\omega} \\frac{I\\_h}{q} \\bigg(\\frac{2\\epsilon\\_0c}{e}\\bigg)^2\\hbar\\omega\\Big|\\vec{E}(\\vec{x},\\omega)\\Big|^2\\end{aligned}$$ \nwhere *α* is the fine-structure constant, Δ*ω* is the bandwidth, and *I**h* is the average hadron beam current (1A for the EIC).\nWe track a test charge through the dipole fields described in subsection [subsec:fringefields], and use its path to integrate Eq. [eqtn:fftE] at various locations on a transverse grid 10m downstream of the entrance pole face of the main arc dipole, where we might place a camera to detect the emitted radiation. We assume a 10% bandwidth for this detector. The result of this is shown in Fig. [fig:radiationdetectionscan]. We see that we may achieve intensities of ∼ 700*μ**W*/*m*2.\nWe repeat the above procedure, but model the effect of signal modification by multiplying the field amplitude produced by each step of the tracker by $\\sqrt{1 + A(s)\\sigma\\_{z,e}/\\sigma\\_{z,h}}$, where *A*(*s*) is the amplitude of signal suppression from Eq. [eqtn:pwrrelative]. We assume 6*μ*m light and use the fit wake of Fig. [fig:wakefitsat] (which includes saturation effects) and a transfer matrix through the relevant drift and concatenated dipole fields to reach an arbitrary location after the kicker. We assume 10m between the end the the kicker and the start of the main arc dipole, with a 1m screen dipole immediately in front of the main arc dipole. *σ**z*, *e*/*σ**z*, *h* is the ratio of the electron to hadron bunch lengths, since only the central hadrons will see the effect of the cooling. We subtract the intensity shown in Fig. [fig:radiationdetectionscan] from that of this modified simulation in order to obtain the intensity of our signal. This is shown in Fig. [fig:radiationintensitychange]. The fractional change is ∼ 0.3%. Integration over 1mm2 for 1 second on-axis will provide 2.0 × 1010 photons from the full bunch, while signal modification in the core protons will reduce this number by 7.1 × 107. The error on the former should follow Poisson statistics, with a numerical value of $\\sqrt{2.0\\times10^{10}} \\approx 1.4\\times10^5$ so that we obtain a signal-to-noise ratio of ∼ 500. However, inefficiencies in the detector will reduce this, so that we may need to integrate for a longer time or over a wider area.\n\n\nWe must also contend with background thermal radiation, whose intensity is given by the well-known blackbody radiation formula:\n \n$$\\begin{aligned}\n\\label{eqtn:thermal}\n S = \\frac{\\hbar}{4\\pi^2c^2}\\frac{\\omega^3}{e^{\\hbar\\omega/k\\_BT}-1} \\Delta \\omega\\end{aligned}$$ \nWithin this same 10% bandwidth at a temperature of 300K, we have a thermal intensity of 10*W*/*m*2, which would swamp our signal. It is therefore necessary to operate at liquid nitrogen temperature (77.29K), where the thermal background is only 1*n**W*/*m*2.\nConclusion\n==========\nWe have derived an expression for signal modification for the case of a CEC cooler. We have also performed simulations of this process, finding good agreement with theoretical expectations. Finally, we have shown that such a signal may be experimentally observed at the few parts-per-thousand level with a cryo-cooled infrared camera set up to observe existing dipole radiation. As the design of the cooler matures, it will be necessary to incorporate the needs of this radiation diagnostic, at least ensuring the existence of a sufficient drift length between the end of the kicker and the first downstream dipole. We will also investigate the possibility of separating the radiation from the core protons from that of the rest of the bunch, which would increase the fractional change in radiation power by a factor of nearly 10, corresponding to the the ratio of hadron to electron bunch lengths. This may require operation at a shorter wavelength. Although we have provided a compelling proof-of-principle argument, we will also need to more carefully consider the details of the fringe fields of the dipole and appropriate compensation.\nAcknowledgements\n================\nThis work was supported by Brookhaven Science Associates, LLC under Contract No. DE-SC0012704 with the U.S. Department of Energy, and by the Department of Energy, contract DE-AC03-76SF00515. We would like to thank the members of the EIC strong hadron cooling group for useful discussions.\nAlternative Derivations of Signal Modification\n==============================================\nWe present two alternative derivations of the theory of signal modification in the purely longitudinal case. In subsection [subsec:thrymike], we adopt a hybrid approach where we integrate the Vlasov equation in the time domain and make use of single-particle statistics. In subsection [subsec:thrystupakov], we back-propagate the phase-space density from the detector to the modulator, in line with the conventions of. In the limit where the hadron delay depends only on its energy offset, ie *M*51 = *M*52 = *M*53 = *M*54 = 0, these results agree with Eq. [eqtn:pwrrelative].\nIntegration of Time-Domain Vlasov Equation\n------------------------------------------\nWe consider the hadron beam entering the modulator as a constant background current, *I*0, with a random Schottky signal, *I**s*(*s*, *t*), overlaid on top, where *s* is the longitudinal position within the cooler and *t* is the time of measurement. After passage through the kicker, the beam will pick up a coherent response current, *I**c*(*s*, *t*), due to the action of the wake, so that the total beam current at position *s* and time *t* is given by\n \n$$\\begin{aligned}\n\\label{eqtn:mike\\_current}\n I(s,t) = I\\_0 + I\\_s(s,t) + I\\_c(s,t)\\end{aligned}$$ \nSince we assume only random noise in the hadron beam when it enters the modulator, we take *I**c*(*s*, *t*) = 0 for *s* < *s**K*, the kicker location. We also consider the respective phase-space densities, *ψ*(*η*, *s*, *t*), of these three currents, with\n \n$$\\begin{aligned}\n\\label{eqtn:mike\\_phis}\n I(s,t) = q\\int\\_{-\\infty}^{\\infty} \\dot{s}(\\eta,s) \\psi(\\eta, s, t) d\\eta\\end{aligned}$$ \nwhere $\\dot{s}(\\eta,s)$ is the local speed of the hadrons as a function of fractional energy deviation, *η*, and longitudinal position in the cooler.\nThe fluctuating part of the Schottky current can be written as\n \n$$\\begin{aligned}\n\\label{eqtn:mike\\_schottky}\n I\\_s(s,t) = \\sum\\_{k=-\\infty}^{\\infty}q \\delta(t - t\\_k - \\tau\\_0(s) - \\eta\\_k\\tau\\_1(s)) - I\\_0\\end{aligned}$$ \nwhere *t**k* is the time that particle *k* arrives at the modulator, at position *s* = 0, *τ*0(*s*) is the time for an on-energy particle to travel from the modulator to point *s*, *τ*1(*s*) is the energy-dependence of this transit time, and *η**k* is the fractional energy deviation of particle *k*.\nWe define a frequency-space version of the Schottky current, *Ĩ**s*(*s*, *ω**m*), with the forward and inverse Fourier transforms\n \n$$\\begin{aligned}\n\\label{eqtn:mike\\_fourier}\n \\tilde{I}\\_{s}(s,\\omega\\_m) &= \\frac{1}{T}\\int\\_{\\tau\\_0(s)-T/2}^{\\tau\\_0(s)+T/2} e^{i\\omega\\_mt}I\\_s(s,t)dt\\\\\\nonumber\n I\\_s(s,t) &= \\sum\\_{m=-\\infty}^{\\infty}e^{-i\\omega\\_m t}\\tilde{I}\\_{s}(s,\\omega\\_m)\\end{aligned}$$ \nwhere *T* is chosen to be large compared to typical values of *η**k**τ*1(*s*) and the wake wavelength divided by *β**c*, so that we may ignore the precise behavior at the endpoints, and where *ω**m* = 2*π**m*/*T*. Particular particles near the edges of this time interval may enter or leave it during the passage through the cooler, but the relatively large size of *T* ensures that they are a negligible fraction of the total, and can be safely ignored. If we assume *N* particles arriving in the cooling system within time *T* at times *t**a*, we obtain for *m* ≠ 0\n \n$$\\begin{aligned}\n\\label{eqtn:mike\\_I\\_sm}\n \\tilde{I}\\_{s}(s, \\omega\\_m) = \\frac{q}{T}\\sum\\_{a=1}^{N} e^{i\\omega\\_m[t\\_a + \\tau\\_0(s) + \\eta\\_a\\tau\\_1(s)]}\\end{aligned}$$ \nIn the kicker, each particle receives a kick, defined by a time-dependent voltage\n \n$$\\begin{aligned}\n\\label{eqtn:mike\\_voltage}\n V(t) &= \\int\\_{-\\infty}^{\\infty}w\\_t(t - \\tau - \\hat{\\tau}\\_0)I\\_s(0, \\tau)d\\tau\\end{aligned}$$ \nwhere *τ̂*0 is the delay of the electron beam in travelling between modulator and kicker and *w**t*(*t*) is related to the wake function of Eq. [eqtn:wakedef] by\n \n$$\\begin{aligned}\n\\label{eqtn:def\\_wake\\_time}\n w\\_t(t) = w(-\\beta c t)\\end{aligned}$$ \nand has an impedance, *Z**t*(*ω*), related to the impedance of Eq. [eqtn:impeddef] by\n \n$$\\begin{aligned}\n\\label{eqtn:def\\_imped\\_time}\n Z\\_t(\\omega) = \\int\\_{-\\infty}^{\\infty}w\\_t(t)e^{i\\omega t}dt = -\\frac{Z(\\omega/\\beta c)}{\\beta}\\end{aligned}$$ \nAfter passing through the kicker, the hadron phase-space density also includes a perturbation described by the coherent response function, obeying the linearized Vlasov equation:\n \n$$\\begin{aligned}\n\\label{eqtn:mike\\_vlasov}\n 0 = &\\frac{\\partial\\psi\\_c(\\eta,s,t)}{\\partial t} + \\frac{\\partial}{\\partial s}(\\dot{s}(\\eta,s)\\psi\\_c(\\eta,s,t))\\\\\\nonumber\n &+ \\frac{qV(t)\\delta(s-s\\_K)\\dot{s}(\\eta,s)}{E\\_0}\\frac{\\partial}{\\partial \\eta}(\\psi\\_0(\\eta))\\end{aligned}$$ \nwhere we have explicitly written the $\\dot{\\eta}$ as a delta function kick from the kicker voltage. We have also pulled the factor of $\\dot{s}$ out of the derivative with respect to energy since we operate in the ultra-relativistic regime and so assume that $\\dot{s}$ is independent of particle energy within the kicker straight section.\nSince there is no coherent response for *s* < *s**K*, integrating the above equation from *s**K*− to *s**K*+ yields\n \n$$\\begin{aligned}\n\\label{eqtn:mike\\_psic\\_real}\n \\psi\\_c(\\eta,s\\_K^+,t) = -\\frac{qV(t)}{E\\_0}\\frac{d\\psi\\_0(\\eta)}{d\\eta}\\end{aligned}$$ \nwhich may be transformed to frequency space as\n \n$$\\begin{aligned}\n\\label{eqtn:mike\\_psic\\_fourier}\n \\tilde{\\psi}\\_{c}(\\eta,s\\_K^+,\\omega\\_m) = -\\frac{q\\tilde{V}(\\omega\\_m)}{E\\_0}\\frac{d\\psi\\_0(\\eta)}{d\\eta}\\end{aligned}$$ \nwhere we have defined a frequency-space voltage, *Ṽ*(*ω**m*), and phase-space density, *ψ̃**c*(*η*, *s*, *ω**m*), using the same procedure as in Eq. [eqtn:mikefourier].\nIntegration of Eq. [eqtn:mikevlasov] to arbitrary *s* > *s**K* yields\n \n$$\\begin{aligned}\n\\label{eqtn:mike\\_int\\_psi}\n \\tilde{\\psi}\\_{c}(\\eta,s,\\omega\\_m) &= \\tilde{\\psi}\\_{c}(\\eta,s\\_K^+,\\omega\\_m)\\\\\\nonumber\n &\\times e^{i\\omega\\_m[\\tau\\_0(s) + \\eta \\tau\\_1(s) - \\tau\\_0(s\\_K) - \\eta \\tau\\_1(s\\_K)]}\\end{aligned}$$ \nIntegrating over particle energy to get the coherent current response, we find\n \n$$\\begin{aligned}\n\\label{eqtn:mike\\_Ic}\n &\\tilde{I}\\_{c}(s,\\omega\\_m) = -\\frac{q^2\\tilde{V}(\\omega\\_m)\\dot{s}\\_0}{E\\_0}\\\\\\nonumber\n &\\times\\int\\_{-\\infty}^{\\infty} \\frac{d\\psi\\_0(\\eta)}{d\\eta}e^{i\\omega\\_m[\\tau\\_0(s) + \\eta \\tau\\_1(s) - \\tau\\_0(s\\_K) - \\eta \\tau\\_1(s\\_K)]} d\\eta\\\\\\nonumber\n &=\\frac{q^2\\tilde{V}(\\omega\\_m)\\dot{s}\\_0}{E\\_0}i\\omega\\_m[\\tau\\_1(s) - \\tau\\_1(s\\_K)]\\\\\\nonumber\n &\\times\\int\\_{-\\infty}^{\\infty} \\psi\\_0(\\eta)e^{i\\omega\\_m[\\tau\\_0(s) + \\eta \\tau\\_1(s) - \\tau\\_0(s\\_K) - \\eta \\tau\\_1(s\\_K)]} d\\eta\\\\\\nonumber\n &=\\tilde{I}\\_{s}(s=0,\\omega\\_m)Z\\_t(\\omega\\_m)e^{i\\omega\\_m\\hat{\\tau}\\_0}\\frac{q^2\\dot{s}\\_0}{E\\_0}i\\omega\\_m[\\tau\\_1(s) - \\tau\\_1(s\\_K)]\\\\\\nonumber\n &\\times\\int\\_{-\\infty}^{\\infty} \\psi\\_0(\\eta)e^{i\\omega\\_m[\\tau\\_0(s) + \\eta \\tau\\_1(s) - \\tau\\_0(s\\_K) - \\eta \\tau\\_1(s\\_K)]} d\\eta\\end{aligned}$$ \nwhere we have assumed that the change in $\\dot{s}(\\eta,s)$ due to the beam energy spread is small relative to the speed of an on-energy particle, $\\dot{s}\\_0$, and made use of Eq. [eqtn:mikeIsm], [eqtn:mikevoltage], and [eqtn:defimpedtime] to write the voltage in the kicker in terms of the frequency-domain Schottky current in the modulator and the impedance *Z**t*(*ω**m*). We use the above to define an effective gain function, *Ĩ**c*(*s*, *ω**m*) = *G*(*s*, *ω**m*)*Ĩ**s*(*s* = 0, *ω**m*). For *m* ≠ 0, so that we may ignore the base current, *I*0, the total current is\n \n$$\\begin{aligned}\n\\label{eqtn:mike\\_It}\n \\tilde{I}\\_{t}(s,\\omega\\_m) = G(s,\\omega\\_m)\\tilde{I}\\_{s}(s=0,\\omega\\_m) + \\tilde{I}\\_{s}(s,\\omega\\_m)\\end{aligned}$$ \nWe define the signal modification by computing\n \n$$\\begin{aligned}\n\\label{eqtn:mike\\_sigsup}\n \\langle|\\tilde{I}\\_{t}(s,\\omega\\_m)|^2\\rangle &= \\langle|\\tilde{I}\\_{s}(s,\\omega\\_m)|^2\\rangle (1 + |G(s,\\omega\\_m)|^2)\\\\\\nonumber\n &+ 2 Re \\langle G(s,\\omega\\_m)\\tilde{I}\\_{s}(s=0,\\omega\\_m)\\tilde{I}^\\*\\_{s}(s,\\omega\\_m)\\rangle\\end{aligned}$$ \nwhere the angle brackets denote the ensemble average and the star refers to complex conjugation.\nUsing Eq. [eqtn:mikeIsm], we can evaluate\n \n$$\\begin{aligned}\n\\label{eqtn:mike\\_I\\_corr}\n \\langle \\tilde{I}\\_{s}&(s=0,\\omega\\_m)\\tilde{I}^\\*\\_{s}(s,\\omega\\_m)\\rangle\\\\\\nonumber\n &=\\frac{q^2}{T^2}\\sum\\_{a=1}^N\\sum\\_{b=1}^{N} e^{i\\omega\\_m[t\\_a - t\\_b - \\tau\\_0(s) - \\eta\\_b\\tau\\_1(s)]}\\\\\\nonumber\n & = \\frac{q^2}{T^2}N \\Big\\langle e^{-i\\omega\\_m[\\tau\\_0(s) + \\eta\\tau\\_1(s)]}\\Big\\rangle\\_{\\eta}\\end{aligned}$$ \nwhere we only kept the diagonal terms due to the uncorrelated nature of the initial particle distribution.\nSimilarly, we find\n \n$$\\begin{aligned}\n\\label{eqtn:miked\\_self\\_corr}\n \\langle |I\\_{s}(s,\\omega\\_m)|^2\\rangle = \\frac{q^2}{T^2}N\\end{aligned}$$ \nWe can now write the fractional signal modification as\n \n$$\\begin{aligned}\n\\label{eqtn:mike\\_rel\\_sigsup}\n \\frac{\\langle|I\\_{t}(s,\\omega\\_m)|^2\\rangle}{\\langle|I\\_{s}(s,\\omega\\_m)|^2\\rangle} &= 1 + |G(s,\\omega\\_m)|^2\\\\\\nonumber\n &+ 2{{\\rm Re}}(Y(s,\\omega\\_m))\\end{aligned}$$ \nwith\n \n$$\\begin{aligned}\n\\label{eqtn:mike\\_g}\n G(s,\\omega\\_m) &= iZ\\_t(\\omega\\_m)\\omega\\_m\\frac{qI\\_0}{E\\_0}[\\tau\\_1(s) - \\tau\\_1(s\\_K)]\\\\\\nonumber\n &\\times e^{i\\omega\\_m[\\hat{\\tau}\\_0 + \\tau\\_0(s) - \\tau\\_0(s\\_K)]}\\Big\\langle e^{i\\omega\\_m\\eta[\\tau\\_1(s) - \\tau\\_1(s\\_K)]} \\Big\\rangle\\_{\\eta}\\end{aligned}$$ \nand\n \n$$\\begin{aligned}\n\\label{eqtn:mike\\_Y}\nY(s,\\omega\\_m) = G(s,\\omega\\_m) e^{-i\\omega\\_m\\tau\\_0(s)} \\Big\\langle e^{-i\\omega\\_m\\eta\\tau\\_1(s)}\\Big\\rangle\\_{\\eta}\\end{aligned}$$ \nwhere the mean current $I\\_0 = q\\dot{s}\\_0\\int\\_{-\\infty}^{\\infty}\\psi\\_0(\\eta)d\\eta$.\nSince the kick from the wake is expected to be small, we ignore the second-order ∣*G*(*s*, *ω**m*)∣2 term. We also assume a Gaussian energy distribution in the beam, with RMS energy error *σ**η*. This allows us to perform the energy averages explicitly, and we obtain the relative signal modification\n \n$$\\begin{aligned}\n\\label{eqtn:mike\\_final}\n\\frac{\\langle|I\\_{t}(s,\\omega\\_m)|^2\\rangle}{\\langle|I\\_{s}(s,\\omega\\_m)|^2\\rangle} - 1 &= 2Re\\Big(iZ\\_t(\\omega\\_m)e^{i\\omega\\_m[\\hat{\\tau}\\_0-\\tau\\_0(s\\_K)]}\\Big)\\\\\\nonumber\n&\\times\\omega\\_m\\frac{qI\\_0}{E\\_0}[\\tau\\_1(s) - \\tau\\_1(s\\_K)]\\\\\\nonumber\n&\\times e^{-\\frac{\\sigma\\_{\\eta}^2\\omega\\_m^2}{2}\\big[\\tau\\_1^2(s) + (\\tau\\_1(s) - \\tau\\_1(s\\_K))^2\\big]}\\end{aligned}$$ \nWe note that *τ*1(*s*) = − 1/*β**c* times the *M*56 element from the modulator to point *s*, *τ̂*0 − *τ*0(*s**K*) = Δ*z*/*β**c*, and *I*0 = *q**n*0*β**c*, where *n*0 is the mean longitudinal hadron density. We identify the wavenumber *k* = *ω**m*/*β**c* and use Eq. [eqtn:defimpedtime] to translate *Z**t*(*ω*) → − *Z*(*k*)/*β*. This translates the fractional signal modification above to\n \n$$\\begin{aligned}\n\\label{eqtn:mike\\_final\\_translated}\n &\\frac{\\langle|I\\_{t}(s,\\beta c k)|^2\\rangle}{\\langle|I\\_{s}(s,\\beta c k)|^2\\rangle} - 1 = -2n\\_0\\frac{q^2c}{E\\_0}k M\\_{56}^{KD}\\\\\\nonumber\n &\\times[{{\\rm Im}}(Z(k))\\cos(k\\Delta z) + {{\\rm Re}}(Z(k))\\sin(k\\Delta z)]\\\\\\nonumber\n&\\times e^{-\\frac{\\sigma\\_{\\eta}^2k^2}{2}\\Big[\\big(M\\_{56}^{MD}\\big)^2 + \\big(M\\_{56}^{KD}\\big)^2\\Big]}\\end{aligned}$$ \nThis agrees with Eq. [eqtn:pwrrelative] in the purely longitudinal limit.\nTheory from Back-Propagation of Phase Space Density\n---------------------------------------------------\nWe now approach the problem of signal modification using the formalism described in, except using the wake and impedance formalism from Eq. [eqtn:wakedef] and [eqtn:impeddef]. This involves back-propagating the beam through the cooler to describe the phase space distribution of the beam at the detector in terms of the known distribution at the modulator. This gives us a longitudinal phase-space density at the detector equal to\n \n$$\\begin{aligned}\n\\label{eqtn:stupakov\\_density}\nf(z,\\eta) = n\\_0F\\_0(&\\eta - \\Delta\\eta(z - M\\_{56}^{KD}\\eta))\\\\\\nonumber\n+ \\delta f^{(m)}(&z - M\\_{56}^{MD}\\eta - \\Delta z + M\\_{56}^{MK}\\Delta\\eta(z - M\\_{56}^{KD}\\eta),\\\\\\nonumber\n&\\eta - \\Delta \\eta (z - M\\_{56}^{KD}\\eta)) \\end{aligned}$$ \nwhere *F*0(*η*) is the energy-dependence of the base longitudinal phase-space density, *n*0 is the mean longitudinal beam density, and *δ**f*(*m*) is some perturbation at the modulator. Note that *z* and *η* are evaluated at the detector, so that the above equation is just writing the original phase space density, *n*0*F*0(*η*(*m*)) + *δ**f*(*m*)(*z*(*m*), *η*(*m*)), in terms of the detector *z*, *η* coordinates.\nWe use Eq. [eqtn:wakedef] and [eqtn:impeddef] to write\n \n$$\\begin{aligned}\n\\label{eqtn:stupakov\\_kick}\n \\Delta \\eta(z) &= \\frac{1}{2\\pi}\\int\\_{-\\infty}^{\\infty} e^{ikz} \\Delta\\tilde{\\eta}(k) dk\\\\\\nonumber\n &= -\\frac{q^2c}{2\\pi E\\_0}\\int\\_{-\\infty}^{\\infty} e^{ikz} Z(k)\\delta\\tilde{n}^{(m)}\\_k dk\\end{aligned}$$ \nwhere *Z*(*k*) is the wake impedance and *δ**ñ**k*(*m*) is the Fourier transform of the hadron longitudinal density perturbation in the modulator, with a general definition\n \n$$\\begin{aligned}\n\\label{eqtn:stupakov\\_n}\n \\delta\\tilde{n}\\_k &= \\int\\_{-\\infty}^{\\infty} d\\eta \\delta \\tilde{f}\\_k(\\eta)\\\\\\nonumber\n &= \\int\\_{-\\infty}^{\\infty} d\\eta \\int\\_{-\\infty}^{\\infty} dz \\delta f(z,\\eta) e^{-ikz}\\end{aligned}$$ \nIf we Taylor expand *F*0 to first order about the energy kick from the kicker and ignore the Δ*η* dependence in the expression for *δ**f*(*m*), since it is already assumed to be a small correction to *F*0, we find the perturbation to the phase space density at the detector:\n \n$$\\begin{aligned}\n\\label{eqtn:stupakov\\_taylor}\n \\delta f(z,\\eta) &= -n\\_0F'\\_0(\\eta)\\Delta\\eta(z - M\\_{56}^{KD}\\eta)\\\\\\nonumber\n &+ \\delta f^{(m)}(z - M\\_{56}^{MD}\\eta - \\Delta z, \\eta)\\end{aligned}$$ \nThe power of signal modification is given by the correlator\n \n$$\\begin{aligned}\n\\label{eqtn:stupakov\\_correlator}\n \\langle\\delta\\tilde{n}\\_k \\delta\\tilde{n}\\_{k'}\\rangle = \\int\\_{-\\infty}^{\\infty} d\\eta d\\eta' \\langle \\delta\\tilde{f}\\_k(\\eta) \\delta\\tilde{f}\\_{k'}(\\eta')\\rangle\\end{aligned}$$ \nWe split the right hand side of Eq. [eqtn:stupakovtaylor] into two parts:\n \n$$\\begin{aligned}\n\\label{eqtn:stupakov\\_split}\n &\\delta f^{(1)}(z,\\eta) \\equiv -n\\_0F'\\_0(\\eta)\\Delta\\eta(z - M\\_{56}^{KD}\\eta)\\\\\\nonumber\n &\\delta f^{(2)}(z,\\eta) \\equiv \\delta f^{(m)}(z - M\\_{56}^{MD}\\eta - \\Delta z, \\eta)\\end{aligned}$$ \nTaking the Fourier transform of the first term, we find\n \n$$\\begin{aligned}\n\\label{eqtn:stupkov\\_f1\\_transform}\n \\delta \\tilde{f}^{(1)}(k) &= -n\\_0F'\\_0(\\eta)\\int\\_{-\\infty}^{\\infty}e^{-ikz}\\Delta\\eta(z - M\\_{56}^{KD}\\eta)dz\\\\\\nonumber\n &=-n\\_0F'\\_0(\\eta)e^{-ikM\\_{56}^{KD}\\eta}\\int\\_{-\\infty}^{\\infty}e^{-ikz'}\\Delta\\eta(z')dz'\\\\\\nonumber\n &=n\\_0F'\\_0(\\eta)\\frac{q^2c}{E\\_0}Z(k)e^{-ikM\\_{56}^{KD}\\eta}\\delta\\tilde{n}^{(m)}\\_k\\end{aligned}$$ \nwhere we have made use of Eq. [eqtn:stupakovkick] in the last step.\nThe Fourier transform of the second term in Eq. [eqtn:stupakovsplit] yields\n \n$$\\begin{aligned}\n\\label{eqtn:stupakov\\_f2\\_transform}\n \\delta\\tilde{f}^{(2)} &= \\int\\_{-\\infty}^{\\infty} e^{-ikz} \\delta f^{(m)}(z - M\\_{56}^{MD}\\eta - \\Delta z, \\eta) dz\\\\\\nonumber\n &= e^{-ikM\\_{56}^{MD}\\eta}e^{-ik\\Delta z}\\int\\_{-\\infty}^{\\infty} e^{-ikz'} \\delta f^{(m)}(z', \\eta) dz'\\\\\\nonumber\n &= e^{-ikM\\_{56}^{MD}\\eta}e^{-ik\\Delta z}\\delta\\tilde{f}^{(m)}\\_k(\\eta)\\end{aligned}$$ \nCalculating the correlator in Eq. [eqtn:stupakovcorrelator], we see that it involves the sum of three terms: ⟨*δ**f̃**k*(1)(*η*)*δ**f̃**k*ʹ(1)(*η*ʹ)⟩, ⟨*δ**f̃**k*(2)(*η*)*δ**f̃**k*ʹ(2)(*η*ʹ)⟩, and ⟨*δ**f̃**k*(1)(*η*)*δ**f̃**k*ʹ(2)(*η*ʹ) + *δ**f̃**k*(2)(*η*)*δ**f̃**k*ʹ(1)(*η*ʹ)⟩. The first of these is quadratic in the wake impedance, and so we will ignore it since we assume that the kick from the wake is small. The second term is evaluated using Eq. 4 of :\n \n$$\\begin{aligned}\n\\label{eqtn:stupakov\\_self\\_corr}\n \\langle \\delta\\tilde{f}^{(2)}\\_k(\\eta) \\delta\\tilde{f}^{(2)}\\_{k'}(\\eta')\\rangle = 2\\pi n\\_0F\\_0(\\eta)\\delta(k+k')\\delta(\\eta-\\eta')\\end{aligned}$$ \nand integration over energy deviations yields\n \n$$\\begin{aligned}\n\\label{eqtn:stupakov\\_self\\_n}\n \\langle\\delta\\tilde{n}\\_k^{(2)}\\delta\\tilde{n}\\_{k'}^{(2)}\\rangle = 2\\pi n\\_0\\delta(k+k')\\end{aligned}$$ \nFor the third term, ⟨*δ**f̃**k*(1)(*η*)*δ**f̃**k*ʹ(2)(*η*ʹ) + *δ**f̃**k*(2)(*η*)*δ**f̃**k*ʹ(1)(*η*ʹ)⟩, we note that the two parts become identical if we swap *k* ↔ *k*ʹ. (We integrate over *η* and *η*ʹ, so the switch of these variables doesn’t matter.) We then focus on the first of these\n \n$$\\begin{aligned}\n\\label{eqtn:stupakov\\_term3}\n \\int\\_{-\\infty}^{\\infty}d\\eta d\\eta'\\langle \\delta\\tilde{f}^{(1)}\\_k(\\eta) \\delta&\\tilde{f}^{(2)}\\_{k'}(\\eta')\\rangle\\\\\\nonumber\n =n\\_0\\frac{q^2c}{E\\_0}Z(k) \\int\\_{-\\infty}^{\\infty} &F'\\_0(\\eta)e^{-ikM\\_{56}^{KD}\\eta}e^{-ik'M\\_{56}^{MD}\\eta'}e^{-ik'\\Delta z}\\\\\\nonumber &\\langle\\delta\\tilde{n}\\_k^{(m)}\\delta\\tilde{f}^{(m)}\\_{k'}(\\eta')\\rangle d\\eta d\\eta'\\end{aligned}$$ \nThe correlator ⟨*δ**ñ**k*(*m*)*δ**f̃**k*ʹ(*m*)(*η*ʹ)⟩ is the integral of Eq. [eqtn:stupakovselfcorr] over *η*, yielding\n \n$$\\begin{aligned}\n\\label{eqtn:stupakov\\_term3b}\n &\\int\\_{-\\infty}^{\\infty}d\\eta d\\eta'\\langle \\delta\\tilde{f}^{(1)}\\_k(\\eta) \\delta\\tilde{f}^{(2)}\\_{k'}(\\eta')\\rangle\\\\\\nonumber\n =&2\\pi n\\_0^2\\frac{q^2c}{E\\_0}Z(k)\\delta(k+k')\\\\\\nonumber\n \\times&\\int\\_{-\\infty}^{\\infty}d\\eta d\\eta' F'\\_0(\\eta)e^{-ikM\\_{56}^{KD}\\eta}e^{-ik'M\\_{56}^{MD}\\eta'}e^{-ik'\\Delta z} F\\_0(\\eta')\\\\\\nonumber\n =&2\\pi ikM\\_{56}^{KD} n\\_0^2\\frac{q^2c}{E\\_0}Z(k)\\delta(k+k')\\\\\\nonumber\n \\times&\\int\\_{-\\infty}^{\\infty}d\\eta d\\eta' F\\_0(\\eta)e^{-ikM\\_{56}^{KD}\\eta}e^{ikM\\_{56}^{MD}\\eta'}e^{ik\\Delta z} F\\_0(\\eta')\\end{aligned}$$ \nIf we take the particular case of a Gaussian energy distribution with RMS *σ**η*, the integrals yield\n \n$$\\begin{aligned}\n\\label{eqtn:stupakov\\_term3c}\n &\\int\\_{-\\infty}^{\\infty}d\\eta d\\eta'\\langle \\delta\\tilde{f}^{(1)}\\_k(\\eta) \\delta\\tilde{f}^{(2)}\\_{k'}(\\eta')\\rangle\\\\\\nonumber\n =&2\\pi ikM\\_{56}^{KD} n\\_0^2\\frac{q^2c}{E\\_0}Z(k)\\delta(k+k')e^{ik\\Delta z}\\\\\\nonumber\n &\\times e^{-\\sigma\\_{\\eta}^2k^2[(M\\_{56}^{KD})^2 + (M\\_{56}^{MD})^2]/2}\\end{aligned}$$ \nAs noted above, the correlator corresponding to ⟨*δ**f̃**k*(2)(*η*)*δ**f̃**k*ʹ(1)(*η*ʹ)⟩ is the same as what we have just computed, with the interchange *k* ↔ *k*ʹ. Noting that the delta function sends *k*ʹ ↔ − *k* and that *Z*(*k*) is the Fourier transform of a real-valued function, so that *Z*(*k*ʹ) = *Z*( − *k*) = *Z*\\*(*k*), we sum the two halves of this correlator to obtain\n \n$$\\begin{aligned}\n\\label{eqtn:stupakov\\_term3\\_full}\n &2\\pi ikM\\_{56}^{KD} n\\_0^2\\frac{q^2c}{E\\_0}[Z(k)e^{ik\\Delta z} - Z^\\*(k)e^{-ik\\Delta z}]\\\\\\nonumber\n &\\times e^{-\\sigma\\_{\\eta}^2k^2[(M\\_{56}^{KD})^2 + (M\\_{56}^{MD})^2]/2}\\delta(k+k')\\\\\\nonumber\n =&-4\\pi kM\\_{56}^{KD} n\\_0^2\\frac{q^2c}{E\\_0}(Re[Z(k)]\\sin(k\\Delta z) + Im[Z(k)]\\cos(k\\Delta z))\\\\\\nonumber\n &\\times e^{-\\sigma\\_{\\eta}^2k^2[(M\\_{56}^{KD})^2 + (M\\_{56}^{MD})^2]/2}\\delta(k+k')\\end{aligned}$$ \nAdding in the result from Eq. [eqtn:stupakovselfn]\n \n$$\\begin{aligned}\n\\label{eqtn:stupakov\\_final}\n\\langle\\delta\\tilde{n}\\_k \\delta\\tilde{n}\\_{k'}\\rangle &= 2\\pi n\\_0\\delta(k+k')\\Big[1 - 2 kM\\_{56}^{KD} n\\_0\\frac{q^2c}{E\\_0}\\\\\\nonumber\n&\\times(Re[Z(k)]\\sin(k\\Delta z) + Im[Z(k)]\\cos(k\\Delta z))\\\\\\nonumber\n&\\times e^{-\\sigma\\_{\\eta}^2k^2[(M\\_{56}^{KD})^2 + (M\\_{56}^{MD})^2]/2}\\Big]\\end{aligned}$$ \nThe fractional signal modification is then\n \n$$\\begin{aligned}\n\\label{eqtn:stupakov\\_final\\_frac}\n\\frac{\\langle\\delta\\tilde{n}\\_k \\delta\\tilde{n}\\_{k'}\\rangle}{\\langle\\delta\\tilde{n}^{(m)}\\_k \\delta\\tilde{n}^{(m)}\\_{k'}\\rangle} &- 1 = -2 kM\\_{56}^{KD} n\\_0\\frac{q^2c}{E\\_0}\\\\\\nonumber\n&\\times(Re[Z(k)]\\sin(k\\Delta z) + Im[Z(k)]\\cos(k\\Delta z))\\\\\\nonumber\n&\\times e^{-\\sigma\\_{\\eta}^2k^2[(M\\_{56}^{KD})^2 + (M\\_{56}^{MD})^2]/2}\\end{aligned}$$ \nWe see that, in the absence of transverse motion, this is identical to Eq. [eqtn:pwrrelative].\nQuadratic Term\n==============\nIn the simplification of Eq. [eqtn:pwr], we only kept the terms linear in the wake amplitude. If we instead keep terms to second order, we arrive at a corrective term equal to\n \n$$\\begin{aligned}\n\\label{eqtn:second\\_order\\_sum}\n\\bigg(k M\\_{56}^{KD}\\frac{q^2}{E\\_0}\\bigg)^2 \\sum\\_{i \\neq a}&e^{-ik\\big[z\\_m^{(i)} - z\\_m^{(a)} + M^{MD}\\_{5u}\\big(\\vec{x}\\_u^{(i)} - \\vec{x}\\_u^{(a)}\\big)\\big]}\\\\\\nonumber\n\\times \\sum\\_{j,\\ell} \\bigg[&-\\frac{1}{2}w(z\\_m^{(i)} + M\\_{5u}^{MK}\\vec{x}\\_u^{(i)} + \\Delta z - z\\_m^{(j)})w(z\\_m^{(i)} + M\\_{5u}^{MK}\\vec{x}\\_u^{(i)} + \\Delta z - z\\_m^{(\\ell)})\\\\\\nonumber\n&-\\frac{1}{2}w(z\\_m^{(a)} + M\\_{5u}^{MK}\\vec{x}\\_u^{(a)} + \\Delta z - z\\_m^{(j)})w(z\\_m^{(a)} + M\\_{5u}^{MK}\\vec{x}\\_u^{(a)} + \\Delta z - z\\_m^{(\\ell)})\\\\\\nonumber\n&+w(z\\_m^{(i)} + M\\_{5u}^{MK}\\vec{x}\\_u^{(i)} + \\Delta z - z\\_m^{(j)})w(z\\_m^{(a)} + M\\_{5u}^{MK}\\vec{x}\\_u^{(a)} + \\Delta z - z\\_m^{(\\ell)})\\bigg]\\end{aligned}$$ \nIf the integral of the wake is zero, taking the average of this quantity over all possible *z**m*(*j*) will yield zero unless *j* = *i*, *a*, or ℓ. Similarly, the only values of ℓ which yield non-trivial results are *i*, *a*, or *j*. Since there are 4 ways to pick *j* and ℓ equal to some combination of *i* and *a* and ∼ *N* > > 4 ways to pick them equal to one another and not equal to *i* or *a*, we only focus on the latter case. Of the three wake cross-terms, only the last one survives, since the others have either *z**m*(*a*) or *z**m*(*i*) appearing only in the complex phase, which would average to zero.\nDoing the sums over the three free parameters, *i*, *a*, and *j*, we find this is equal to\n \n$$\\begin{aligned}\n\\label{eqtn:second\\_order\\_int}\n&N^3\\bigg(k M\\_{56}^{KD}\\frac{q^2}{E\\_0}\\bigg)^2 \\int\\_{-L/2}^{L/2} dz\\_m^{(i)}dz\\_m^{(a)}dz\\_m^{(j)}/L^3\\\\\\nonumber\n&\\times\\int\\_{-\\infty}^{\\infty}d^5\\vec{x}^{(i)}d^5\\vec{x}^{(a)}\\rho\\big(\\vec{x}^{(i)}\\big)\\rho\\big(\\vec{x}^{(a)}\\big) \\\\\\nonumber\n&\\times e^{-ik\\big[z\\_m^{(i)} - z\\_m^{(a)} + M^{MD}\\_{5u}\\big(\\vec{x}\\_u^{(i)} - \\vec{x}\\_u^{(a)}\\big)\\big]}\\\\\\nonumber\n&\\times w(z\\_m^{(i)} + M\\_{5u}^{MK}\\vec{x}\\_u^{(i)} + \\Delta z - z\\_m^{(j)})\\\\\\nonumber\n&\\times w(z\\_m^{(a)} + M\\_{5u}^{MK}\\vec{x}\\_u^{(a)} + \\Delta z - z\\_m^{(j)})\\end{aligned}$$ \nApproximating the longitudinal integrals over *z**m*(*i*) and *z**m*(*a*) as extending from − ∞ to ∞, making the change of variables *z*ʹ ≡ *z**m*(*i*) + *M*5*u**M**K**x⃗**u*(*i*) + Δ*z* − *z**m*(*j*) and *z*ʺ ≡ *z**m*(*a*) + *M*5*u**M**K**x⃗**u*(*a*) + Δ*z* − *z**m*(*j*), and using Eq. [eqtn:impeddef], we obtain\n \n$$\\begin{aligned}\n\\label{eqtn:second\\_order\\_int2}\n&N\\bigg(n\\_0 k M\\_{56}^{KD}\\frac{q^2c}{E\\_0}\\bigg)^2 \\int\\_{-L/2}^{L/2} dz\\_m^{(j)}/L\\\\\\nonumber\n&\\times\\int\\_{-\\infty}^{\\infty}d^5\\vec{x}^{(i)}d^5\\vec{x}^{(a)}\\rho\\big(\\vec{x}^{(i)}\\big)\\rho\\big(\\vec{x}^{(a)}\\big) \\\\\\nonumber\n&\\times e^{-ik\\big[\\big(M^{MD}\\_{5u} - M^{MK}\\_{56}\\big)\\big(\\vec{x}\\_u^{(i)} - \\vec{x}\\_u^{(a)}\\big)\\big]}Z(k)Z(-k)\\\\\\nonumber\n&=N\\bigg(n\\_0 k M\\_{56}^{KD}\\frac{q^2c}{E\\_0}\\bigg)^2 |Z(k)|^2\\\\\\nonumber\n&\\times\\int\\_{-\\infty}^{\\infty}d^5\\vec{x}^{(i)}d^5\\vec{x}^{(a)}\\rho\\big(\\vec{x}^{(i)}\\big)\\rho\\big(\\vec{x}^{(a)}\\big) \\\\\\nonumber\n&\\times e^{-ik\\big[\\big(M^{MD}\\_{5u} - M^{MK}\\_{56}\\big)\\big(\\vec{x}\\_u^{(i)} - \\vec{x}\\_u^{(a)}\\big)\\big]}\\\\\\nonumber\n&=N\\bigg(n\\_0 k M\\_{56}^{KD}\\frac{q^2c}{E\\_0}\\bigg)^2 |Z(k)|^2e^{-k^2 \\sum\\_{u\\neq5} \\sigma^2\\_{\\hat{x}\\_u} \\big(\\hat{M}^{MD}\\_{5u} - \\hat{M}^{Mk}\\_{5u}\\big)^2}\\end{aligned}$$ \nwhere we have used the fact that *Z*( − *k*) = *Z*\\*(*k*) for real wakes, the definition *n*0 = *N*/*L*, and the transformed transfer matrices and beam sizes of Eq. [eqtn:mtilde] - [eqtn:sigmas]. This yields a corrected version of Eq. [eqtn:pwrrelative]:\n \n$$\\begin{aligned}\n\\label{eqtn:pwr\\_relative\\_correct}\n\\frac{\\Delta |\\tilde{n}(k)|^2}{|\\tilde{n}(k)|^2} &= -2n\\_0\\frac{q^2c}{E\\_0}kM\\_{56}^{KD}\\\\\\nonumber\n&\\times[{{\\rm Re}}(Z(k))\\sin(k\\Delta z) + {{\\rm Im}}(Z(k))\\cos(k\\Delta z)]\\\\\\nonumber\n&\\times e^{-\\frac{k^2}{2} \\sum\\_{u\\neq5} \\sigma^2\\_{\\hat{x}\\_u}\\Big[\\big(\\hat{M}^{MD}\\_{5u}\\big)^2 + \\big(\\hat{M}^{MD}\\_{5u} - \\hat{M}^{MK}\\_{5u}\\big)^2\\Big]}\\\\\\nonumber\n&+\\bigg(n\\_0 k M\\_{56}^{KD}\\frac{q^2c}{E\\_0}\\bigg)^2 |Z(k)|^2e^{-k^2 \\sum\\_{u\\neq5} \\sigma^2\\_{\\hat{x}\\_u} \\big(\\hat{M}^{MD}\\_{5u} - \\hat{M}^{MK}\\_{5u}\\big)^2}\\end{aligned}$$ \nSimilar results are seen in the derivations of Appendix [app:alternative] if we keep the ∣*G*∣2 term of Eq. [eqtn:mikerelsigsup] or the ⟨*δ**f̃**k*(1)(*η*)*δ**f̃**k*ʹ(1)(*η*ʹ)⟩ term of Appendix [subsec:thrystupakov].\n\nSchottky Signal Modification as a Diagnostic Tool for Coherent Electron Cooling\n===============================================================================\nCoherent electron cooling is a promising technique to cool high-intensity hadron bunches by imprinting the noise in the hadron beam on a beam of electrons, amplifying the electron density modulations, and using them to apply cooling kicks to the hadrons. The typical size for these perturbations can be on the *μ*m scale, allowing us to extend the reach of classical stochastic cooling by several orders of magnitude. However, it is crucial to ensure that the electron and hadron beams are longitudinally aligned within this same *μ*m scale. In order to provide fast feedback for this process, we discuss the extension of signal suppression to coherent electron cooling, and show in both theory and simulation that certain components of the spectral noise in the hadron beam will be predictably modified at the several percent level, which may be detected by observations of the radiation of the hadron beam.\nIntroduction\n============\nIn high-intensity hadron storage rings, intrabeam scattering (IBS) and beam-beam effects will degrade the beam emittance over the length of the store, limiting machine luminosity. In particular, at the planned Electron-Ion Collider (EIC), the IBS times are expected to be at the timescale of a couple hours, and so some form of strong hadron cooling is necessary to achieve the physics goals.\nThe proposed method in this case is microbunched electron cooling (MBEC), a particular form of coherent electron cooling (CeC). This was first introduced in, and the theory has since been developed extensively in. The premise of MBEC is that the hadron beam to be cooled copropagates with an electron beam in a straight “modulator” section, during which time the hadrons will provide energy kicks to the electrons. The two beams are then separated, and the electrons pass through a series of amplifiers to change this initial energy modulation into a density modulation and amplify it. The hadrons travel through their own chicane before meeting the electrons again in a straight “kicker” section. Here, the amplified density modulations in the electron beam provide energy kicks to the hadrons. By tuning the hadron chicane so that the hadron delay in travelling from the modulator to the kicker is energy-dependent, we may arrange it so that the energy kick that the hadron receives in the kicker tends to correct initial energy offsets. If the chicane also gives the hadrons a delay dependent on their transverse phase-space coordinates and if there is non-zero dispersion in the kicker, then the transverse emittance of the hadron beam can also be cooled. In the current design of an MBEC Cooler for the EIC, the typical scale of the electron density modulations at the top energy will be ∼ 1*μ*m. This corresponds to about 4 orders of magnitude higher bandwidth than can be achieved with microwave stochastic cooling, allowing the cooling of dense hadron bunches, but also making alignment a challenge.\nIt is important that the hadron arrives in the kicker at the same time as the density perturbations which it had induced in the electron beam, or else it will receive entirely uncorrelated energy kicks. Comparing the ∼ 100m distance between modulator and kicker to the ∼ 1*μ*m density perturbations in the electron beam, we see that the transit times of the electrons and hadrons must be maintained at a level of ten parts per billion. In order to commission and operate such a system, it is necessary to have some way to quickly measure the relative alignment of the electron and hadron beams at the sub-micron scale. Directly observing cooling would require waits on the timescale of hours, which would make commissioning painful and prevent any sort of fast feedback during operations. The method proposed here is to make use of “signal modification,” an extension of the well-known signal suppression of microwave stochastic cooling to the case of MBEC cooling. The principle of this method is that after the hadron beam has received its cooling kicks, it will propagate to a “detector” where the power of the hadron beam at particular wavelengths may be measured. If the hadron beam is well-aligned with the electron beam, this will produce a predictable change in the spectral content of the hadron beam on a single-pass basis.\nAs our model, we assume that the cooling section consists of a modulator, where the hadron beam imprints on the electrons; a kicker, where the electrons provide an energy kick back to the hadrons; and a detector, where we will observe the density spectrum in the hadron beam. See Fig. [fig:layout].\n\nIn Section [sec:thry], we provide a theoretical derivation of signal modification. In Section [sec:sim], we discuss simulation tools to model this process, and find good agreement with the theoretical predictions. In Section [sec:detection], we discuss what will be needed to measure such a signal experimentally. We present our outlook in Section [sec:conclude] and conclude.\nNote that, although this paper focuses on MBEC, the general form of these results will hold for coherent electron cooling with other amplification mechanisms.\nTheory\n======\nWe derive here the theory of signal modification by directly propagating the particles themselves from the modulator to the detector with arbitrary 6-dimensional transfer matrices. Alternative derivations using the longitudinal Vlasov equation are presented in Appendix [app:alternative]. In subsection [subsec:thrydecohere], we comment on decoherence of the signal when observing a range of frequencies and in subsection [subsec:multikick], we discuss an extension where the kicks are distributed throughout the length of the kicker.\nWe use phase-space coordinates *x*, *x*ʹ, *y*, *y*ʹ, *z*, and *η*, where the first four are the transverse positions and angles, *z* is the particle’s longitudinal position in the bunch, with positive *z* values corresponding to the head of the bunch, and *η* is the fractional energy offset of the particle.\nIn order to characterize the cooling process, we follow and define a longitudinal wake function in the 1D approximation, such that the energy kick a hadron receives in the kicker is the convolution of this wake function with the longitudinal distribution of hadrons in the modulator. Explicitly, the fractional energy kick Δ*η* received by a hadron at longitudinal position *z* within the bunch at the kicker is given by\n \n$$\\begin{aligned}\n\\label{eqtn:wake\\_def}\n\\Delta\\eta(z) = \\frac{q^2}{E\\_0} \\int\\_{-\\infty}^{\\infty}w(z+\\Delta z-z')n(z') dz'\\end{aligned}$$ \nwhere *q* is the hadron charge, *E*0 is the nominal hadron energy, *n*(*z*) is the longitudinal hadron density in the modulator, and Δ*z* is the difference in modulator-to-kicker longitudinal delay between the hadron and electron beams. We also identify a corresponding impedance\n \n$$\\begin{aligned}\n\\label{eqtn:imped\\_def}\n Z(k) = -\\frac{1}{c}\\int\\_{-\\infty}^{\\infty} w(z) e^{-ikz} dz\\end{aligned}$$ \nIn order for these simple longitudinal wakes to hold, we require coherent oscillation of the beams, and therefore the electric field at our frequency of interest must be large compared to the beam sizes. In particular, we require :\n \n$$\\begin{aligned}\n\\label{eqtn:1D}\n \\Sigma\\_{\\perp} \\lesssim \\frac{\\beta\\gamma\\lambda}{2}\\end{aligned}$$ \nwhere Σ⊥ is the transverse beam size, *β* is the relativistic beta, *γ* is the relativistic gamma, and *λ* is the beam oscillation wavelength. Examining the values in Tab. [tab:param], we see that the largest beam size is that of the protons in the modulator and kicker (0.95mm), and the gamma factor is 293, so our approximation is good for wavelengths above ∼ 6.5*μ*m. As we will see, we will be interested mainly in wavelengths of 6*μ*m, at the limit of applicability of the above. However, we note that the requirement of Eq. [eqtn:1D] is likely not a strict one—as it was shown in, in the case when there is not amplification, the 3D theory gives exactly the same result as the 1D model which we employ here.\nIn order for us to neglect Landau damping in the electron beam amplifiers, one usually requires :\n \n$$\\begin{aligned}\n\\label{eqtn:landau\\_damp}\n k \\ll k\\_D,\\end{aligned}$$ \nwhere *k* is the wavenumber of our signal, *k**D* ≡ *ω**p*/*c**σ**β* is the Debye wavenumber, and *σ**β* is the spread in relativistic beta. This electron beam has *k**D* = 4 × 106m− 1 in the amplifiers, larger than the *k* = 106m− 1 of our microbunching. One also has to keep in mind that the concept of Landau damping, with the condition of Eq. [eqtn:landaudamp], is only valid asymptotically, after many plasma oscillations, which is not the case of the microbunching coherent cooling where the electron beam executes $\\frac{1}{4}$ of the plasma oscillation in the amplifier sections. A more accurate analysis of when one can ignore the Landau damping for this case was carried out in Section IV of.\nIn order to ensure that the microbunching is not washed out by the energy spread and divergence of the beam within any of the straight sections, we require :\n \n$$\\begin{aligned}\n \\lambda \\gg \\frac{2L}{\\beta^3\\gamma^2}\\sigma\\_{\\eta}\\end{aligned}$$ \nand\n \n$$\\begin{aligned}\n \\lambda \\gg \\frac{L}{\\beta^2}\\sigma\\_{x'}^2\\end{aligned}$$ \nwhere *σ**η* is the fractional energy spread of the beam, *σ**x*ʹ is the beam divergence, and *L* is the length of a drift. Comparison with the parameters in Tab. [tab:param] shows that these are met for both the electrons and hadrons in all the straight sections.\nWe assume that there is no transverse correlation in the electron or hadron beam. Further discussion of the effects which this might have is available in.\nWe focus our attention on the peak region of the electron and hadron beams and take the limit of a longitudinally infinite and uniform plasma. Since the typical wake wavelength is on the order of a few microns, and the typical bunch lengths are a few mm or longer, this is a reasonable approximation. Finally, we assume that the hadron beam enters the modulator with no correlated structures on the scale of the wake wavelength. Although such structures will be generated within the kicker, their characteristic size is on the micron scale, much less than the millimeter scale longitudinal motion per turn, washing out any memory of the kick by the time the beam enters the modulator again.\nAs illustrated in Fig. [fig:layout], we take the hadrons to have transfer matrix *M**M**K* between modulator and kicker and *M**K**D* between kicker and detector, with the transfer matrix between modulator and detector given by *M**M**D*.\nWe treat our particles as existing in a region of length *L*, much larger than any length scale associated with the wake function, and assume periodic boundary conditions, so that we may arbitrarily shift the limits of integration in our integrals. In this model, we also consider the full 6-dimensional evolution of the hadron beam and ignore any collective effects during beam transport except for the electron-hadron interactions characterized by the wake function, as discussed above. We write the evolution of a hadron’s position between modulator and detector as\n \n$$\\begin{aligned}\n\\label{eqtn:delays}\nz\\_d^{(i)} = &z\\_m^{(i)} + M^{MD}\\_{5u}\\vec{x}\\_u^{(i)}\\\\\\nonumber\n+ &M^{KD}\\_{56}\\sum\\_j \\frac{q^2}{E\\_0} w(z\\_m^{(i)} + M^{MK}\\_{5u}\\vec{x}\\_u^{(i)} + \\Delta z - z\\_m^{(j)})\\end{aligned}$$ \nwhere *z**m*(*i*) is the longitudinal position of particle *i* in the modulator, *z**d*(*i*) is its position within the detector, and *x⃗*(*i*) are the phase-space coordinates of particles *i* in the modulator. We use the convention that the repeated “*u*” subscript refers to sums over the 5 phase-space coordinates *excluding* the longitudinal position. The summation over *j* is over all particles within the hadron beam. The first two terms of this equation describe the modulator to kicker beam evolution by a simple transfer matrix, while the final term gives the contribution to delay due to the extra energy kick our particle receives in the kicker due to the wakes of all particles in the beam, including itself.\nAt the detector, the longitudinal density of the hadron beam is\n \n$$\\begin{aligned}\n\\label{eqtn:rho\\_d}\nn(z) = \\sum\\_i \\delta(z - z\\_d^{(i)})\\end{aligned}$$ \nwith corresponding density in Fourier space\n \n$$\\begin{aligned}\n\\label{eqtn:rho\\_tilde\\_d}\n\\tilde{n}(k) &= \\int\\_{-\\infty}^{\\infty}\\sum\\_i e^{-ikz} \\delta(z - z\\_d^{(i)}) dz\\\\\\nonumber\n&=\\sum\\_i e^{-ikz\\_d^{(i)}}\\end{aligned}$$ \nThe power in the hadron spectrum at a given wavenumber is then given by\n \n$$\\begin{aligned}\n\\label{eqtn:pwr}\n|\\tilde{n}(k)|^2 = &\\sum\\_{i,a}e^{-ik\\big[z\\_d^{(i)} - z\\_d^{(a)}\\big]}\\\\\\nonumber\n=N + &\\sum\\_{i \\neq a}e^{-ik\\big[z\\_m^{(i)} - z\\_m^{(a)} + M^{MD}\\_{5u}\\big(\\vec{x}\\_u^{(i)} - \\vec{x}\\_u^{(a)}\\big)\\big]}\\\\\\nonumber\n&\\times e^{-ikM^{KD}\\_{56}q^2/E\\_0\\sum\\_j w\\big(z\\_m^{(i)} + M^{MK}\\_{5u}\\vec{x}\\_u^{(i)} + \\Delta z - z\\_m^{(j)}\\big)}\\\\\\nonumber\n&\\times e^{ikM^{KD}\\_{56}q^2/E\\_0\\sum\\_j w\\big(z\\_m^{(a)} + M^{MK}\\_{5u}\\vec{x}\\_u^{(a)} + \\Delta z - z\\_m^{(j)}\\big)}\\end{aligned}$$ \nwhere we have substituted in the expression for *z**d*(*i*) given in Eq. [eqtn:delays] and used the fact that the *i* = *a* terms in the sum are all equal to 1, giving us the *N* out front, where *N* is the number of particles in the length-*L* section of the beam.\nTypically, the kick from the wake is small, and so we may Taylor-expand the final two exponentials above to linear order. We thereby obtain\n \n$$\\begin{aligned}\n\\label{eqtn:pwr\\_taylor}\n|\\tilde{n}(k)|^2 &= N + \\sum\\_{i \\neq a}e^{-ik\\big[z\\_m^{(i)} - z\\_m^{(a)} + M^{MD}\\_{5u}\\big(\\vec{x}\\_u^{(i)} - \\vec{x}\\_u^{(a)}\\big)\\big]}\\\\\\nonumber\n\\times \\big[1 &- ikM^{KD}\\_{56}\\frac{q^2}{E\\_0}\\sum\\_j w\\big(z\\_m^{(i)} + M^{MK}\\_{5u}\\vec{x}\\_u^{(i)} + \\Delta z - z\\_m^{(j)}\\big)\\\\\\nonumber\n&+ ikM^{KD}\\_{56}\\frac{q^2}{E\\_0}\\sum\\_j w\\big(z\\_m^{(a)} + M^{MK}\\_{5u}\\vec{x}\\_u^{(a)} + \\Delta z - z\\_m^{(j)}\\big)\\big]\\end{aligned}$$ \nwith the effect of second-order terms considered in Appendix [app:secondorder].\nWe now wish to take the expectation value of the beam power, requiring integrals over the 12 phase space coordinates of particles *i* and *a* and the longitudinal position of particle *j*. However, note that the dependence on *z**m*(*j*) appears only in the argument of the wake functions. If the total integral of the wake is zero, then, if particle *j* is distinct from both particles *i* and *a*, this integral will evaluate to 0. We then need only consider the terms *j* = *i* and *j* = *a* in those sums. The beam power can then be written as\n \n$$\\begin{aligned}\n\\label{eqtn:pwr\\_taylor2}\n|\\tilde{n}(k)|^2 &= N + \\sum\\_{i \\neq a}e^{-ik\\big[z\\_{ia} + M^{MD}\\_{5u}\\big(\\vec{x}\\_u^{(i)} - \\vec{x}\\_u^{(a)}\\big)\\big]}\\\\\\nonumber\n\\times [1 &- ikM^{KD}\\_{56}\\frac{q^2}{E\\_0} w(M^{MK}\\_{5u}\\vec{x}\\_u^{(i)} + \\Delta z)\\\\\\nonumber\n&+ ikM^{KD}\\_{56} \\frac{q^2}{E\\_0}w(M^{MK}\\_{5u}\\vec{x}\\_u^{(a)} + \\Delta z)\\\\\\nonumber\n&- ikM^{KD}\\_{56} \\frac{q^2}{E\\_0}w(z\\_{ia} + M^{MK}\\_{5u}\\vec{x}\\_u^{(i)} + \\Delta z)\\\\\\nonumber\n&+ ikM^{KD}\\_{56} \\frac{q^2}{E\\_0}w(-z\\_{ia} + M^{MK}\\_{5u}\\vec{x}\\_u^{(a)} + \\Delta z)]\\end{aligned}$$ \nwhere we have made the definition *z**i**a* ≡ *z**m*(*i*) − *z**m*(*a*). Since we assume a homogeneous hadron bunch, *z**m*(*i*) and *z**m*(*a*) themselves are irrelevant and *z**i**a* has the same probability distribution as them.\nWe note in the above formula that the first three terms have their only *z**i**a* dependence in the leading exponential, and so performing an average over all *z**i**a* will be zero. We therefore need only focus on the 4th and 5th terms. Approximating the *N*(*N* − 1) terms in the above sum as *N*2, the relevant integral for the 4th term is\n \n$$\\begin{aligned}\n\\label{eqtn:4th\\_term}\n&-N^2ikM\\_{56}^{KD} \\int\\_{-L/2}^{L/2} dz\\_{ia}/L \\int\\_{-\\infty}^{\\infty} d^5\\vec{x}^{(i)} d^5\\vec{x}^{(a)} \\rho\\big(\\vec{x}^{(i)}\\big) \\rho\\big(\\vec{x}^{(a)}\\big)\\\\\\nonumber\n&\\times \\frac{q^2}{E\\_0}w(z\\_{ia} + M^{MK}\\_{5u}\\vec{x}\\_u^{(i)} + \\Delta z)\ne^{-ik\\big[z\\_{ia} + M^{MD}\\_{5u}\\big(\\vec{x}\\_u^{(i)} - \\vec{x}\\_u^{(a)}\\big)\\big]}\\end{aligned}$$ \nwhere $\\rho\\big(\\vec{x}\\big)$ the hadron phase-space density in the modulator within the 5 phase-space coordinates excluding longitudinal position.\nApproximating the longitudinal integral as extending from − ∞ to ∞, using the impedance of Eq. [eqtn:impeddef], and making an appropriate change of variables to *z*ʹ ≡ *z**i**a* + *M*5*u**M**K**x⃗**u*(*i*) + Δ*z*, the longitudinal integral in Eq. [eqtn:4thterm] may be evaluated, leaving us with\n \n$$\\begin{aligned}\n\\label{eqtn:4th\\_term\\_2}\n&N^2ikM\\_{56}^{KD} \\frac{q^2c}{E\\_0}\\frac{1}{L} Z(k) e^{ik\\Delta z}\\\\\\nonumber\n&\\times\\int\\_{-\\infty}^{\\infty} d^5\\vec{x}^{(i)} d^5\\vec{x}^{(a)} \\rho\\big(\\vec{x}^{(i)}\\big) \\rho\\big(\\vec{x}^{(a)}\\big)\\\\\\nonumber\n&\\times e^{-ik\\big[M^{MD}\\_{5u}\\big(\\vec{x}\\_u^{(i)} - \\vec{x}\\_u^{(a)}\\big) - M^{MK}\\_{5u}\\vec{x}\\_u^{(i)}\\big]}\\end{aligned}$$ \nTo perform the remaining integrals, we write the evolution of the phase-space coordinates explicitly in terms of action-angle variables and Courant-Snyder parameters at the start of the transfer matrix, finding\n \n$$\\begin{aligned}\n\\label{eqtn:action\\_angle}\nM\\_{5u}\\vec{x}\\_u &= M\\_{51}(\\sqrt{2J\\_x\\beta\\_x}\\cos(\\phi\\_x)+D\\_x\\eta)\\\\\\nonumber\n&+ M\\_{52}(-\\sqrt{2J\\_x/\\beta\\_x}[\\sin(\\phi\\_x)+\\alpha\\_x\\cos(\\phi\\_x)]+D'\\_x\\eta)\\\\\\nonumber\n&+ M\\_{53}(\\sqrt{2J\\_y\\beta\\_y}\\cos(\\phi\\_y)+D\\_y\\eta)\\\\\\nonumber\n&+ M\\_{54}(-\\sqrt{2J\\_y/\\beta\\_y}[\\sin(\\phi\\_y)+\\alpha\\_y\\cos(\\phi\\_y)]+D'\\_y\\eta)\\\\\\nonumber\n&+ M\\_{56}\\eta\\\\\\nonumber\n&\\\\\\nonumber\n&= (M\\_{51}-\\frac{\\alpha\\_x}{\\beta\\_x}M\\_{52})\\sqrt{2J\\_x\\beta\\_x}\\cos(\\phi\\_x)\\\\\\nonumber\n&- M\\_{52}\\sqrt{2J\\_x/\\beta\\_x}\\sin(\\phi\\_x)\\\\\\nonumber\n&+ (M\\_{53}-\\frac{\\alpha\\_y}{\\beta\\_y}M\\_{54})\\sqrt{2J\\_y\\beta\\_y}\\cos(\\phi\\_y)\\\\\\nonumber\n&- M\\_{54}\\sqrt{2J\\_y/\\beta\\_y}\\sin(\\phi\\_y)\\\\\\nonumber\n&+ (D\\_xM\\_{51} + D'\\_xM\\_{52} + D\\_yM\\_{53} + D'\\_yM\\_{54} + M\\_{56})\\eta\\\\\\nonumber\n&\\\\\\nonumber\n\\equiv & \\hat{M}\\_{51}\\hat{x} + \\hat{M}\\_{52}\\hat{x}' + \\hat{M}\\_{53}\\hat{y} + \\hat{M}\\_{54}\\hat{y}' + \\hat{M}\\_{56}\\hat{\\eta}\\end{aligned}$$ \nwith\n \n$$\\begin{aligned}\n\\label{eqtn:m\\_tilde}\n&\\hat{M}\\_{51} \\equiv M\\_{51}-\\frac{\\alpha\\_x}{\\beta\\_x}M\\_{52}\\\\\\nonumber\n&\\hat{M}\\_{52} \\equiv M\\_{52}\\\\\\nonumber\n&\\hat{M}\\_{53} \\equiv M\\_{53}-\\frac{\\alpha\\_y}{\\beta\\_y}M\\_{54}\\\\\\nonumber\n&\\hat{M}\\_{54} \\equiv M\\_{54}\\\\\\nonumber\n&\\hat{M}\\_{56} \\equiv D\\_xM\\_{51} + D'\\_xM\\_{52} + D\\_yM\\_{53} + D'\\_yM\\_{54} + M\\_{56}\\end{aligned}$$ \nFor a Gaussian beam, the *x̂*, *x̂*ʹ, *ŷ*, *ŷ*ʹ, and *η̂* are normally distributed with\n \n$$\\begin{aligned}\n\\label{eqtn:sigmas}\n &\\sigma\\_{\\hat{x}} = \\sqrt{\\epsilon\\_x\\beta\\_x}\\\\\\nonumber\n &\\sigma\\_{\\hat{x}'} = \\sqrt{\\epsilon\\_x/\\beta\\_x}\\\\\\nonumber\n &\\sigma\\_{\\hat{y}} = \\sqrt{\\epsilon\\_y\\beta\\_y}\\\\\\nonumber\n &\\sigma\\_{\\hat{y}'} = \\sqrt{\\epsilon\\_y/\\beta\\_y}\\\\\\nonumber\n &\\sigma\\_{\\hat{\\eta}} = \\sigma\\_{\\eta}\\end{aligned}$$ \nwhere the *ε* are the horizontal and vertical emittances and *σ**η* is the fractional energy spread. In this case, the remaining ten integrals in Eq. [eqtn:4thterm2] may be performed, yielding\n \n$$\\begin{aligned}\n\\label{eqtn:4th\\_term\\_final}\n&ikM\\_{56}^{KD}\\frac{q^2c}{E\\_0} \\frac{N^2}{L} Z(k) e^{ik\\Delta z}\\\\\\nonumber\n&\\times e^{-\\frac{k^2}{2} \\sum\\_{u\\neq5} \\sigma^2\\_{\\hat{x}\\_u}\\Big[\\big(\\hat{M}^{MD}\\_{5u}\\big)^2 + \\big(\\hat{M}^{MD}\\_{5u} - \\hat{M}^{MK}\\_{5u}\\big)^2\\Big]}\\end{aligned}$$ \nA similar procedure may be applied to the fifth term of Eq. [eqtn:pwrtaylor2], resulting in\n \n$$\\begin{aligned}\n\\label{eqtn:5th\\_term\\_final}\n&-ikM\\_{56}^{KD} \\frac{q^2c}{E\\_0}\\frac{N^2}{L} Z(-k) e^{-ik\\Delta z}\\\\\\nonumber\n&\\times e^{-\\frac{k^2}{2} \\sum\\_{u\\neq5} \\sigma^2\\_{\\hat{x}\\_u}\\Big[\\big(\\hat{M}^{MD}\\_{5u}\\big)^2 + \\big(\\hat{M}^{MD}\\_{5u} - \\hat{M}^{MK}\\_{5u}\\big)^2\\Big]}\\end{aligned}$$ \nMaking use of the fact that, for real wakes, *Z*( − *k*) = *Z*\\*(*k*), and defining *n*0 ≡ *N*/*L* as the mean linear density of the hadrons, we may sum Eqs. [eqtn:4thtermfinal] and [eqtn:5thtermfinal] and incorporate them back into Eq. [eqtn:pwrtaylor2], obtaining\n \n$$\\begin{aligned}\n\\label{eqtn:pwr\\_final}\n|\\tilde{n}(k)|^2 &= N - 2Nn\\_0\\frac{q^2c}{E\\_0}kM\\_{56}^{KD}\\\\\\nonumber\n&\\times[{{\\rm Re}}(Z(k))\\sin(k\\Delta z) + {{\\rm Im}}(Z(k))\\cos(k\\Delta z)]\\\\\\nonumber\n&\\times e^{-\\frac{k^2}{2} \\sum\\_{u\\neq5} \\sigma^2\\_{\\hat{x}\\_u}\\Big[\\big(\\hat{M}^{MD}\\_{5u}\\big)^2 + \\big(\\hat{M}^{MD}\\_{5u} - \\hat{M}^{MK}\\_{5u}\\big)^2\\Big]}\\end{aligned}$$ \nThen, the fractional change in beam power as a function of electron/hadron misalignment is\n \n$$\\begin{aligned}\n\\label{eqtn:pwr\\_relative}\n\\frac{\\Delta |\\tilde{n}(k)|^2}{|\\tilde{n}(k)|^2} &= -2n\\_0\\frac{q^2c}{E\\_0}kM\\_{56}^{KD}\\\\\\nonumber\n&\\times[{{\\rm Re}}(Z(k))\\sin(k\\Delta z) + {{\\rm Im}}(Z(k))\\cos(k\\Delta z)]\\\\\\nonumber\n&\\times e^{-\\frac{k^2}{2} \\sum\\_{u\\neq5} \\sigma^2\\_{\\hat{x}\\_u}\\Big[\\big(\\hat{M}^{MD}\\_{5u}\\big)^2 + \\big(\\hat{M}^{MD}\\_{5u} - \\hat{M}^{MK}\\_{5u}\\big)^2\\Big]}\\end{aligned}$$ \nNote that if ${{\\rm Re}}(Z(k)) = 0$ and ${{\\rm Im}}(Z(k)) > 0$ (which is the case considered below, see Eq. [eqtn:impedapprox]), and if *M*56*K**D* > 0, then perfect alignment of the hadrons and electrons, ie Δ*z* = 0, results in Δ∣*ñ*(*k*)∣2 < 0, corresponding to noise suppression below the shot noise in the beam. Such noise suppression has been previously studied theoretically in and observed experimentally in. Our result Eq. [eqtn:pwrrelative] is an agreement with the theoretical analysis of.\nDecoherence\n-----------\nThe above posits that the amount of signal modification has a purely sinusoidal dependence on the electron/hadron misalignment, ie,\n \n$$\\begin{aligned}\n\\label{eqtn:decoherence\\_ideal}\n \\frac{\\Delta |\\tilde{n}(k)|^2}{|\\tilde{n}(k)|^2} = A(k) \\cos(k\\Delta z + \\theta\\_0)\\end{aligned}$$ \nwhere *A* is some amplitude and *θ*0 is a phase, equal to 0 for an antisymmetric wake.\nHowever, the above derivation assumes an observation at a pure single frequency. If we take the more realistic case that we sample some range of frequencies with bandwidth Δ*k*, the signal amplitude will be\n \n$$\\begin{aligned}\n\\label{eqtn:decoherence\\_true}\n \\frac{\\Delta |\\tilde{n}(k)|^2}{|\\tilde{n}(k)|^2} &\\approx A(k)\\frac{1}{\\Delta k}\\int\\_{k-\\Delta k/2}^{k+\\Delta k/2} \\cos(k'\\Delta z + \\theta\\_0) dk'\\\\\\nonumber\n &= A(k)\\cos(k\\Delta z + \\theta\\_0) \\frac{\\sin(\\Delta k\\Delta z/2)}{\\Delta k\\Delta z/2}\\end{aligned}$$ \nso that the amplitude of the oscillations will decay over lengths of ∼ 1/Δ*k*.\nIntegrated Kicker\n-----------------\nThe above analysis assumes that the full kick to the hadrons takes place at a single point in the kicker. However, in the more realistic case where the kick is applied over the full length of the kicker, we may change Eq. [eqtn:delays] to\n \n$$\\begin{aligned}\n &z\\_d^{(i)} = z\\_m^{(i)} + M^{MD}\\_{5u}\\vec{x}\\_u^{(i)}\\\\\\nonumber\n&+ \\frac{1}{N\\_s}\\sum\\_{s=1}^{N\\_s}M^{K\\_sD}\\_{56}\\sum\\_j \\frac{q^2}{E\\_0} w(z\\_m^{(i)} + M^{MK\\_s}\\_{5u}\\vec{x}\\_u^{(i)} + \\Delta z - z\\_m^{(j)})\\end{aligned}$$ \nwhere we now split our kick in the kicker into *N**s* fractional kicks at locations *K**s*.\nThe rest of the analysis may be carried through in the same way as before, and we arrive at\n \n$$\\begin{aligned}\n \\frac{\\Delta |\\tilde{n}(k)|^2}{|\\tilde{n}(k)|^2} &= -2n\\_0\\frac{q^2c}{E\\_0}\\frac{1}{N\\_s}\\sum\\_{s=1}^{N\\_s}kM\\_{56}^{K\\_sD}\\\\\\nonumber\n&\\times[{{\\rm Re}}(Z(k))\\sin(k\\Delta z) + {{\\rm Im}}(Z(k))\\cos(k\\Delta z)]\\\\\\nonumber\n&\\times e^{-\\frac{k^2}{2} \\sum\\_{u\\neq5} \\sigma^2\\_{\\hat{x}\\_u}\\Big[\\big(\\hat{M}^{MD}\\_{5u}\\big)^2 + \\big(\\hat{M}^{MD}\\_{5u} - \\hat{M}^{MK\\_s}\\_{5u}\\big)^2\\Big]}\\end{aligned}$$ \nHowever, we have not observed any significant difference numerically between using the above equation and Eq. [eqtn:pwrrelative], with the single kick taken at the kicker center.\nSimulation\n==========\nIn order to check the validity and limits of the above theory, we make use of simulation. We first examine the case of a perfectly linear simulation, where the fractional energy kick to a given hadron in the kicker is simply the convolution of the wake function with the longitudinal hadron distribution in the modulator. We then turn our attention to a more detailed simulation, where the electrons and hadrons are tracked with a particle-in-cell code in order to incorporate saturation effects. In this and future sections, we consider the cooling parameters currently planned for 275 GeV protons in the EIC, listed in Tab. [tab:param]. The electron optics are assumed to be kept roughly constant within the modulator, kicker, and amplifiers through the use of focusing. Due to their much higher energy, the hadrons see the modulator and kicker as drifts, with the optics parameters specified at the center. Electron chicanes are defined from the end of one straight section to the start of the next one, while the proton *M*56 and phase advances are evaluated from modulator center to kicker center.\n[!hbt]\n| | |\n| --- | --- |\n| ***Geometry*** | |\n| Modulator Length (m) | 45 |\n| Kicker Length (m) | 45 |\n| Number of Amplifier Straights | 2 |\n| Amplifier Straight Lengths (m) | 37 |\n| ***Proton Parameters*** | |\n| Energy (GeV) | 275 |\n| Protons per Bunch | 6.9e10 |\n| Average Current"},"data_id":{"kind":"string","value":"arxiv_0000358"}}},{"rowIdx":95,"cells":{"content":{"kind":"string","value":"We introduce a model of a controlled random graph process. In this model, the edges of the complete graph *K**n* are ordered randomly and then revealed, one by one, to a player called Builder. He must decide, immediately and irrevocably, whether to purchase each observed edge. The observation time is bounded by parameter *t*, and the total budget of purchased edges is bounded by parameter *b*. Builder’s goal is to devise a strategy that, with high probability, allows him to construct a graph of purchased edges possessing a target graph property P, all within the limitations of observation time and total budget. We show the following:\n* Builder has a strategy to achieve *k*-vertex-connectivity at the hitting time for this property by purchasing at most *c**k**n* edges for an explicit *c**k* < *k*; and a strategy to achieve minimum degree *k* (slightly) after the threshold for minimum degree *k* by purchasing at most (1 + *ɛ*)*k**n*/2 edges (which is optimal).\n* Builder has a strategy to create a Hamilton cycle at the hitting time for Hamiltonicity by purchasing at most *C**n* edges for an absolute constant *C* > 1; this is optimal in the sense that *C* cannot be arbitrarily close to 1. This substantially extends the classical hitting time result for Hamiltonicity due to Ajtai–Komlós–Szemerédi and Bollobás.\n* Builder has a strategy to create a perfect matching by time (1 + *ɛ*)*n*log*n*/2 while purchasing at most (1 + *ɛ*)*n*/2 edges (which is optimal).\n* Builder has a strategy to create a copy of a given *k*-vertex tree if *t* ≥ *b* ≫ max{(*n*/*t*)*k* − 2, 1}, and this is optimal;\n* For ℓ = 2*k* + 1 or ℓ = 2*k* + 2, Builder has a strategy to create a copy of a cycle of length ℓ if $b\\gg\\max \\{n^{k+2}/t^{k+1},n/\\sqrt{t}\\}$, and this is optimal.\nIntroduction\n============\nThe model\n---------\nThe random graph process, introduced by Erds and Rényi , is a stochastic process that starts with an empty *n*-vertex graph and, at each step, gains a new uniformly selected random edge. At any fixed time *t*, the process is distributed as the uniform random graph *G*(*n*, *t*). A **graph property** is a family of graphs that is closed under isomorphisms. It is **monotone** if it is closed under addition of edges. A vast body of literature is concerned with finding *thresholds* for various monotone graph properties in the random graph process, namely, with finding time *t**c* such that the random graph process belongs to P with high probability (**whp**; that is, with probability tending to 1 as *n* → ∞) whenever *t* is much (or somewhat) larger than *t**c* and does not belong to P **whp** if *t* is much (or somewhat) smaller than *t**c*.\nIn many cases, if one observes the random graph process at time *t* above the threshold, the graph has the desired property but, in fact, contains a much sparser subgraph that has the property. For example, one of the outstanding results in this model regards a threshold for Hamiltonicity: Komlós and Szemerédi and, independently, Bollobás showed that if 2*t*/*n* − log*n* − loglog*n* tends to infinity, then the random graph process, at time *t*, contains a Hamilton cycle **whp**. Evidently, not all ∼ *n*log*n*/2 observed edges must be included in the resulting graph for it to be Hamiltonian (as any Hamilton cycle only uses *n* edges). Nevertheless, when an edge arrives, it is generally hard to determine whether it will be crucial for Hamiltonicity.\nThe above motivates the following “online” version of building a subgraph of the random graph process. We think of it as a one-player game, where the player (“**Builder**”) has a limited “budget”. Edges “arrive” one at a time in random order and are presented to Builder. Whenever he *observes* an edge, he must immediately and irrevocably decide whether to *purchase* it. A non-purchased edge is thrown away and never reappears. The **time** (total number of presented edges) and the **budget** (maximum number of edges Builder can purchase) are both capped. The question is as to whether or not, under the given time and budget constraints, Builder has a strategy that allows him to obtain, **whp**, a particular monotone graph property in the graph of purchased edges.\nTo demonstrate the model, consider the property of connectedness. Let *τ**C* be the (random) time at which the random graph process becomes connected. Evidently, if Builder wishes to construct a connected subgraph of the random graph process, he must purchase at least *n* − 1 edges. However, in this case purchasing *n* − 1 edges suffices: Builder’s strategy would be to purchase an edge if and only if it decreases the number of connected components in his graph. That way, Builder maintains a forest, which becomes connected exactly at time *τ**C*. Therefore, in this example, Builder does not have to “pay” for having to make decisions online. However, this is not always the case. For example, if Builder wants to purchase a triangle and wishes to do so while observing *o*(*n*2) edges, he must purchase (many) more than three edges (see ).\n \nWe denote the underlying random graph process at time *t* (namely, after *t* edges have been presented to Builder) by *G**t* (where the number of vertices, *n*, is implicit). As we mentioned earlier, *G**t* is distributed as the uniform random graph *G*(*n*, *t*). The **hitting time** for a monotone graph property P is the (random) minimum time *t* for which *G**t* has P. Builder’s graph at time *t*, denoted *B**t*, is a subgraph of *G**t* on the same vertex set that consists of the edges purchased by Builder by time *t*. A **(*t*, *b*)-strategy** of Builder is a (potentially random) function that, for any *s* ≤ *t*, decides whether to purchase the edge presented at time *s*, given *B**s* − 1, under the limitation that *B**s* has at most *b* edges.\nOur results\n-----------\nOur first result discusses a strategy for constructing a subgraph with a given minimum degree and, for *k* ≥ 3, of a given vertex-connectivity, as quickly as possible. For every positive integer *k*, denote by *τ**k* the hitting time for minimum degree *k* in the random graph process. It was proved by Erds and Rényi that $\\tau\\_k=\\frac{n}{2}(\\log{n}+(k-1)\\log\\log{n}+O(1))$ **whp** (see ). Let *κ* denote vertex-connectivity.\n[Minimum degree and vertex-connectivity at the hitting time][thm:mindeg:hit] For every positive integer *k* there exists a constant *o**k* ∈ (*k*/2, 3*k*/4] such that the following holds. If $b\\ge o\\_k n+\\omega(\\sqrt{n}\\log{n})$ then there exists a (*τ**k*, *b*)-strategy *B* of Builder such that \nlim*n* → ∞P(*δ*(*B**τ**k*) ≥ *k*) = 1. \n If *k* ≥ 3, the same strategy guarantees \nlim*n* → ∞P(*κ*(*B**τ**k*) ≥ *k*) = 1. \nFor *k* = 1, as we mentioned in the introduction, a sufficient and necessary budget for connectivity at the hitting time is *n* − 1. For *k* = 2, below implies that a budget of *O*(*n*) suffices for 2-connectivity at the hitting time for minimum degree 2. The constant *o**k* in is explicit and computable (see in ; the first three values in the sequence *o**k* are $\\frac{3}{4}$, $\\frac{11}{8}$ and $\\frac{63}{32}$), and is roughly *k*/2 when *k* is large (see ). We believe that at the hitting time *τ**k*, that constant is optimal (see and the discussion following it). However, if we allow the time to be optimal only asymptotically, then an asymptotically optimal budget suffices.\n[Minimum degree][thm:mindeg] Let *k* be a positive integer and let *ɛ* > 0. If *t* ≥ (1 + *ɛ*)*n*log*n*/2 and *b* ≥ (1 + *ɛ*)*k**n*/2 then there exists a (*t*, *b*)-strategy *B* of Builder such that \nlim*n* → ∞P(*δ*(*B**t*) ≥ *k*) = 1. \nThe above theorem is tight in the following sense: if *t* ≤ (1 − *ɛ*)*n*log*n*/2 then the underlying random graph process has, **whp**, isolated vertices; and if *b* < *k**n*/2 then every strategy will fail since Builder will not be able to purchase enough edges to obtain the required minimum degree. We remark that after an earlier version of this paper appeared online, a vertex-connectivity version of that holds for every *k* ≥ 2 has been proved by Lichev (see ). follows from below (for *k* = 1) and from Lichev’s result (for *k* ≥ 2). We keep it here due to its short and simple proof (see ). Note that is *not* implied by Lichev’s result, as implies a strategy that succeeds **whp** *at the hitting time* *τ**k*, whereas Lichev’s strategy typically requires *ω*(*n*) additional steps (which can, as we demonstrate later in and the discussion following it, make a big difference). An asymptotic version of Lichev’s result, according to which for every *ɛ* > 0 there exists *k*0 such that for every *k* ≥ *k*0 there exists a strategy to build a *k*-vertex-connected graph in time (1 + *ɛ*)*n*log*n*/2 and with budget (1 + *ɛ*)*k**n*/2 follows from together with the observation that *o**k* ∼ *k*/2 when *k* is large ().\nWe continue to Hamiltonicity. The classical *hitting-time* result of Bollobás and independently of Ajtai, Komlós, and Szemerédi states that the random graph process, **whp**, becomes Hamiltonian as soon as its minimum degree reaches 2. As we reminded earlier, *τ*2 = Θ(*n*log*n*), while a Hamilton cycle has only *n* edges. The following result states that there exists an online algorithm choosing only linearly many edges from the first *τ*2 edges of the random graph process that still obtains a Hamilton cycle **whp**. Moreover, as we show in the proof, this algorithm is *extremely* simple, deterministic, and requires only local information. The theorem recovers – and substantially strengthens – the aforementioned classical result.\n[Hamiltonicity at the hitting time][thm:ham:hit] There exists *C* > 1 for which there exists a (*τ*2, *C**n*)-strategy *B* of Builder with \nlim*n* → ∞P(*B**τ*2is Hamiltonian) = 1. \nThe constant *C* we obtain in is large, and we made no serious effort to optimise it. A natural question is whether it could be as low as 1 + *ɛ* for every *ɛ* > 0. The answer, found by Anastos (see the concluding remarks there), is negative. As part of the proof of, we show that the so-called *random *k*-nearest neighbour graph* (see ) – for large enough constant *k* – is **whp** Hamiltonian (). This partially solves an open problem of the first author \\*Problem 45. To prove, we prove a stronger hitting time version of () which states that the *k*-nearest neighbour graph, considered as a process and *stopped as the minimum degree becomes 2*, is **whp** Hamiltonian.\nWe remark that an immediate corollary of (using ) is that if *t* ≥ (1 + *ɛ*)*n*log*n*/2 (namely, slightly after the Hamiltonicity threshold) and *b* ≥ *C**n* (namely, when the budget is inflated by a constant), then there exists a (*t*, *b*)-strategy of Builder that succeeds (**whp**) in building a Hamilton cycle. In an earlier version of this paper that appeared online, we also proved a complementary statement: if *t* ≥ *C**n*log*n*/2 and *b* ≥ (1 + *ɛ*)*n*, then there exists a “successful” (*t*, *b*)-strategy (see ). After that early version appeared online, Anastos proved, using different and more involved techniques, that Hamiltonicity can be achieved in asymptotically optimal time *and* budget. Namely, he showed that in the above statement, one could take *C* = *C*(*ɛ*) = 1 + *ɛ*. He states this result as a special case in a model that generalises both the budget model discussed here and the so-called Achlioptas process (see discussion in ). We leave here our original proof due to its relative shortness and simplicity (see ). Note that does not follow from the work of Anastos.\nFor perfect matchings, we show that there exists an asymptotically-optimal-time asymptotically-optimal-budget strategy.\n[Perfect matchings][thm:pm] Suppose *n* is even. For every *ɛ* > 0 if *t* ≥ (1 + *ɛ*)*n*log*n*/2 and *b* ≥ (1 + *ɛ*)*n*/2 then there exists a (*t*, *b*)-strategy *B* of Builder such that \nlim*n* → ∞P(*B**t*has a perfect matching) = 1. \nAn earlier version of this paper that appeared online contained a non-optimal version of (analogous to ). Recently, Anastos proved a more general version of (see remark after ), using different and more involved techniques. The proof we provide here is more elementary.\nThe next two theorems discuss optimal strategies for purchasing small subgraphs. We resolve the problem whenever the target subgraph is a fixed tree or a cycle.\n[Small trees][thm:trees] Let *k* ≥ 3 be an integer and let *T* be a *k*-vertex tree. If *t* ≥ *b* ≫ max{(*n*/*t*)*k* − 2, 1} then there exists a (*t*, *b*)-strategy *B* of Builder such that \nlim*n* → ∞P(*T* ⊆ *B**t*) = 1 \n and if *b* ≪ (*n*/*t*)*k* − 2 then for any (*t*, *b*)-strategy *B* of Builder, \nlim*n* → ∞P(*T* ⊆ *B**t*) = 0. \nFor a visualization of see.\n[t!]\n[ xmin=0, xmax=1.5, ymin=0, ymax=1, x=5cm,y=5cm, samples=100, axis y line=center, axis x line=middle, xlabel=log*n**t*, xtick=0,1/2,2/3,3/4,4/5,1, xticklabels=0,1/2,2/3,,4/5,1, ylabel=log*n**b*, ytick=0,1/2,2/3,3/4,4/5,1, yticklabels=0,1/2,2/3,,4/5,1, every axis x label/.style= at=(ticklabel\\* cs:1.05), anchor=west,, every axis y label/.style= at=(ticklabel\\* cs:1.05), anchor=south,, reverse legend, ] ;\n;\n;\n;\n;\n[fig:trees]\nWe show in the proof of that if *t* ≫ *n* then, in fact, there exists a (*t*, *b*)-strategy for *b* = *k* − 1 that succeeds **whp** in building a copy of a *k*-vertex tree.\n[Short cycles][thm:cycles] Let *k* ≥ 1 be an integer and let *H* = *C*2*k* + 1 or *H* = *C*2*k* + 2. Write $b^\\*=b^\\*(n,t,k)=\\max\\{n^{k+2}/t^{k+1},n/\\sqrt{t}\\}$. If *t* ≫ *n* and *b* ≫ *b*\\* then there exists a (*t*, *b*)-strategy *B* of Builder such that \nlim*n* → ∞P(*H* ⊆ *B**t*) = 1, \n and if *t* ≪ *n* or *b* ≪ *b*\\* then for any (*t*, *b*)-strategy *B* of Builder, \nlim*n* → ∞P(*H* ⊆ *B**t*) = 0. \nFor a visualization of see. For discussion on the difficulty arising in handling graphs with more than one cycle see.\n[t]\n[ xmin=0, xmax=1, ymin=0, ymax=1, x=5cm,y=5cm, samples=100, axis y line=center, axis x line=middle, xlabel=log*n**t*, xtick=0,1/9,1/7,1/5,1/3,1, extra x ticks=0, xticklabels=0,,,6/5,4/3,2, extra x tick labels=1, ylabel=log*n**b*, ytick=1,1/2,1/2-1/18,1/2-1/10,1/3,0, yticklabels=1,1/2,,2/5,1/3,0, every axis x label/.style= at=(ticklabel\\* cs:1.05), anchor=west,, every axis y label/.style= at=(ticklabel\\* cs:1.05), anchor=south,, reverse legend, ] ;\n; ;\n; ;\n; ;\n; ;\n[fig:cycles]\nTools and techniques\n--------------------\n#### Expanders\n*Expanders* are graphs in which (sufficiently) small sets expand. Namely, these are graphs in which the neighbourhood of each small set is larger than that set by a constant factor[4](#fn4). It is well known that connected expanders are helpful in finding Hamilton cycles (see ); more concretely, connected non-hamiltonian expanders have many “boosters”, namely, non-edges whose addition to the graph creates a graph that is either Hamiltonian or whose longest path is longer (for a comprehensive account, we refer the reader to ). Thus, expanders will play a crucial role in the proof of (and also in the proof of ): Builder will attempt to construct (sparse, and therefore cheap) connected expanders within the random graph process. A standard method for obtaining sparse expanders in random graphs is choosing an appropriate random (sub)graph model. Natural candidates are discussed in the following paragraphs.\n#### Random *k*-out graphs\nIn the goals described in, Builder must achieve a certain minimum degree in his (spanning) graph. The (standard) random graph process is quite wasteful in this regard, as to avoid isolated vertices, a superlinear number of edges must arrive. Thus, a wise Builder would instead construct a much sparser subgraph of the random graph process with the desired minimum degree. A classical sparse graph with a given minimum degree is the *random regular graph* (see, e.g., in ). However, such a graph is generally very hard to construct online. A much simpler alternative is the so-called *random *k*-out graph* (see, e.g., in ). In the *k*-out graph, each vertex chooses, random, independently, and without repetitions, *k* neighbours to connect to. Thus, the total number of edges in a *k*-out graph is at most *k**n*, and the minimum degree is at least *k*. Unfortunately, *k*-out graphs are not generally subgraphs of the random graph process at the hitting time for minimum degree *k*. We will therefore analyse a different model, which can be considered the undirected counterpart of the *k*-out graph (see the next paragraph). Nevertheless, we will exploit the simplicity of the *k*-out graph to analyse its (slightly more complicated) undirected counterpart.\n#### Random *k*-nearest neighbour graphs\nSuppose the edges of the complete graph are endowed with (potentially random) “lengths”. The graph made up only of the *k* shortest edges incident with each vertex is called the *random *k*-nearest neighbour graph*, and has been studied in various, primarily geometric, contexts (see, e.g., ). When the weights are independent uniform random variables supported on [0, 1], the model becomes a symmetric random graph[5](#fn5), which we denote by O*k* (see ). To prove, we devise a simple strategy that emulates O*k*. Since the minimum degree of O*k* is obviously *k*, the statement would follow from a theorem of Cooper and Frieze according to which the number of edges of O*k* is at most *c**n* for some concrete constant *c* = *c*(*k*) < *k* which is roughly *k*/2 when *k* is large (see ).\nBy observing that O*k* is stochastically dominated by the random *k*-out graph (), we prove that it is a connected () expander (), and use that to prove that it is Hamiltonian (). We use that helpful fact in several places: when aiming to create a Hamilton cycle at the hitting time and with inflated budget, we construct a subgraph of O*k* that is already Hamiltonian; when creating a Hamilton cycle under an asymptotically optimal budget, we emulate copies of O*k* in small parts of the graph, to allow us to *absorb* a long path into a Hamilton cycle; finally, we use a similar approach for the construction of perfect matchings.\n#### High-level arguments for the containment of fixed graphs\nWe give brief proof outlines for. The proof for the upper bound on the budget threshold for trees is essentially inductive: given a fixed *k*-vertex tree *T*, we let *T*ʹ be obtained from *T* by removing a leaf. The inductive argument is that Builder can construct sufficiently many copies of *T*ʹ while leaving enough time to extend some of them into copies of *T*. The proof for the lower bound is based on a similar idea: we show that Builder cannot construct a connected component of size at least *k*. In order to construct such a component, he needs to construct enough smaller components so that in the remaining time, he will, **whp**, connect two of them so that the resulting component will be large enough.\nThe strategy for obtaining a cycle of length ℓ = 2*k* + 1 goes through the construction of many “traps”, namely, non-edges whose addition to Builder’s graph would create the desired cycle. An optimised way to construct many such traps quickly is by constructing *r* *d*-ary trees of depth *k* (for a correct choice of *r*, *d*). If *d* is large, most pairs of leaves in such a tree are connected by a path of length 2*k*, and thus form a trap. The argument for a cycle of length ℓ = 2*k* + 2 is similar. We complement the upper bound with two lower bounds; each matches the upper bound in a different regime. We first show a “universal” lower bound, based on the straightforward observation that the number of traps Builder has in his graph is bounded by *O*(*b*2), as he has at most 2*b* vertices of positive degree in his graph. A more involved lower bound fits the earlier regimes (lower values of *t*). To this aim, we observe that Builder’s graph is typically Θ(1)-degenerate. Then, we use this observation, together with an estimate on the number of paths of a fixed length that contain a vertex of “high” degree, to bound from above the number of paths of length ℓ (and hence the number of traps).\nRelated work\n------------\nThe study of random graph processes is at the heart of the theory of random graphs, providing a dynamic point of view on their evolution. One of the main questions is to determine the thresholds of monotone increasing graph properties or their (more refined) hitting times. Classical results of this sort include the thresholds of minimum degree and vertex-connectivity *k* , the appearance of a “giant” component , Hamiltonicity and the containment of fixed subgraphs . For a comprehensive coverage of the topic, we refer the reader to the books of Bollobás , Janson, Łuczak and Ruciński , and Frieze and Karoński . In our context, an obvious necessary condition for the existence of a winning strategy for Builder is that the “time” is above the threshold (or at least at the hitting time), as otherwise, the underlying graph process is not guaranteed to have the desired property **whp**. Evidently, if *b* = *t* (namely, if the budget equals the time), and *t* is (sufficiently) above the threshold of the target property, then Builder has a winning strategy: the naive one that purchases each observed edge. Since in many cases Builder can do better, the model can be seen as an extension of the “standard” random graph process.\nIn the last couple of decades, partly inspired by the remarkable work of Azar, Broder, Karlin and Upfal on *balanced allocations*, there has been a growing interest in *controlled* random processes. In the context of graph processes, an algorithm is provided with a random flow of edges (usually, but not always, the random graph process) and with (the *offline* version) or without (the *online* version) “peeking into the future”, makes a decision that handles that flow by accepting/rejecting edges, by colouring them, or by other means. We mention several related models that fall into this category, mainly in the online version.\nThe *Achlioptas process*, proposed by Dimitris Achlioptas in 2000, is perhaps the most studied controlled random graph process. In this process, the algorithm is fed by a stream of random *k*-sets of edges (with or without repetitions) and should pick, immediately and irrevocably (“online”), a single edge to accept, rejecting the others. The algorithm’s goal is to make the graph of accepted edges satisfy some monotone increasing (decreasing) graph property while minimising (maximising) the total number of rounds. Early work on this model treated the original question posed by Achlioptas of *avoiding* a giant component for as long as possible. Other works considered the opposite of the original question, namely, the question of accelerating the appearance of the giant (see also for a general framework). The model has also been studied for other objectives, such as avoiding or creating a fixed subgraph or obtaining a Hamilton cycle . Several variants have also been studied, such as an *offline* version, in which the algorithm sees all *k*-sets of edges at the beginning of the process (see, e.g., ); and a *memoryless* version, in which the algorithm’s decision may not depend on its previous decisions (see ). A common generalisation of the Achlioptas process and the process considered in this paper, where an algorithm should pick *at most* a single edge to accept/purchase subject to a restricted “budget”, was recently studied by Anastos in (see remarks after ).\nThe *semi-random process*, proposed by the third author in 2016, is a variation of the Achlioptas process in which the algorithm is fed by a stream of random spanning stars (with repetitions) instead of *k*-sets of edges. As with the Achlioptas process, the algorithm must immediately and irrevocably pick a single edge to accept. Work on this model treats monotone properties such as containment of fixed subgraphs , minimum degree and vertex-connectivity , perfect matchings , Hamilton cycles and bounded degree spanning graphs .\nAnother model that fits in this setting was studied by Frieze and Pegden and by Anastos under the name “purchasing under uncertainty”. In their model, whenever an edge arrives, it is given an independent random “cost”, and the algorithm has to decide whether to purchase that edge, aiming to pay the minimum total cost required to obtain a desired graph property.\nIn a Ramsey-type version of controlled random processes, incoming edges are coloured (irrevocably) by an online algorithm. The algorithm aims to avoid, or to create, a monochromatic property. In , the triangle-avoidance game for up to 3 colours is discussed. This was extended to any fixed cyclic graph and any number of colours in . The model was further studied in the contexts of the giant component and Hamilton cycles .\nFinally, we would like to mention a related adaptation of the *two-stage optimisation with recourse* framework for the *minimum spanning tree* model . Here, every edge of the complete graph is given an independent random “Monday” cost and another independent random “Tuesday” cost. The algorithm sees all Monday costs and decides (immediately and irrevocably) which edges to incorporate into its (future) spanning tree. Then, Tuesday costs are revealed, and the algorithm uses them to complete his (current) forest into a spanning tree. The algorithm’s goal is to minimise the total cost of edges in his constructed tree.\nPaper organisation and notation\n-------------------------------\nThe rest of the paper is organised as follows. In we introduce some preliminaries. is devoted to the random *k*-nearest neighbours graph and to the *k*-out graph. contains the proofs of, and contains the proofs of. In the final section,, we mention a few relevant open problems.\nThroughout the paper, all logarithms are in the natural basis. If *f*, *g* are functions of *n* we write *f* ≼ *g* if *f* = *O*(*g*), *f* ≪ *g* if *f* = *o*(*g*), *f* ≍ *g* if *f* = Θ(*g*), and *f* ∼ *g* if *f* = (1 + *o*(1))*g*. For two vertices *u*, *v* of a graph we write *u* ∼ *v* to denote that they are neighbours. For simplicity and clarity of presentation, we often make no particular effort to optimise the constants obtained in our proofs and omit floor and ceiling signs whenever they are not crucial.\nPreliminaries\n=============\nConcentration inequalities\n--------------------------\nWe will make use of the following version of Chernoff bounds (see, e.g., in, \\*Chapter 2).\n[Chernoff bounds][thm:chernoff] Let *n* ≥ 1 be an integer and let *p* ∈ [0, 1], let *x* ∼ Bin(*n*, *p*), and let *μ* = E*x* = *n**p*. Then, for every *a* > 0, \n$${\\mathbb{P}}(x\\le \\mu-a) \\le \\exp\\left(-\\frac{a^2}{2\\mu}\\right),\\qquad\n {\\mathbb{P}}(x\\ge \\mu+a) \\le \\exp\\left(-\\frac{a^2}{2(\\mu+a/3)}\\right).$$ \nThe following are trivial yet useful bounds.\n[cl:bin:lowtail] Let *n* ≥ 1 be an integer and let $p\\in[0,\\frac{1}{2}]$, and let *x* ∼ Bin(*n*, *p*). Write *q* = 1 − *p* and let 1 ≤ *k* ≤ *n**p*/*q* be an integer. Then\n \n$${\\mathbb{P}}(x\\le k) \\le \\left(\\frac{enp}{kq}\\right)^k e^{-np}.$$ \nBy the binomial theorem, for every *α* ∈ (0, *p*/*q*], \n$$(1+\\alpha)^n\n = \\sum\\_{i=0}^n \\binom{n}{i}\\alpha^i\n \\ge \\sum\\_{i=0}^k\n \\binom{n}{i}\n \\left(\\frac{p}{q}\\right)^i\n \\left(\\frac{\\alpha q}{p}\\right)^k,\n $$ \n hence \n$$ {\\mathbb{P}}(x\\le k)\n = \\sum\\_{i=0}^k \\binom{n}{i}\\left(\\frac{p}{q}\\right)^i q^n\n \\le \\frac{(1+\\alpha)^n p^k}{\\alpha^kq^k} \\cdot (1-p)^n\n \\le \\frac{e^{\\alpha n} p^k}{\\alpha^kq^k} \\cdot e^{-np}.\n $$ \n Taking *α* = *k*/*n* ≤ *p*/*q* we obtain \n$$ {\\mathbb{P}}(x\\le k)\n \\le \\left(\\frac{enp}{kq}\\right)^k \\cdot e^{-np}.\\qedhere\n $$ \n[cl:bin:uptail] Let *x* ∼ Bin(*n*, *p*) with *μ* = *n**p* and let 1 ≤ *k* ≤ *n*. Then\n \n$${\\mathbb{P}}(x\\ge k) \\le \\left(\\frac{enp}{k}\\right)^k.$$ \nFor a proof see, e.g., .\nThe random graph process\n------------------------\nRecall that *G**t* denotes the random graph process at time *t*. The next lemma essentially states that in *G**t*, where *t* is superlinear (but not too large), there are no small dense sets.\n[lem:span] Suppose that *R* ≫ 1 and *n*− 5*R*2*t*3 ≤ *α* < 1/8. Then, in *G**t*, **whp**, every vertex set of size *r* ≤ *R* spans at most 3*r* edges.\nLet $N=\\binom{n}{2}$ and *p* = *t*/*N*. For a set *S* of *s* edges the probability that *S* is contained in *G**t* is at most *p**s*. By the union bound, the probability that there exists a set *U* of size *r* that spans at least 3*r* edges is at most\n \n$$\\binom{n}{r}\\binom{\\binom{r}{2}}{3r}p^{3r}\n \\le \\left(n r^2p^3\\right)^r.$$ \nWe note that *n**R**p*3 ≍ *n*− 5*R**t*3 ≤ *α*/*R* ≪ 1 and *n**R*2*p*3 ∼ 8*n*− 5*R*2*t*3 ≤ 8*α*. By the union bound over 2 ≤ *r* ≤ *R*, the probability that there exists such a set is at most \n$$\\sum\\_{r=2}^R \\left(n r^2p^3\\right)^r\n \\le \\sum\\_{r=2}^{\\sqrt{R}} \\left(nRp^3\\right)^r\n + \\sum\\_{r=\\sqrt{R}}^{R} \\left(nR^2p^3\\right)^r\n \\le \\sum\\_{r=2}^{\\sqrt{R}} (o(1))^r\n + \\sum\\_{r=\\sqrt{R}}^R (8\\alpha^r) = o(1),$$ \n and the claim follows.\nThe lemma easily implies the following useful fact.\n[cl:degen] Suppose 1 ≪ *b*2 ≤ *n*5/(40*t*3). Then, **whp**, every subgraph of *G**t* with at most *b* edges is 6-degenerate.\nLet *B* be a subgraph of *G**t* with at most *b* edges. Thus, *B* has at most 2*b* non-isolated vertices. Apply with *R* = 2*b* and *t* (we can do so as *b* ≫ 1 and *n*− 5(2*b*)2*t*3 ≤ 1/10). We therefore obtain that, with high probability, for every *q* ≥ 0, every subgraph of *B* on *q* vertices spans at most 3*q* edges, and hence has minimum degree of at most 6, and the result follows.\nWe introduce more notation. For *d* ≥ 1, let *X**d**t* be the set of vertices in *G**t* with degree less than *d*, and let *Y**d**t* be its complement. The next lemmas state that in time linear in *n* there is a transition between “most vertices are of degree less than *d*” and “most vertices are of degree at least *d*”.\n[lem:gnt:large] Let *d* ≥ 1 be an integer. Then, deterministically, ∣*Y**d**t*∣ ≤ 2*t*/*d*.\nIt follows from *t* = ∣*E*(*G**t*)∣ ≥ ∣*Y**d**t*∣ ⋅ *d*/2.\n[lem:gnt:small] For every integer *d* ≥ 1, if *t* ≥ 6*d**n* then ∣*X**d**t*∣ ≤ *n*/100 **whp**.\nWhen proving statements about *G**t* it is often convenient to prove them for *G* ∼ *G*(*n*, *p*) with $p\\sim t/\\binom{n}{2}$ instead. We could then translate the results back to *G**t* using standard methods (see, e.g., in ).\nLet *t* = 6*d**n* (the result would follow for *t* ≥ 6*d**n* due to monotonicity). Write $p=12d/n\\sim t/\\binom{n}{2}$ and *m* = *n*/100. If ∣*X**d**t*∣ > *m* then there exists a vertex set *V*0 with ∣*V*0∣ = *m* such that $|E(V\\_0,\\overline{V\\_0})|<md$. Let $x=|E(V\\_0,\\overline{V\\_0})|$ and note that *x* ∼ Bin(*m*(*n* − *m*), *p*). By and the union bound over the choice of *V*0, the probability that the statement of the lemma does not hold is at most\n \n$$\\begin{aligned}\n \\binom{n}{m}{\\mathbb{P}}(x<md)\n &\\le \\left(\\frac{en}{m}\\right)^m \\left(\\frac{em(n-m)p}{md(1-p)}\\right)^{md} e^{-m(n-m)p}\\\\\n &\\le \\left(\\frac{en}{m}\\right)^m \\left(\\frac{enp}{d}\\right)^{md} e^{-0.99mnp}\n \\le \\left(100e\\cdot\\left(12e\\cdot e^{-11}\\right)^d\\right)^m = o(1).\\qedhere\n \\end{aligned}$$ \nWe will also use the next two known results about *τ**d*.\n[][lem:gnt:deg:hit] For every fixed *d* ≥ 1, *τ**d* = *n*(log*n* + (*d* − 1)loglog*n* + *x*)/2, where *x* = *O*(1) **whp**.\nFor a proof of the next lemma see, e.g., .\n[lem:gnt:small:far] Let *d* ≥ 1 be an integer. Then, **whp**, ∣*X**d**τ*2∣ ≤ *n*0.5, the distance between two vertices in *X**d**τ*2 is at least 5, and no vertex in *X**d**τ*2 lies in a cycle of length at most 4.\nFinally, we would need the following lemma on the size of the *k*-core of a random graph. The ***k*-core** of a graph is its (unique) maximal subgraph of minimum degree at least *k*.\n[lem:gnt:core] For every *k* ≥ 1 the size of the *k*-core of *G**t*, for *t* = 8*k**n*, is **whp** at least *n*/2.\nWe prove the statement for *G*(*n*, *p*) with $p=16 k/n\\sim t/\\binom{n}{2}$. First we show that **whp** the number of edges between any set of size *n*/2 and its complement is at least *k**n*. Indeed, let *V*0 be a fixed vertex set of size *n*/2. Then, $x=|E(V\\_0,\\overline{V\\_0})|\\sim{{\\mathsf{Bin}}}\\left(n^2/4,p\\right)$, hence E*x* = 4*k**n*. By Chernoff bounds (), ${\\mathbb{P}}(x\\le kn) \\le \\exp(-\\frac{9}{8}kn)$. By the union bound over all choices of *V*0, the probability that there exists such a set is at most $2^n\\cdot\\exp(-\\frac{9}{8}kn)=o(1)$.\nNow, suppose the *k*-core is smaller than *n*/2. Consider the process of removing vertices of degree smaller than *k* one by one. By definition of the *k*-core, this process lasts more than *n*/2 steps. Stop the process exactly after *n*/2 steps. At this point, the remaining set ∣*V*0∣ is of size *n*/2. On the other hand, we have removed less than *k* ⋅ *n*/2 edges, hence $|E(V\\_0,\\overline{V\\_0})|<kn$, and we have seen that **whp** there is no such set.\nRotations, expanders, and boosters\n----------------------------------\nIn our proofs we shall repeatedly use the so-called *rotation–extension* technique of Pósa . Given a longest path *P* = (*v*0, …, *v**t*) we say that *P*ʹ obtained from *P* by an **elementary rotation** of *P* (with *v*1 fixed) if *P*ʹ = (*v*0, …, *v**i* − 1, *v**i*, *v**j*, *v**j* − 1, …, *v**i* + 1) (*j* > *i*). We let R(*P*) denote the set of endpoints of paths obtained from *P* by a (finite) sequence of elementary rotations. We will use the following classical lemma of Pósa:\n[Pósa’s lemma ][lem:Posa] Let *G* be a graph and let *P* be a longest path in *G*. Then ∣*N*(R(*P*))∣ ≤ 2∣R(*P*)∣ − 1.\nSay that a graph *G* = (*V*, *E*) is an ***R*-expander** if every set *U* ⊆ *V* with ∣*U*∣ ≤ *R* has ∣*N*(*U*)∣ ≥ 2∣*U*∣. For an *n*-vertex graph *G* denote by *λ*(*G*) the length of a longest path in *G*, or *n* if *G* is Hamiltonian. A non-edge *e* of *G* is called a **booster** if *λ*(*G* + *e*) ≥ min{*λ*(*G*) + 1, *n*}. The following lemma is a corollary of. For a proof see, e.g., .\n[lem:boosters] Let *G* be a connected *R*-expander which contains no Hamilton cycle. Then *G* has at least (*R* + 1)2/2 boosters.\nWe use by *sprinkling*: we begin with a connected expander and add random edges that hit boosters and advance the expander towards Hamiltonicity. Occasionally, the expander we handle is not guaranteed to be connected. This is sometimes not a problem, as random edges can first connect the expander, and then hit boosters. We summarise this in the following lemma:\n[Sprinkling][lem:sprinkling] Let *β* > 0 and let *G* be an *n*-vertex *β**n*-expander. Then there exists *C* = *C*(*β*) > 0 such that the addition of *C**n* random non-edges to *G* makes it **whp** Hamiltonian.\nSince *G* is a *β**n*-expander, each of its components is of size at least 3*β**n*, hence its number of connected components is at most 1/(3*β*). Assume first that *G* is not connected (and hence *β* ≤ 1/6). Let (*e*1, …, *e*ℓ) be a random permutation of the non-edges of *G*. For 0 ≤ *i* ≤ ℓ, write *G**i* = *G* ∪ {*e*1, …, *e**i*}. As long as *G**i* is not connected, there are at least 3*β*(1 − 3*β*)*n*2 ≥ *β*2*n*2 edges that connect distinct components, hence the probability that a random edge connects distinct components is at least *β*2. Let *t*1 ≫ 1 be an integer, and consider the first *t*1 random edges. For *i* ∈ [*t*1] let *x**i* be the indicator of the event that the *i*’th edge connects distinct components, or that *G**i* − 1 is already connected. Let *x* = ∑*i* = 1*t*1*x**i* and note that *x* ≥ 1/(3*β*) implies that *G**t*1 is connected. Since *x* stochastically dominates a binomial random variable with *t*1 attempts and success probability *β*2, it follows from that the probability that *x* ≤ 1/(3*β*) is *o*(1). Thus, after adding *t*1 random non-edges, the graph becomes **whp** connected.\nNow, using and the monotonicity of expansion, we know that *G**t*1 is either Hamiltonian (in which case we are done) or it has at least $\\beta^2\\binom{n}{2}$ boosters. Thus, the probability that a random edge hits a booster is at least *β*2. Let *t*2 = *τ**n* be an integer with *τ* > *β*− 2, and consider the next *t*2 random edges. For *i* ∈ {*t*1 + 1, …, *t*1 + *t*2} let *y**i* be the indicator of the event that the *i*’th edge hits a booster, or that *G**i* − 1 is already Hamiltonian. Let *y* = ∑*i* = 1*t*1*y**i*, and note that *y* ≥ *n* implies that *G**t*1 + *t*2 is Hamiltonian. Since *y* stochastically dominates a binomial random variable with *t*2 attempts and success probability *β*2, it follows from that the probability that *y* ≤ *n* is *o*(1). Thus, after adding *t*2 further random non-edges, the graph becomes **whp** Hamiltonian.\nWe also use to show that in Hamiltonian expanders, the set of endpoints of Hamilton paths whose other endpoint is fixed is large.\n[lem:manyends] Let *G* be a Hamiltonian *R*-expander and let *v* be a vertex of *G*. Then, the number of endpoints of Hamilton paths of *G* whose other endpoint is *v* is at least *R*.\nSince *G* is Hamiltonian, there exists a Hamilton path *P* with *v* as an endpoint. By, ∣*N*(R(*P*))∣ ≤ 2∣R(*P*)∣ − 1; thus ∣R(*P*)∣ > *R*.\nWe show that the greedy strategy works “well” for purchasing a large *k*-matching; namely, a subgraph of maximum degree *k* in which all but a few vertices are of degree *k*. This will be useful in the proofs of and.\n[lem:kmatching] Let *k* be a positive integer and let *ɛ* > 0. Then, there exists *C* = *C*(*k*, *ɛ*) > 0 such that if *t* ≥ *C**n* and *b* ≥ (*k* − *ɛ*)*n*/2 then there exists a (*t*, *b*)-strategy of Builder (that succeeds **whp**) that purchases a graph with maximum degree *k* in which all but at most *ɛ**n* vertices are of degree *k*.\nBuilder follows the greedy strategy; that is, he purchases every edge both of whose endpoints are of degree below *k*. Observe that this strategy ensures that the maximum degree of Builder’s graph is at most *k* at any stage. Let *U* denote the (dynamic) set of vertices of degree below *k*. Let *C* = *k**ɛ*− 2 and let *t* ≥ *C**n*. For *i* = 1, …, *t* let *x**i* be the indicator of the event that the *i*’th edge arriving is contained in *U* (and thus purchased by Builder) *or* ∣*U*∣ ≤ *ɛ**n*. The probability for this event is at least ∼ *ɛ*2. Thus, *x* : = ∑*x**i* stochastically dominates a binomial random variable with mean ∼ *ɛ*2*C**n* = *k**n*. Therefore, the probability that *x* < (*k* − *ɛ*)*n*/2 is, by Chernoff bounds (), *o*(1). On the event that *x* ≥ (*k* − *ɛ*)*n*/2, either ∣*U*∣ ≤ *ɛ**n*, or Builder has purchased at least (*k* − *ɛ*)*n*/2 edges, in which case it follows that ∣*U*∣ ≤ *ɛ**n*. Obviously, by that time Builder has purchased at most (*k* − *ɛ*)*n*/2 edges.\nGiven a vertex set we may use below to construct a spanning graph of an arbitrary fixed minimum degree (see ). That requires, however, that the time will be long enough so that every vertex will have the desired degree in the underlying graph process. If we are satisfied with a *small* graph of a given minimum degree, we can do it in linear time. The following corollary follows directly from.\n[cor:core] Let *k* ≥ 1 be an integer and let *b* = *t* = 8*k**n*. Then there exists a (*t*, *b*)-strategy of Builder (that succeeds **whp**) that purchases a graph whose *k*-core is of size at least *n*/2.\nSuppose the edges of the complete graph are endowed with independent uniform random “lengths” in [0, 1]. The (random) graph obtained by retaining (only) the *k* shortest edges incident with each vertex is called the **random *k*-nearest neighbour graph** (see ). We denote it by O*k* (or by O*k*(*n*) if we wish to emphasize that it is on *n* vertices). As we remarked earlier, the choice of uniform distribution supported on [0, 1] is arbitrary; any distribution without atoms would yield the same random graph (see ); we use this fact in the proof of. The following theorem of Cooper and Frieze (stated here in a weaker form) will be useful for us in several places in this paper.\n[\\*Theorem 1.1][thm:ok:size] For every constant *k*, the number of edges of O*k* is **whp** \n$$kn - \\frac{n(n-1)}{2n-3}\\sum\\_{1\\le i\\le j\\le k}\n \\frac{\\binom{n-2}{i-1}\\binom{n-2}{j-1}}{2^{\\delta(i,j)}\\binom{2n-4}{i+j-2}}\n + O(\\sqrt{n}\\log{n}),$$ \n where *δ*(*i*, *j*) is the Kronecker delta.\nIt would be useful for us to state in an “asymptotic” form. To this end, we complement the analysis of Cooper and Frieze by obtaining an asymptotic estimate of the expected size of O*k*. Let\n \n$$\\label{eq:ok}\n o\\_k = k-\\frac{1}{4}\\sum\\_{i,j=0}^{k-1}\\binom{i+j}{i}2^{-(i+j)}.$$ \n[cor:ok:size] For every integer *k* ≥ 1 the number of edges of O*k* is **whp** $o\\_k n + O(\\sqrt{n}\\log{n})$. Moreover, *k*/2 < *o**k* ≤ 3*k*/4, and lim*k* → ∞*o**k*/*k* = 1/2.\nFirst note that for every nonnegative integer *k*,\n \n$$\\label{eq:coins}\n \\sum\\_{i=0}^k \\binom{i+k}{i}2^{-(i+k)} = 1.$$ \nIndeed, consider a sequence of fair coin flips that stops whenever *k* + 1 heads *or* *k* + 1 tails are encountered. Let *x* be the number of tails if the sequence stopped by encountering *k* + 1 heads, or the the number heads otherwise. Thus, *x* is supported on {0, …, *k*} and ${\\mathbb{P}}(x=i) = \\binom{i+k}{i}2^{-(i+k)}$. For an integer *k* ≥ 0 write $f(k)=\\sum\\_{i,j=0}^k\\binom{i+j}{i}2^{-(i+j)}$ (so *o**k* = *k* − *f*(*k* − 1)/4). Note that for *k* ≥ 1, by, \n$$f(k) = f(k-1) + 2\\sum\\_{i=0}^k\\binom{i+k}{i}2^{-(i+k)} - \\binom{2k}{k}2^{-2k}\n = f(k-1) + 2 - \\Theta(k^{-1/2}).$$ \n Thus, *f*(0) = 1, *f*(*k*) ≤ 1 + 2*k* for *k* ≥ 0 and\n \n$$\\label{eq:coins:lim}\n \\lim\\_{k\\to\\infty} \\frac{f(k)}{k} = 2.$$ \nIt follows from that the number of edges of O*k* is, **whp**, \n$$\\begin{aligned}\n & \\left(k - \\frac{1}{2} \\sum\\_{1\\le i\\le j\\le k} \\binom{i+j-2}{i-1}2^{-(i+j-2)-\\delta(i,j)}\\right)n\n + O(\\sqrt{n}\\log{n})\\\\\n &= \\left(k - \\frac{1}{4} \\sum\\_{i,j=0}^{k-1} \\binom{i+j}{i} 2^{-(i+j)}\\right)n\n + O(\\sqrt{n}\\log{n})\n = o\\_kn + O(\\sqrt{n}\\log{n}).\n \\end{aligned}$$ \n Evidently, *f*(*k*) is increasing; thus, $\\frac{k}{2}+\\frac{1}{4} \\le o\\_k\\le 3k/4$. In addition, implies that *o**k*/*k* → 1/2 as *k* → ∞.\nOne of the main results of , that we conveniently use here, is that for *k* ≥ 3, O*k* is **whp** *k*-vertex-connected. Cooper and Frieze also showed (there) that the probability that O2 is connected is bounded away from 0 and 1.\n[\\*Theorem 1.4][thm:ok:conn] For *k* ≥ 3, O*k* is **whp** *k*-vertex-connected.\nOur main goal in this section is to prove that for large enough *k*, O*k* is **whp** Hamiltonian.\n[thm:ok:ham] There exists *k**H* > 0 such that for every *k* ≥ *k**H*, O*k* is **whp** Hamiltonian.\nAs mentioned in the introduction, this partially resolves \\*Problem 45. It is conjectured that *k**H* = 3 or *k**H* = 4.\nIn fact, we need (and hence prove) a stronger result that immediately implies. To state it, consider the following equivalent way of generating O*k* (see ). Given a uniform random ordering of the edges of the complete graph *K**n*, we consider them one by one. When an edge arrives, we add it to O*k* if and only if one of its endpoints is incident to a vertex that is — at this point — of degree less than *k*. When we only add edges that arrived by time *t* for some fixed $0\\le t\\le \\binom{n}{2}$, we obtain a subgraph of O*k* that we denote by O*k**t*. This process yields a coupling between the random graph process at time *t* and O*k**t*. Under this coupling, which we call the **natural coupling**, O*k**t* is a subgraph of *G**t*. For convenience, we denote O*k*, *d* = O*k**τ**d* for every 0 ≤ *d* ≤ *k*. Note that as long as the minimum degree of the random graph process is at most 1, it cannot contain a Hamilton cycle; thus, for *t* < *τ*2, O*k**t* is not Hamiltonian.\n[thm:ok:ham:hit] There exists *k**H* > 0 such that for every *k* ≥ *k**H*, O*k*, 2 is **whp** Hamiltonian.\nAs mentioned in the introduction, a classical result due to Bollobás and independently Ajtai, Komlós, and Szemerédi is a hitting-time result, which states that, **whp**, the random graph process will contain a Hamiltonian cycle as soon as its minimum degree reaches 2. refines that result by showing that there exists an *online* algorithm that can choose a small number of edges (linear in the number of vertices) from the first *τ*2 edges of the random graph process that induce a Hamiltonian cycle **whp**. Furthermore, this algorithm is deterministic, simple, and only requires local information, namely, the degrees of the endpoints of the considered edge.\nThe rest of this section is devoted to the proof of, which implies, in turn, (see ). While the proof shares some ideas with previous proofs of the hitting time result for Hamiltonicity of the random graph process (see, e.g., ), it still has its twists and turns to achieve the goal. In particular, we find a way around the common method of *random sparsification* accompanied by *sprinkling* (which is not available to us in this setting) by casting our argument directly on the (already sparse) graph and finding boosters inside it using a sophisticated argument. We begin with a couple of helpful observations. The following is immediate:\n[obs:ok:deg] For integers 0 ≤ *d* < *k*, time *t* and a vertex *v*, under the natural coupling between *G**t* and O*k**t* one has *d*O*k**t*(*v*) = *d* if and only if *d**G**t*(*v*) = *d*.\nThe next observation that we will need for this goal is that O*k* is stochastically dominated by the **random *k*-out graph** (see, e.g., in ): the random graph whose edges are generated independently for each vertex by a random choice of *k* distinct edges incident to that vertex. For brevity, denote it by G*k*. Note that G*k* has roughly *k**n* edges, whereas, as stated in, O*k* is nearly twice as sparse (for large *k*).\n[obs:ok:kout] O*k* is stochastically dominated by G*k*.\nFor every ordered pair of distinct vertices *u*, *v* let **w**(*u*, *v*) be a uniform [0, 1] random weight assigned to the ordered pair (*u*, *v*). Then, G*k* is obtained by adding the edge {*u*, *v*} whenever **w**(*u*, *v*) is one of the *k* smallest weights in the family {**w**(*u*, *w*)}*w*. For every (unordered) pair of distinct vertices *u*, *v* let **x**{*u*, *v*} = min{**w**(*u*, *v*), **w**(*v*, *u*)}. Observe crucially that while **x** is not uniform, it has a continuous density function on [0, 1] and, more importantly, is independent for distinct edges. Thus, O*k* is obtained by including the edge {*u*, *v*} whenever **x**{*u*, *v*} is one of the *k* smallest weights in the family {**x**{*u*, *w*}}*w*. Let us now show that under this coupling, O*k* ⊆ G*k*. Let {*u*, *v*} ∈ O*k* and assume without loss of generality that **x**{*u*, *v*} = **w**(*u*, *v*). Pick a vertex *w* for which **x**{*u*, *v*} < **x**{*u*, *w*} (there are at least *n* − 1 − *k* such vertices). Then, **w**(*u*, *v*) = **x**{*u*, *v*} < **x**{*u*, *w*} ≤ **w**(*u*, *w*). It follows that there are at least *n* − 1 − *k* vertices *w* for which **w**(*u*, *v*) < **w**(*u*, *w*), hence {*u*, *v*} ∈ G*k*.\nWe now show that for *k* ≥ 12, O*k*, 2 is **whp** an expander[6](#fn6).\n[lem:ok:exp] There exists *β* > 0 such that for *k* ≥ 12, O*k*, 2 is **whp** a *β**n*-expander.\nWe prove it for *k* = 12 and derive the statement for every *k* ≥ 12 due to monotonicity. Fix *b* = *k*/11 > 1. We first prove that there exists *γ* > 0 for which, **whp**, no vertex set *S* in O*k*, 2 of cardinality *s* ≤ *γ**n* spans more than *b**s* edges. Since O*k*, 2 is stochastically dominated by O*k* and (by ) O*k* is stochastically dominated by G*k*, we may prove that statement for G*k*. Note that the probability that a given edge appears in G*k* is at most twice the probability that a given vertex chooses a given incident edge (which is *k*/(*n* − 1)). By the union bound over the choice of *S* and the choices of the spanned edges, that probability is at most \n$$\\sum\\_{s=2}^{\\gamma n} \\binom{n}{s}\\binom{\\binom{s}{2}}{bs}\n \\left(\\frac{3k}{n}\\right)^{bs}\n \\le \\sum\\_{s=2}^{\\gamma n}\n \\left[\n \\frac{en}{s} \\left(\\frac{es}{2b}\\cdot\\frac{3k}{n}\\right)^b\n \\right]^s\n = \\sum\\_{s=2}^{\\gamma n}\\left(\\Theta\\left(\\left(\\frac{s}{n}\\right)^{b-1}\\right)\\right)^s.$$ \n If *n*/log*n* ≤ *s* ≤ *γ**n* then the *s*’th summand is at most *c**n*/log*n* for some *c* < 1 (for small enough *γ* > 0). Otherwise, it is (*o*(1))*s*. Thus, the sum is *o*(1).\nBack to O*k*, 2, let *X* = *X**k**τ*2 be the set of vertices of degree smaller than *k* at time *τ*2. Suppose first that there exists a set *B* of size at most *β**n* for *β* = *γ*/5 with $B\\cap X={\\varnothing}$ and ∣*N*(*B*)∣ ≤ 4∣*B*∣. Set *S* = *B* ∪ *N*(*B*) and note that *s* = ∣*S*∣ ≤ 5*β**n* = *γ**n* while ∣*E*(*S*)∣ ≥ *k**β**n*/2 > *b**s*, contradicting the above. Now, let *A* with ∣*A*∣ ≤ *β**n*. Let *A**X* = *A* ∩ *X* and *A**Y* = *A* \\ *X*. Let further *S**X* = *A**X* ∪ *N*(*A**X*) and *S**Y* = *A**Y* ∪ *N*(*A**Y*). By the above, ∣*N*(*A**Y*)∣ ≥ 4∣*A**Y*∣. By, ∣*N*(*A**X*)∣ ≥ 2∣*A**X*∣ and every vertex in *A**Y* (or in general) has at most one neighbour in *S**X*. Hence, in particular, ∣*N*(*A**Y*) \\ *S**X*∣ ≥ ∣*N*(*A**Y*)∣ − ∣*A**Y*∣ ≥ 3∣*A**Y*∣. Thus, \n∣*N*(*A*)∣ = ∣*N*(*A**X*) \\ *A**Y*∣ + ∣*N*(*A**Y*) \\ *S**X*∣ ≥ (2∣*A**X*∣ − ∣*A**Y*∣) + 3∣*A**Y*∣ ≥ 2∣*A*∣, \n as required.\nOur next step is to show that O*k*, 2 is also typically connected (for large enough *k*). For that, we need the following lemma. As we mentioned earlier, Cooper and Frieze showed that O2 is connected with probability that is bounded away from 0 and 1. Since O3 ⊈ O*k*, 2 for any *k*, we cannot use.\n[lem:ok:quad] Let *α* > 0 and let *F* be a fixed set of edges in *K**n* with ∣*F*∣ ≥ *α**n*2. Then, there exist *k*0 = *k*0(*α*), *η* = *η*(*α*) and *c* = *c*(*α*) > 0 such that for every *k* ≥ *k*0, if *t* = *η**k**n* then ${\\mathbb{P}}(F\\cap E({\\mathcal{O}}\\_k^{t})={\\varnothing})\\le \\exp(-ckn)$.\nLet $\\gamma=\\sqrt{\\alpha}$, and *η* = *γ*/3. Take *k* > 6(1 − log*γ*)/*γ*2, and *t*0 = *η**k**n*. Let *E*0 = {*e*1, …, *e**t*0} be the first *t*0 edges of the underlying random graph process (in this order) and let *F*0 = *F* ∩ *E*0. Let *V*0 = ⋃*F*0 be the set of vertices covered by the edges of *F*0. We observe that typically *V*0 cannot be too small; indeed, for a given set *S* to be ⋃*F*0 we need every edge of *F* that is not contained in *S* to appear after time *t*0. By the union bound over all choices of *V*0 of size at most *γ**n*, \n$$\\begin{aligned}\n {\\mathbb{P}}(|V\\_0|\\le\\gamma n)\n &\\le \\binom{n}{\\gamma n} \\left(1-\\frac{t\\_0}{\\binom{n}{2}}\\right)^{|F|-\\binom{\\gamma n}{2}}\n \\le \\exp((\\gamma-\\gamma\\log\\gamma-\\eta k(2\\alpha-\\gamma^2))n)\\\\\n &\\le \\exp((\\gamma-\\gamma\\log{\\gamma}-\\gamma^3k/3)n)\n \\le \\exp(-ckn),\n \\end{aligned}$$ \n for *c* = *γ*3/6 > 0. Let *F*1 = *F*0 ∩ *E*(O*k**t*0). Note that if $F\\_1={\\varnothing}$ then when an edge in *F*0 arrived both its endpoints had degree at least *k*; hence all vertices of *V*0 have degree at least *k* in O*k**t*0. But in this case, ∣*V*0∣ ≥ *γ**n* implies that *γ**n**k*/2 ≤ ∣*E*(O*k**t*0)∣ ≤ *t*0 = *η**k**n* = *γ**k**n*/3, a contradiction. Thus, \n$${\\mathbb{P}}(F\\cap E({\\mathcal{O}}\\_k^t)={\\varnothing})\n \\le {\\mathbb{P}}(F\\cap E({\\mathcal{O}}\\_k^{t\\_0})={\\varnothing})\n \\le {\\mathbb{P}}(F\\_1={\\varnothing})\n = \\exp(-ckn).\n \\qedhere$$ \n[lem:ok:large] For every *β* > 0 there exists *k*0 = *k*0(*β*) such that for every *k* ≥ *k*0, **whp**, every two disjoint vertex sets in O*k*, 2 with ∣*A*∣, ∣*B*∣ ≥ *β**n* are connected by an edge.\nLet *α* = *β*2 and let *k*0, *η*, *c* be the constants guaranteed by. Let H(*β*) = − *β*log*β* − (1 − *β*)log(1 − *β*), and set *k*1 = 3H(*β*)/*c*. Take *k* ≥ max{*k*0, *k*1} and *t* = ⌈*η**k**n*⌉. Fix two disjoint vertex sets *A*, *B* with ∣*A*∣, ∣*B*∣ = *β**n*. Let *F* = *E*(*A*, *B*), so ∣*F*∣ ≥ *α**n*2. By, the probability that in O*k**t* does not contain an edge from *F* (and hence the probability that in O*k**t* the sets *A*, *B* are not connected by an edge) is at most exp( − *c**k**n*). By the union bound over all sets *A*, *B*, the probability that in O*k**t* there are two disjoint sets of size at least *β**n* that are not connected by an edge is at most \n$$\\binom{n}{\\beta n}^2 e^{-ckn}\n = \\exp(((2+o(1))H(\\beta)-ck)n) = \\exp(-\\Omega(n)) = o(1).$$ \n The result follows since **whp** *t* ≤ *τ*2, implying O*k**t* ⊆ O*k*, 2.\n[cor:ok:exp] There exists *k*0 > 0 such that for every *k* ≥ *k*0, O*k*, 2 is **whp** an $\\frac{n}{4}$-expander.\nLet *β* be the constant guaranteed by (we may (and will) assume that *β* < 1/4), and let *k*0 = *k*0(*β*) be the constant guaranteed by. Let *k* ≥ max{*k*0, 12}. Thus, by, O*k*, 2 is **whp** a *β**n*-expander; and, by, O*k*, 2 has **whp** the property that any two disjoint vertex sets of size at least *β**n* are connected by an edge. Assume both properties hold and let *A* be a vertex set with ∣*A*∣ ≤ *n*/4. If ∣*A*∣ ≤ *β**n* then ∣*N*(*A*)∣ ≥ 2∣*A*∣ since O*k*, 2 is a *β**n*-expander. Otherwise, there is an edge between *A* and any other set of size at least *β**n*, hence ∣*N*(*A*)∣ ≥ *n* − *β**n* − ∣*A*∣ ≥ (3/4 − *β*)*n* ≥ *n*/2 ≥ 2∣*A*∣.\n[cor:ok:conn] There exists *k*0 > 0 such that for every *k* ≥ *k*0, O*k*, 2 is **whp** connected.\nLet *k*0 be the constant guaranteed by and let *k* ≥ *k*0. From it follows that every connected component of O*k*, 2 is, **whp**, of size at least 3*n*/4, implying that the graph is connected.\nWe are now ready to prove.\n[Proof of ] Let *k*0 be the constant from, let ℓ = max{*k*0, 500} and let *k* = 100ℓ. Note that by proving the statement of the theorem for *k*, due to monotonicity of O*k*, 2 in the first parameter, we prove it for every *k*ʹ ≥ *k*. Let *t*0 = 6ℓ*n* and write *a* = 6ℓ/*k*, so *t*0 = *a**k**n*. Consider the first *t*0 edges of the random graph process (and note that *t*0 ≪ *τ*2 **whp**). When an edge *e* arrives we colour it *red* if it is incident to a vertex of degree *d* < ℓ, or *black* otherwise. Let *L* = *Y*ℓ*t*0, and note that by, ∣*L*∣ ≥ 0.99*n* **whp**. We now consider the next edges of the random graph process, until time *τ*2. When an edge *e* arrives we colour it *red* if it is incident to a vertex of degree *d* < ℓ (note that these red edges are *not* contained in *L*), or *black* if it is any other edge not contained in *L*. If the arriving edge is contained in *L*, we do not record its identity, but rather its arrival time. In particular, we count these edges. We observe crucially that the exact identity of such an edge does not affect the rest of the colouring process; indeed, any permutation of the edges of the random graph process that fixes the coloured edges (by time *τ*2) and the edges not in *L* is feasible. Let *m* be the number of edges we counted in *L* and let *t*1, …, *t**m* be their arrival times (*t*0 < *t*1 < … < *t**m* < *τ*2). Note that the probability of an edge to fall inside *L* before *m* edges fell there (without conditioning on the rest of the process) is at least *p* : = 0.98 (since both *m* and the number of existing edges in *L* are *o*(*n*2)). Thus, for any *b* = *O*(1), the number of edges falling into *L* between *t*0 and *t*0 + *b**n* stochastically dominates a binomial random variable with *b**n* attempts and success probability *p*. Take *b* = 500, and *m*ʹ = ⌊*b**n**p*/2⌋. It follows that *m*ʹ < *m* and *t**i* < *t*0 + *b**n* for every 1 ≤ *i* ≤ *m*ʹ **whp**.\nLet *F* be the set of non-edges in *L* at time *τ*2. Conditioning on the entire colouring process (including *m* and the times *t*1, …, *t**m*), the set of edges that fall into *L* at times *t*1, …, *t**m* is distributed as a uniformly sampled edge set of size *m* of *F*. Let *f*1, …, *f**m* be a random ordering of this random set of edges. Write *F**i* = {*f*1, …, *f**i*} for 0 ≤ *i* ≤ *m*. We add the random edges in *F**m* one by one (edge *f**i* at time *t**i*), so the distribution of *f**i* is uniform over *F* \\ *F**i* − 1. Some of these edges will be coloured *blue* according to a rule explained below. For *i* ∈ [*m*], let *K**i* = *Y**k**t**i* − 1 ∩ *L* be the set of vertices of degree at least *k* in *L* just before time *t**i*. We note by, and the discussion above that ∣*K**i*∣ ≤ 2*t**i*/*k* ≤ 2(*a* + *b*/*k*)*n* for every *i* ∈ [*m*ʹ], **whp**. Note that the graph that consists of the red edges at time *τ*2 is Oℓ, 2. Denote by *H**i* the graph that consists of the red edges and the blue edges among *F**i*. Thus, this is a supergraph of Oℓ, 2. By, and since expansion (and connectivity) is monotone, either *H**i* is Hamiltonian, or there are at least *n*2/32 boosters with respect to it. Suppose *H**i* is not Hamiltonian, and let *Z**i* be the set of boosters with respect to *H**i*. Let $Z'\\_i=Z\\_i\\cap \\binom{L}{2}{\\setminus}\\binom{K\\_i}{2}{\\setminus}F\\_i$. These are the boosters that are (a) contained in *L*, (b) have at least one endpoint of degree less than *k* at the time of arrival, and (c) have not been revealed earlier. It follows that \n$$|Z'\\_i|\\ge |Z\\_i| - |E(X\\_\\ell^{t\\_0},V)| - \\left|\\binom{K\\_i}{2}\\right| - |F\\_i|\n \\ge \\left(\\frac{1}{32}-\\frac{1}{100}-2\\left(a+\\frac{b}{k}\\right)^2 - o(1)\\right)n^2.$$ \n \nThis is at least *n*2/100 for ℓ ≥ 500 and *k* = 100ℓ. When considering the edge *f**i* + 1 we colour it blue if it is in *Z*ʹ*i*, or if it is in $\\binom{L}{2}{\\setminus}\\binom{K\\_i}{2}$ and *H**i* is already Hamiltonian. But the probability that it is in *Z**i*ʹ is at least ∣*Z**i*ʹ∣/(∣*F*∣ − ∣*F**i*∣) ≥ *q* : = 1/100, independently of the past. Thus, the number of blue edges stochastically dominates a binomial random variable with *m*ʹ ∼ *b**n**p*/2 attempts and success probability *q*, and hence mean ∼ *q**b**n**p*/2 > 2*n*. By Chernoff bounds (), the number of blue edges is at least *n* **whp**. We finish the proof by noting that if *H**i* is not Hamiltonian and *f**i* + 1 is coloured blue then *f**i* + 1 is a booster with respect to *H**i* that is contained in O*k*, 2. Thus, *H**m*ʹ (and thus *H**m*; and thus O*k*, 2) is **whp** Hamiltonian.\nThe proof above yields the following corollary.\n[cor:ok:ham] Let *k**H* be the constant from. Then, for every *k* ≥ *k**H*, O*k*, 2 is **whp** a Hamiltonian $\\frac{n}{4}$-expander.\nSpanning subgraphs\n==================\nMinimum degree and vertex-connectivity\n--------------------------------------\nIt follows from that Builder can construct a graph with minimum degree *k* by purchasing (sufficiently) more than *o**k**n* edges, while observing only enough edges to guarantee a sufficient minimum degree in the underlying random graph process.\n[Proof of ] We describe Builder’s strategy. Builder purchases any edge touching at least one vertex of degree less than *k*. Since by time *τ**k* the random graph process has, by definition, minimum degree of at least *k*, the obtained graph is exactly O*k*. We conclude by observing that by Builder purchased, **whp**, at most $o\\_kn+O(\\sqrt{n}\\log{n})$ edges, and that for *k* ≥ 3, by, O*k* is *k*-vertex-connected.\n[Proof of ] Let *ɛ*ʹ = *ɛ*/2. We describe Builder’s strategy in stages.\n#### Stage I (Constructing a *k*-matching)\nDuring the first *C**n* steps, for *C* = *C*(*ɛ*ʹ, *k*), Builder constructs a *k*-matching, in which all but at most *ɛ*ʹ*n* vertices are of degree *k*. This is possible, **whp**, by.\n#### Stage II (Handling low-degree vertices)\nDenote by *V*0 the set of vertices of degree smaller than *k* in Builder’s graph at the end of Stage I. By the above, **whp**, ∣*V*0∣ ≤ *ɛ*ʹ*n*. Builder now emulates the O*k* model with respect to these vertices; namely, Builder purchases any edge contained in *V*0 that is incident to at least one vertex of degree less than *k* in *G*[*V*0]. Since in the random graph process, by time *t*, the minimum degree is logarithmic in *n*, Builder will observe at least *k* additional edges incident to each vertex of *V*0. He will purchase a subset of these (thus, at most *ɛ*ʹ*k**n* edges).\nHamilton cycles\n---------------\n### Hitting time, inflated budget\n[Proof of ] Let *k* be a large enough constant so that O*k*, 2 is **whp** Hamiltonian (such *k* is guaranteed to exist by ). Builder emulates O*k*, 2 by purchasing every edge that is incident to a vertex of degree less than *k*. He completes this in time *τ*2 while purchasing at most *k**n* edges.\n### Inflated time, optimal budget\nIn this section, we prove the following theorem:\n[Hamiltonicity][thm:ham] For every *ɛ* > 0 there exists *C* > 1 such that the following hold. If *t* ≥ *C**n*log*n*/2 and *b* ≥ (1 + *ɛ*)*n* then there exists a (*t*, *b*)-strategy *B* of Builder such that \nlim*n* → ∞P(*B**t*is Hamiltonian) = 1. \n#### Proof of\nWe describe Builder’s strategy in (four) stages. Set *ɛ*ʹ = *ɛ*/40. We describe a (*t*, *b*)-strategy for *b* = (1 + *ɛ*)*n* and *t* < *n*log*n*/*ɛ*ʹ.\n#### Stage I (Growing disjoint paths)\nLet *C* be a large constant to be chosen later. In the first stage, Builder grows many sublinear paths, covering together (1 − *ɛ*ʹ)*n* vertices. Let *s*ʹ ∼ *n*/log1/3*n* be an integer and let *s*0 = *n*/log1/2*n* ≪ *s*ʹ. Builder grows simultaneously paths *P*1, …, *P**s*ʹ as follows. He begins by letting each *P**i* be a (distinct) vertex *v*0*i*. He then claims every edge that extends one of the paths without intersecting with other paths. Formally, let *ν* = (1 − *ɛ*ʹ)*n*/*s*ʹ. If at a given stage Builder has the paths (*v*01, …, *v*ℓ11), …, (*v*0*s*ʹ, …, *v*ℓ*s*ʹ*s*ʹ) on the vertex set *V**P* then he claims an observed edge if and only if it is of the form {*v*ℓ*j**j*, *w*} for some *j* = 1, …, *s*ʹ, ℓ*j* < *ν* and $w\\notin V\\_P$. Builder stops if all but at most *s*0 of the paths are of length at least *ν* (in which case the stage is “successful”), or when he has observed $t\\_1=\\frac{C}{{\\varepsilon}'}n\\log^{1/3}{n}$ edges (in which case the stage “fails”), whichever comes first. For 1 ≤ *j* ≤ *s*ʹ and 1 ≤ *i* ≤ *t*1, let *y**i**j* be the indicator that Builder has purchased the *i*’th observed edge, and that this edge extends the path *P**j*. For convenience, if ∣*V**P*∣ ≥ (1 − *ɛ*ʹ)*n* we set *y**i**j* to 1. Observe that for every *j*, *i*, ${\\mathbb{P}}(y^j\\_i=1) \\ge (n-|V\\_P|)/\\binom{n}{2}>{\\varepsilon}'/n$. If the stage fails then there is an *s*0-subset *S*0 ⊆ [*s*ʹ] such that for every *j* ∈ *S*0, the path *P**j* ends up with length less than *ν*. Write *y*0 = ∑*j* ∈ *s*0∑*i* = 1*t*1*y**i**j*. Thus, if the stage fails then *y*0 < *ν**s*0. But *y*0 stochastically dominates a binomial random variable with *t*1 attempts and success probability *p*1 = *ɛ*ʹ*s*0/*n*. Set *μ* = *t*1*p*1 ∼ *C**s*0log1/3*n* ≫ *s*ʹ. By the union bound over the choices of an *s*0-subset of the paths and using Chernoff bounds (), for large enough *C*, the probability that the stage fails is at most \n$$\\binom{s'}{s\\_0} {\\mathbb{P}}(y\\_0 < \\nu s\\_0)\n \\le 2^{s'} \\exp(-\\mu/10) = o(1).$$ \n#### Stage II (Connecting the paths)\nAt this stage Builder tries to connect most of the *s* = *s*ʹ − *s*0 ∼ *s*ʹ complete paths to each other, eventually obtaining an almost-spanning path in his graph. For convenience, we ree-numerate the complete paths by [*s*]. Let *t*2 = *n*log1/2*n*. During the next *t*2 observed edges, Builder purchases an edge connecting the last vertex of one of his paths to the first vertex of another. Formally, he purchases every observed edge of the form {*v*ℓ*i**i*, *v*0*j*} for *i* ≠ *j*. We will need the following lemma:\n[\\*Lemma 4.4][lem:DFS] Let *s*, *k* ≥ 1 and let *D* be an *s*-vertex digraph in which for every two disjoint *A*, *B* ⊆ *V*(*D*) with ∣*A*∣, ∣*B*∣ ≥ *k*, *D* contains an edge between *A* and *B*. Then, *D* contains a directed path of length *s* − 2*k* + 1.\nWe use the lemma on an auxiliary digraph *D* on the vertex set [*s*] that is defined as follows. We observe *t*2 random edges. Whenever Builder observes (and purchases) an edge of the form {*v*ℓ*i**i*, *v*0*j*}, we add the edge (*i*, *j*) to *D*. We now have to claim that *D* satisfies the requirements of the lemma for *k* = *ɛ*ʹ*s*.\n[cl:D:exp] **Whp**, every two disjoint *A*, *B* ⊆ [*s*] with ∣*A*∣, ∣*B*∣ ≥ *ɛ*ʹ*s* satisfy ${E\\_D(A,B)\\ne{\\varnothing}}$.\nFix disjoint *A*, *B* with ∣*A*∣, ∣*B*∣ ≥ *ɛ*ʹ*s*. The event that $E\\_D(A,B)={\\varnothing}$ implies that none of the *t*2 observed edges hits a pair {*v*ℓ*i**i*, *v*0*j*} for *i* ∈ *A* and *j* ∈ *B*. There are at least *ɛ*ʹ2*s*2 such edges, hence that probability is at most \n$$\\binom{\\binom{n}{2}-{\\varepsilon}'^2 s^2}{t\\_2}/\\binom{\\binom{n}{2}}{t\\_2}\n \\le \\exp\\left(-{\\varepsilon}'^2 s^2 t\\_2/\\binom{n}{2}\\right)\n = \\exp(-\\Theta(n\\log^{-1/6}{n})).$$ \n Here we used the general bound $\\binom{a-b}{c}/\\binom{a}{c}\\le\\exp(-bc/a)$. By the union bound over the choices of *A*, *B*, the probability that the event in the statement does not hold is at most \n$$\\binom{n}{{\\varepsilon}' s}^2 \\exp(-\\Theta(n\\log^{-1/6}{n}))\n \\le \\exp({\\varepsilon}' s\\log\\log{n} -\\Theta(n\\log^{-1/6}{n})) = o(1).\\qedhere$$ \nimply that *D* has, **whp**, a directed path of length (1 − 2*ɛ*ʹ)*s*. By the way we have defined *D*, this implies the existence of a path *Q* of length (1 − 3*ɛ*ʹ)*n* in Builder’s graph. To analyse the number of purchased edges at this stage, we note that at any stage an edge is purchased with probability ≍ *s*2/*n*2, hence the expected number of purchased edges is ≍ *t*2*s*2/*n*2 ≍ *n*/log1/6*n*. By Chernoff bounds (), the number of purchased edges is sublinear **whp**.\n#### Stage III (Preparing the ground)\nLet *q*1, *q*2 be the endpoints of *Q*. Let *V*1 ∪ *V*2 = *V* \\ *V*(*Q*) be a partition of the vertices outside *Q* (so ∣*V**i*∣ = 3*ɛ*ʹ*n*/2 for *i* = 1, 2). Builder performs the next two tasks simultaneously.\n##### (Connecting the endpoints of *Q* to *V*1, *V*2)\nBuilder claims the first observed edge from *q**i* to *V**i* for each *i* = 1, 2 (unless such an edge already exists). After *c**n*log*n* steps, for any *c* > 0, there will be **whp** a neighbour of each of *q**i* in *V**i*. Call this neighbour *w**i*.\n##### (Constructing expanders on *V*1, *V*2)\nBuilder purchases a copy of O12 in *V**i*. Observing that every new edge falls inside *V**i* with probability at least 2*ɛ*ʹ2, we conclude, using, that this could be done **whp** by observing at most $\\frac{1}{{\\varepsilon}'}n\\log{n}$ edges and purchasing at most 36*ɛ*ʹ*n* of them. By, the obtained graphs are both *β**n*-expanders for some *β* > 0. For *i* = 1, 2 let *B**i* = *B*[*V**i*] denote Builder’s graph on the vertex set *V**i*.\n#### Stage IV (Sprinkling)\nBy there exists *C*4 = *C*4(*β*, *ɛ*ʹ) > 0 such that by observing at most *C*4*n* edges and purchasing at most $\\frac{3}{2}{\\varepsilon}'n$ edges among those landing inside each *V**i*, both *B*1, *B*2 become **whp** Hamiltonian. Conditioning on that event, denote the sets of endpoints of Hamilton cycles in *B**i* whose other endpoint is *w**i* by *Y**i* for *i* = 1, 2. By, ∣*Y**i*∣ ≥ *β**n*.\n#### Stage V (Closing a Hamilton cycle)\nTo close a Hamilton cycle, Builder purchases an edge between *Y*1 and *Y*2. The probability of failing to do so after observing log*n* steps is *o*(1).\nPerfect matchings\n-----------------\nIn this section we prove. Recall the constant *k**H* from, and let *ɛ*ʹ = *ɛ*/*k**H*.\n#### Stage I (Constructing a *k*-core)\nLet *k* be some large (even) constant to be determined later (in Stage VI). Choose *ɛ*1 = *ɛ*ʹ/(10*k*) and let *V*1 be an arbitrary vertex set of size ∼ *ɛ*1*n*. During the next *t*1 ∼ 9*k**n* steps Builder purchases every edge that lands inside *V*1. By that time, **whp**, at least 8*k**ɛ*1*n* (and at most 10*k**ɛ*1*n* = *ɛ*ʹ*n*) edges are being purchased. Thus, by, **whp**, the obtained graph contains a *k*-core on vertex set *U*1 ⊆ *V*1 of size at least *ɛ*1*n*/2. Write *η*1 = ∣*U*1∣/*n* and note that *η*1 ≤ *ɛ*1 ≤ *ɛ*ʹ/10.\n#### Stage II (Constructing a large matching)\nLet *ɛ*2 = *ɛ*ʹ/3. Let *Y* = *V* \\ *U*1; so ∣*Y*∣ ∼ (1 − *η*1)*n*. Let *C*2 = *C*2(1, *ɛ*2) be the constant guaranteed by. Builder follows a (*t*2, *b*2)-strategy proposed by to construct a matching *M*2 with vertex set *X*2 ⊆ *Y* where ∣*X*2∣ ≥ ∣*Y*∣(1 − *ɛ*2) ∼ (1 − *ɛ*2)(1 − *η*1)*n*. The lemma guarantees that this is doable, **whp**, for *t*2 = *C*2*n* and *b*2 ≤ *n*/2. Write *η*2 = 1 − ∣*X*2∣/*n*, and note that *η*1 ≤ *η*2 ≤ *η*1 + *ɛ*2.\n#### Stage III (Extending the large matching)\nDenote the vertices outside *X*2 ∪ *U*1 by *W*2. Recall that ∣*W*2∣ = *n* − (∣*X*2∣ + ∣*U*1∣) ∼ (*η*2 − *η*1)*n*. Write *η*3 = *η*2 − *η*1 ≤ *ɛ*2. During the next *t*3 ∼ *ɛ*ʹ*n*log*n* steps Builder (partially) constructs O*k**H* on *W*2. The number of purchased edges is **whp** at most *k**H**η*3*n* ≤ *k**H**ɛ*2*n* ≤ *ɛ**n*/3. Let *S* be the set of vertices of degree less than *k**H* in *W*2 at the end of this process. We now show that, **whp**, ∣*S*∣ ≤ *n*1 − *δ*3 for *δ*3 = *η*3*ɛ*ʹ/2. Indeed, let *w* ∈ *W*2. The probability that the next edge contains *w* and is contained in *W*2 is *p*3 ∼ 2*η*3/*n*. Thus, the total number of observed edges incident to *w* stochastically dominates a binomial random variable with *t*3 attempts and success probability *p*3. Consequently, the probability that edges containing *w* were observed less than *k**H* times is, by, at most *n*− *η*3*ɛ*ʹ. Hence, by Markov’s inequality, P(∣*S*∣ ≥ *n*1 − *δ*3) ≤ *η*3*n*1 − *η*3*ɛ*ʹ/*n*1 − *δ*3 ≪ 1. We argue that *B*[*W*2] contains a matching that covers all but 2*k**H**n*1 − *δ*3 vertices. Indeed, if we had continued constructing the copy of O*k**H* without any time restrictions, we would had – according to – a (nearly) perfect matching there. But, deterministically, we would have purchased at most *k**H*∣*S*∣ additional edges to achieve that goal. Thus, **whp**, the current largest matching covers all but at most 2*k**H**n*1 − *δ*3 vertices in *W*2. Denote this matching by *M*3, and let *X*3 = *V*(*M*3).\n#### Stage IV (Building stars)\nWe append the matching constructed in Stage III to the matching constructed in Stage II; namely, we set *M* = *M*2 ∪ *M*3, *X* = *X*2 ∪ *X*3 and *W* = *V* \\ (*X* ∪ *U*1). We recall that ∣*W*∣ ≤ *n*1 − *δ*4 for some 0 < *δ*4 < *δ*3 and ∣*X*∣ ∼ (1 − *η*1)*n*. Let *K* > 1/(2*η*1*ɛ*ʹ) be an integer. In this stage Builder attempts to construct disjoint *K*-leaf stars at each vertex of *W*, with leaves inside *X*, where each edge of *M* contains at most one leaf. For *w* ∈ *W* let *x**w* denote the number of leaves in the star that is rooted at *w* that Builder managed to purchase by the end of the stage. We show that **whp** for every *w* ∈ *W*, *x**w* = *K*, hence the stage is successful. Indeed, whenever an edge arrives it has probability 2/*n* to be incident to *w*. The other end of such an edge is incident to an available edge of *M* with probability at least (∣*X*∣ − 2*K*∣*W*∣)/*n* ∼ 1 − *η*1. Thus, *x**w* stochastically dominates a binomial random variable with *t*4 = (1 + *ɛ*ʹ)*n*log*n*/2 attempts and success probability *p*3 = (1 − 2*η*1) ⋅ 2/*n*. Therefore, by (since *ɛ*ʹ < 1 and 2*η*1 < *ɛ*ʹ), \n$${\\mathbb{P}}(x\\_w<K)\n \\le \\left(\\frac{et\\_3p\\_3}{K(1-p\\_3)}\\right)^K e^{-t\\_3p\\_3}\n = (\\Theta(\\log{n}))^K \\cdot n^{-(1+{\\varepsilon}')(1-2\\eta\\_1)}\n \\ll n^{-1}.$$ \n By the union bound over all *o*(*n*) vertices of *W*, the probability that any *x**w* is below *K* is *o*(1).\n#### Stage V (Building 3-paths)\nFor every *w* ∈ *W* let *a*1*w*, …, *a**K**w* ∈ *X* be the leaves of the stars rooted at *w* that were constructed in the previous stage. For *i* = 1, …, *K*, let *b**i**w* ∈ *X* be the neighbour of *a**i**w* in *M*. Builder now purchases every edge that connects *b**i**w* (for some *w* ∈ *W* and *i* ∈ [*K*]) to a vertex *u* ∈ *U*1, as long as none of *b*1*w*, …, *b**K**w*, *u* was touched earlier during this stage. Fix *w* ∈ *W*. The probability that in a given step an edge between (an untouched vertex of) *U*1 and {*b*1*w*, …, *b**K**w*} appears is at least $p\\_4=K(|U\\_1|-|W|)/\\binom{n}{2}\\sim 2K\\eta\\_1/n$. Thus, the probability that in *t*4 = *ɛ*ʹ*n*log*n* steps such an edge does not appear is at most *e*− *p*4*t*4 = *n*− 2*K**η*1*ɛ*ʹ. Noting that *K* > (2*η*1*ɛ*ʹ)− 1 we get that this probability is *o*(*n*− 1), hence, by the union bound over all *o*(*n*) vertices of *W*, we match one leaf in the star rooted at *w* with a unique vertex in *U*1 for every *w* ∈ *W* **whp**.\n#### Stage VI (Sprinkling)\nWe note that each of the 3-paths constructed in the previous stage can be used as an augmenting path. Namely, we let *M*ʹ be the matching obtained by *M* by replacing every edge *e* ∈ *M* that is contained in such a 3-path (as a middle edge) by the other two edges of that path. Let *U*ʹ be the set of vertices in *U*1 matched in *M*ʹ, and write *U*\\* = *U*1 \\ *U*ʹ. Thus, *M*ʹ matches all vertices but those in *U*\\*. We now show that *B*[*U*\\*] is **whp** an expander. For that, note that *U*ʹ is a uniformly chosen subset of size ∣*W*∣ ≤ *n*1 − *δ*4 of *U*1. We first argue that *δ*(*B*[*U*\\*]) ≥ *k*/2 **whp**. Indeed, fix *u* ∈ *U*1, and let *u*1, …, *u**k* be arbitrary neighbours of *u* in *B*[*U*1]. The probability that *d*(*u*, *U*\\*) ≤ *k*/2 is at most the probability that *k*/2 of these *k* neighbours end up in *U*ʹ. By the union bound, this is at most $\\binom{k}{k/2}n^{-\\delta\\_4}$ \n$$\\binom{k}{k/2} \\frac{\\binom{|U\\_1|-k/2}{|W|-k/2}}{\\binom{|U\\_1|}{|W|}}\n \\le \\left(4\\cdot\\frac{|W|}{|U\\_1|}\\right)^{k/2}\n \\asymp n^{-\\delta\\_4k/2}.$$ \n Making sure *k* > 2/*δ*4, this is *o*(*n*− 1), and hence, by the union bound, **whp**, *δ*(*B*[*U*\\*]) ≥ *k*/2. We remark that there is no circular dependency of the constants here.\nWe recall that by (taking *t* = *t*1 ∼ 9*k**n* and *R* = 3*β*6*n* for *β*6 = 3− 5*k*− 3/2) the random graph process *G**t*, and hence also *B*[*U*\\*], typically does not have a set of size *r* ≤ 3*β*6*n* that spans more than 3*r* edges. This, together with the minimum degree of *B*[*U*\\*], implies expansion; indeed, let *A* ⊆ *U*\\* be a vertex set with ∣*A*∣ = *a* ≤ *R*/3 = *β*6*n*. Suppose ∣*N*(*A*)∣ < 2∣*A*∣ and write *S* = *A* ∪ *N*(*A*) (so *r* = ∣*S*∣ < 3*a* ≤ *R*). But the number of edges spanned by *S* is at least *δ*(*B*[*U*\\*])∣*A*∣/2 ≥ *k**a*/4 > *k**r*/12 ≥ 3*r* (making sure *k* ≥ 36), a contradiction to. Thus, *B*[*U*\\*] is a *β*6*n*-expander. By there exists *C*6 = *C*6(*β*6) such that adding to *B*[*U*\\*] *C*6*η*1*n* random non-edges makes it, **whp**, Hamiltonian. Letting *t*6 = *C*6ʹ*n*/*η*1 for *C*6ʹ > *C*6, it follows from that after observing *t*6 edges, at least *C*6*η*1*n* of them land, **whp**, inside *U*\\* (whose size is ∼ *η*1*n*). Builder purchases each of these edges. Thus, by, Builder’s resulting graph on *U*\\* contains, **whp**, a Hamilton cycle. That Hamilton cycle contains a perfect matching *M*6. Appending *M*6 to *M*, we get a perfect matching in Builder’s graph.\nSmall subgraphs\n===============\nWe introduce the following terminology. For a fixed graph *H* we call a non-edge *e* in Builder’s graph an ***H*-trap** (or, simply, a trap) if *B* + *e* contains a copy of *H*. Following the terminology of “hitting” boosters, we say that Builder **hits** a trap if he encounters (and therefore purchases) such a non-edge.\nTrees\n-----\n### 1-statement\n[Proof of the 1-statement of ]\nLet *T*1 ⊆ … ⊆ *T**k* be a sequence of subtrees of *T*, *T**i* having *i* vertices (so *T**k* = *T*). We first note that if *t* ≫ *n* then a budget *b* = *k* − 1 suffices with high probability. Indeed, suppose Builder has constructed *T**i* (note that he builds *T*1 with a zero budget). To build *T**i* + 1, Builder waits for one of at least ∼ *n* edges whose addition to his current copy of *T**i* would create a copy of *T**i* + 1. By time *t*/(*k* − 1) Builder observes (and purchases) such an edge **whp**. Thus, by time *t* Builder constructs, **whp**, a copy of *T*.\nWe may thus assume from now on that *t* ≤ *c**n* for 0 < *c* < 1/5 and *t* ≥ *b* ≫ (*n*/*t*)*k* − 2. In particular, *t* ≫ *n*1 − 1/(*k* − 1). For *i* = 1, …, *k* define \n$$s\\_i = \\frac{b}{k-1}\\cdot\\left(\\frac{t}{(k-1)n}\\right)^{i-2},$$ \n and note that *s**i* is a strictly monotone decreasing sequence with *n* ≥ *s*1 > *s**k* ≫ 1. We describe Builder’s strategy in stages (for convenience, the first stage will be indexed by 2). We assume that at the beginning of stage *i*, *i* = 2, …, *k*, Builder has built *s**i* − 1 vertex-disjoint copies of *T**i* − 1. The assumption trivially holds for *i* = 2, since *s*1 ≤ *n*. At stage *i* ≥ 2, Builder purchases any edge extending one of the *s**i* − 1 copies of *T**i* − 1 he currently has to a copy of *T**i*, so that the resulting trees are still vertex-disjoint. He continues doing so only as long as he does not have *s**i* copies of *T**i* (in which case this stage is successful) or if he had observed more than *t*/(*k* − 1) edges during this stage (in which case this stage fails, and thus the entire strategy fails). For stage *i* and 1 ≤ *j* ≤ *t*/(*k* − 1) let *y**j**i* denote be the event that the *j*’th observed edge of stage *i* is being purchased, or that *s**i* copies of *T**i* have already been built. Being at stage *i*, there are at least *s**i* − 1 − *s**i* potential trees to extend, each can be extended by observing one of at least *n* − 2*b* edges. Thus, noting that *s**i* − 1 − *s**i* ≥ *s**i* − 1(1 − *c*/(*k* − 1)) and *n* − 2*b* ≥ *n* − 2*t* ≥ *n*(1 − 2*c*), we have \n$${\\mathbb{P}}(y^i\\_j) \\ge\n \\frac{(s\\_{i-1}-s\\_i)(n-2b)}{\\binom{n}{2}}\n \\ge \\frac{2(1-\\frac{c}{k-1})s\\_{i-1}\\cdot(1-2c)}{n}.$$ \n Write *c*ʹ = 2(1 − *c*/(*k* − 1))(1 − 2*c*) and note that for small enough *c* > 0 (*c* < 1/5 suffices) we have *c*ʹ > 1. Therefore, *y**i* = ∑*j**y**j**i* stochastically dominates a binomial random variable with mean $\\frac{t}{k-1}\\cdot\\frac{c's\\_{i-1}}{n} = c's\\_i$. By Chernoff bounds (), the stage is successful **whp**. Thus, Builder builds, **whp**, *s**k* ≥ 1 copies of *T**k* = *T* by the end of *k* − 1 stages, that is, by time *t*.\n### 0-statement\n[Proof of the 0-statement of ] We assume *b* ≪ (*n*/*t*)*k* − 2. It follows that *t* ≪ *n*. We may also assume that *b* ≪ *t*; indeed, if *b* ≍ *t* then *t* ≪ (*n*/*t*)*k* − 2, hence *t* ≪ *n*1 − 1/(*k* − 1). This implies that **whp** no tree on *k* vertices exists in the graph of the observed edges. We prove that any (*t*, *b*)-strategy of Builder fails, **whp**, to build a connected component of size *k*. Define a sequence (*r**i*)*i* = 1*k* − 1 by *r*1 = *n* and *r**i* = *b*(3*k**t*/*n*)*i* − 2 for *i* = 2, …, *k* − 1. We argue that **whp** Builder cannot build more than *r**i* connected components of size *i*. The proof is by induction on *i*. The case *i* = 1 is trivial. The case *i* = 2 follows deterministically by the budget restriction, as *r*2 = *b*. To build a connected component of size 2 < *i* < *k*, Builder must observe an edge connecting a connected component of size 1 ≤ *j* < *i* with a connected component of size *i* − *j* ≥ *j*. By induction, the probability that the next observed edge is such is at most \n$$\\frac{\\sum\\_{j=1}^{{\\left\\lfloor{i/2}\\right\\rfloor}} (jr\\_j \\cdot (i-j)r\\_{i-j})}{\\binom{n}{2}-t}\n \\le \\frac{inr\\_{i-1} + \\Theta\\left(b^2(t/n)^{i-4}\\right)}{\\binom{n}{2}-t}\n \\le \\frac{2.5kr\\_{i-1}}{n}.$$ \n Here we used the fact that *b* ≪ *t*, hence *b*2(*t*/*n*)*i* − 4 ≪ *n**r**i* − 1. Thus, by Chernoff bounds (), during *t* rounds the number of such edges is, **whp**, smaller than *r**i* − 1 ⋅ 3*k**t*/*n* = *r**i*. To build a tree of order *k*, Builder must construct a connected component of size at least *k*. For that, Builder must observe an edge connecting a connected component of size 1 ≤ *i* ≤ *k* − 1 with a connected component of size max{*i*, *k* − *i*} ≤ *j* ≤ *k* − 1 (in his graph). The probability that the next observed edge is such is at most \n$$\\frac{\\sum\\_{i=1}^{k-1}\\sum\\_{j=\\max\\{i,k-i\\}}^{k-1} (ir\\_i\\cdot jr\\_j)}{\\binom{n}{2}-t}\n \\preceq \\frac{r\\_{k-1}}{n}.$$ \n Hence, during *t* rounds the expected number of edges Builder observes that create such a connected component is ≼ *r**k* − 1*t*/*n* ≪ 1; thus, Builder fails to construct such a component **whp**.\nCycles\n------\n### 1-statement\n[Proof of the 1-statement of ]\nLet *b* ≫ *b*\\* = *b*\\*(*n*, *t*, *k*). As *b*\\* ≪ *n*, we may assume *b* ≪ *n*. Choose *n* ≪ *t*ʹ ≪ *t* such that it still holds that *b* ≫ *b*ʹ : = *b*\\*(*n*, *t*ʹ, *k*). Set $s=n^2/(b't')=\\min\\{(t'/n)^k,n/\\sqrt{t'}\\}$, and note that 1 ≪ *s* ≪ *b*. Indeed, *b*/*s* ≫ *b*ʹ/*s* = *n*2/(*s*2*t*ʹ) ≥ 1. Denote *d* = *c**s*1/*k* for *c* < 1/(*k* + 5) and note that *d* ≫ 1. Note further that *d**n* = *c**s*1/*k**n* ≤ *c**t*ʹ and that *c**k* < 1/6.\n#### Stage I (Growing *d*-ary trees)\nBuilder performs Stage I in *k* rounds, each lasting *d**n* steps. Set *r* = *b*/*s* ≫ 1. Builder chooses 2*k**r* vertices arbitrarily, call them *roots*. Denote by *P*0 = {*v*1, …, *v*2*k**r*} the set of these roots, and let *L*0 = *R*0 = *P*0 (we think of each root as being both on the right side of a tree and on the left side of a tree). Suppose at the beginning of round *i* ∈ [*k*] we have a set *J**i* − 1 ⊆ [*P*0] with ∣*J**i* − 1∣ = 2*k* − (*i* − 1)*r* and trees (*T**j*)*j* ∈ *J**i* − 1, where the tree *T**j* is rooted at *v**j* and is of depth *i* − 1 (so at the beginning of round 1 these trees are isolated roots). Write *L**i**j* for the set of vertices in *L**i* which are in tree *j*, and define *R**i**j* analogously. Builder’s goal at round *i* is to “extend” *half* of these trees, by attaching *d*∣*L**i* − 1*j*∣ leaves to vertices in *L**i* − 1*j* and *d*∣*R**i* − 1*j*∣ (distinct) leaves to vertices in *R**i* − 1*j*. Each of the new leaves should not be already covered by Builder’s graph. A leaf attached to *L**i* − 1*j* will be placed in *L**i**j*, and a leaf attached to *R**i* − 1*j* will be placed in *R**i**j*. Note that if *i* = 1 then *L**i* − 1*j* = *R**i* − 1*j* = {*v**j*}, in which case we attach *two* *d*-leaf stars at each root *v**j*. Let *J**i* ⊆ *J**i* − 1 be the subset of indices for which *T**j* has been successfully extended. Set *L**i* = ⋃*j* ∈ *J**i**L**i**j* and *R**i* = ⋃*j* ∈ *J**i**R**i**j*. Note that if Builder’s strategy has not failed by the beginning of round *i*, ∣*J**i* − 1∣ = 2*k* − (*i* − 1)*r* and ∣*L**i* − 1*j*∣ = ∣*R**i* − 1*j*∣ = *d**i* − 1 for every *j* ∈ *J**i* − 1. For *i* ∈ [*k*] and *j* ∈ *J**i* − 1 let *x**i*(*j*) be the number of edges observed at round *i* with one endpoint in *L**i* − 1*j* and the other outside the set of vertices covered by the edges of Breaker’s graph at the time of observation. If ∣*x**i*(*j*)∣ ≥ *d*∣*L**i* − 1*j*∣ then Builder purchases the first ∣*L**i* − 1*j*∣ of those observed edges and this stage is “left-successful” for tree *j*. Similarly, let *y**i*(*j*) be the number of edges observed at round *i* between *R**i* − 1*j* and the uncovered part of the graph. If ∣*y**i*(*j*)∣ ≥ *d*∣*R**i* − 1*j*∣ then Builder purchases the first ∣*R**i* − 1*j*∣ of those observed edges and this stage is “right-successful” for tree *j*. If a stage is both left-successful and right-successful for a tree *j* then *T**j* is successfully extended. As we remarked earlier, after extending successfully 2*k* − *i**r* trees, Builder stops.\nLet us analyse this strategy. For 0 ≤ *i* ≤ *k*, denote by *b**i* the number of covered vertices by the end of step *i*. Observe that \n*b**i* ≤ ∑*i*ʹ = 0*i*2*k* + 1 − *i*ʹ*r**d**i*ʹ ≤ (1 + *o*(1))2*k* − *i* + 1*r**d**i* \n (since *d* ≫ 1). In particular, *b**k* < 3*r**d**k* = 3*c**k**b* ≪ *n*. Take *i* ∈ [*k*] and assume the strategy was successful so far. Thus, *x**i*(*j*) (and also *y**i*(*j*)) stochastically dominates a binomial random variable with *d**n* attempts and success probability at least $|L\\_{i-1}^j|\\cdot(n-b\\_i)/\\binom{n}{2} \\sim 2d^{i-1}/n$. Thus, by Chernoff bounds (), P(*x**i*(*j*) < *d*∣*L**i* − 1*j*∣) = P(*x**i*(*j*) < *d**i*) < exp( − *d**i*/5) ≪ 1. By Markov’s inequality, **whp** for at least half of the trees indexed by *J**i* − 1 Builder succeeds in extending the tree. The total time that passes until the end of this stage is at most *k**d**n* ≤ *c**k**t*ʹ < *t*ʹ ≪ *t*, and the total budget is at most 3*c**k**b* < *b*/2.\n#### Stage II (Connecting leaves)\nNote that for every *j* ∈ *J**k*, if *u* ∈ *L**k**j* and *v* ∈ *R**k**j* then *u*, *v* are both leaves of *T**j* of distance 2*k* from each other. Thus, by adding the edge {*u*, *v*}, a cycle of length ℓ is formed. Thus, the number of traps is ∣*J**k*∣ ⋅ ∣*L**k**j*∣ ⋅ ∣*R**k**j*∣ = *r**d*2*k* ≍ *r**s*2 = *b**s* ≫ *n*2/*t*ʹ. Thus, after time *t* − *t*ʹ ≫ *t*ʹ we hit a trap **whp**.\n#### Even cycles\nWe briefly discuss the modifications needed to obtain the result for cycles of length 2*k* + 2. Instead of choosing *P*0, Builder greedily constructs a matching *M*0 of size 2*k**r*; this is doable (**whp**) since the first *C**r* observed edges (recalling that *r* ≪ *n*) are mostly disjoint, and, in particular, for large enough *C*, contain a matching of size 2*k**r*. Then, for each edge in *M*0, Builder puts one vertex in *L*0 and one vertex in *R*0. Builder continues in the same fashion as in the case of odd cycles, where the only difference is that instead of *L*0 = *R*0 we have $L\\_0\\cap R\\_0={\\varnothing}$. We finish by noting that an edge between *u* ∈ *L**k**j* and *v* ∈ *R**k**j* now closes a cycle of length 2*k* + 2.\n### -statement\nLet *H* = *C*ℓ for ℓ = 2*k* + 1 or ℓ = 2*k* + 2. The 0-statement in follows from combining the next two claims together with the known fact that if *t* ≪ *n* then *G**n*, *t*, **whp**, contains no copy of *H*.\n[cl:cycles:lb:univ] If $b\\ll n/\\sqrt{t}$ then for any (*t*, *b*)-strategy *B* of Builder, \nlim*n* → ∞P(*H* ⊆ *B**t*) = 0. \n[cl:cycles:lb:spec] If *b* ≪ *n**k* + 2/*t**k* + 1 then for any (*t*, *b*)-strategy *B* of Builder, \nlim*n* → ∞P(*H* ⊆ *B**t*) = 0. \n[Proof of ] We may assume *t* ≪ *n*2. Consider a strategy for building *H*. Let *B*ʹ be Builder’s graph before obtaining the first copy of *H*. We show that at this point, the time it takes Builder to hit an *H*-trap (and hence construct a copy of *H*) is typically much larger than *t*. Let *X* denote the number of *H*-traps at this point. Observe that every *H*-trap is a non-edge of *B*ʹ, both whose endpoints are vertices of positive degree. Evidently, the number of vertices of positive degree is at most 2*b*, thus *X* ≤ 4*b*2. Thus, the probability of hitting a trap in *t* steps is at most $tX/(\\binom{n}{2}-t)\\preceq tb^2/n^2\\ll 1$.\nFor the proof of we will need a couple of lemmas. In what follows, we may assume that *t* ≤ *n*(*k* + 1)/(*k* + 1/2), as otherwise $n^{k+2}/t^{k+1}\\le n/\\sqrt{t}$ and the claim follows from.\n[cl:badpaths] Let ℓ ≥ 1 be an integer, write *d* = 2*t*/*n* ≫ 1 and let *C* > 1. Then, **whp**, the number of paths of length ℓ in *G**t* that contain a vertex whose degree is at least *C**d* is *o*(*n*/*d*).\nLet *L* be the set of all vertices of degree at least *C**d*, and let *Q* be the set of all paths containing a vertex from *L*. By Chernoff bounds (), \n$${\\mathbb{E}}|Q| \\le n^{\\ell+1} \\left(\\frac{t}{n^2}\\right)^\\ell (\\ell+1) \\cdot{\\mathbb{P}}(d(v)\\ge Cd-2)\n \\le nd^\\ell e^{-cd} \\ll n/d,$$ \n for some constant *c* = *c*(*C*) > 0, and the result follows from Markov’s inequality.\nWe turn to count the number of copies of a path of length ℓ in a *z*-degenerate graph *G*0 with maximum degree Δ.\n[lem:path:embed] Let *z*, *q*, Δ, ℓ ≥ 1 be integers. Let *G*0 be a *z*-degenerate *q*-vertex graph with maximum degree Δ. Then, the number of paths of length ℓ in *G*0 is at most *q* ⋅ 2ℓ*z*⌈ℓ/2⌉Δ⌊ℓ/2⌋.\nLet *v*1, …, *v**q* be an ordering of the vertices of *G*0 such that for 1 ≤ *i* ≤ *q*, *v**i* has at most *z* neighbours among *v*1, …, *v**i* − 1. 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Wang, JCAP **1310**, 011 (2013) [arXiv:1210.0569 [hep-th]].\nA. Gruzinov, Phys. Rev. D **70**, 063518 (2004) [astro-ph/0404548].\nY. Urakawa *et al.*, in preparation\n---\n1. The conservation of *ζ* in such a setup also can be understood as a direct consequence of the *δ**N* formalism .[↩](#fnref1)\n2. Here, we also assume that the (spatially averaged) background universe is the FRW universe. This excludes, say, the solid inflation case, where the anisotropic pressure does not vanish in the large scale limit . (See also Ref. .)[↩](#fnref2)\n\nConservation of *ζ* with radiative corrections from heavy field\n===============================================================\nIntroduction\n============\nInflation provides us with a natural experimental instrument to explore the high energy physics. Measurements of the temperature anisotropies and polarization of the cosmic microwave background can constrain the Hubble parameter *H* at the time when the fluctuation was generated. The current data puts an upper bound on *H* at around 1014 GeV , which is much higher than the accessible energy scale in particle accelerators. The precise measurements of the primordial perturbations generated during inflation may place a constraint on the theory of high energy physics independently of the particle experiments.\nIn string theory, compactification of the extra dimensions typically yields a number of scalar fields, which may have masses bigger than the Hubble parameter during inflation. Investigating a possible imprint of these massive fields might allow us to explore the high energy physics behind. While one field model is consistent with the current data , there is still room to include a contamination of such massive fields, which act as isocurvature modes. If such a massive field has a mass much bigger than the Hubble scale, integrating out the massive field only gives local contributions to the effective action for the inflaton (relevant works can be found, e.g., in Refs. ). In such a case, since we are ignorant of the high energy theory, it is impossible to disentangle the radiative corrections of the massive field. However, if one of the isocurvature modes has a mass of order *H*, the radiative correction may yield a distinctive non-local contribution.\nChen and Wang studied an impact of a massive field on the primordial curvature perturbation *ζ* in Ref. (see also Ref. ). In their setup, the inflaton has a non-minimal coupling with the massive field, which yields the cross-correlation between them. As emphasized in Ref. , where a more extensive analysis, including higher spin fields, was done, the massive field leaves more direct information in the squeezed configuration of the correlation functions, which has a soft external leg, than in other configurations. The massive scalar field with 0 < *m*/*H* ≤ 3/2 decays as *η*Δ− with\n \n$$\\begin{aligned}\n & \\Delta \\equiv \\frac{3}{2} - \\sqrt{\\frac{9}{4}- \\frac{m^2}{H^2}}\n \\label{Delta-} \\end{aligned}$$ \nat large scales. Then, as was computed in Refs. the contribution of the massive field to the squeezed bi-spectrum, when the shorter mode *k* crosses the Hubble scale, is given by\n \n$$\\begin{aligned}\n & \\langle \\zeta\\_q \\zeta\\_k \\zeta\\_k \\rangle \\propto P\\_\\zeta(q) P\\_\\zeta(k)\n(q/k)^{\\Delta}\\,, \\qquad \\quad (q/k \\ll 1)\\,, \\label{Eq:bizeta}\\end{aligned}$$ \nwhere *P**ζ*(*k*) is the power spectrum of *ζ*. Notice that (*q*/*k*)Δ encodes the evolution between the Hubble crossing time for the mode *k* and the one for *q*. For *m* > 3*H*/2, the massive field oscillates, while decaying, as $\\eta^{\\tilde\\Delta\\_{\\pm}}$ with\n \n$$\\begin{aligned}\n & \\tilde{\\Delta}\\_{\\pm} \\equiv \\frac{3}{2} \\pm i \\sqrt{\\frac{m^2}{H^2}-\n \\frac{9}{4}}\\,, \\label{Deltapm}\\end{aligned}$$ \nwhich gives the same momentum dependence as in Eq. ([Eq:bizeta]) except that Δ− is replaced with $\\tilde{\\Delta}\\_\\pm$.\nWhen the curvature perturbation stops evolving after the Hubble crossing time of the shorter mode *k*, Eq. ([Eq:bizeta]) gives the squeezed bispectrum at the end of inflation. As far as the massive fields do not contribute to the background evolution (an example where a massive field modulates the background evolution was studied, e.g., in Refs. ) and the tree level contribution is concerned, the curvature perturbation is conserved after all modes cross the Hubble scale [1](#fn1). In this case, Eq. ([Eq:bizeta]) indeed gives the bi-spectrum at the end of inflation . The argument in Ref. is based on the symmetry of the de Sitter spacetime. Therefore, one may speculate that the loop correction of the massive field may also be still given by Eq. ([Eq:bizeta]), while the scaling dimension Δ will no longer be given as in Eq. ([Delta-]) nor Eq. ([Deltapm]). In this generalization, a non-trivial point may be in showing the conservation of *ζ* after the Hubble crossing. In Refs. , the conservation of *ζ* was addressed, including the loop correction of *ζ* in the setup of single field inflation.\nIn this paper, we address the conservation of the curvature perturbation *ζ* which is affected by loop corrections of a heavy field *χ*, assuming that the heavy field does not contribute to the classical background trajectory. The constant non-decaying mode of *ζ* is called the adiabatic mode. To compute the evolution of the curvature perturbation, we integrate out the heavy field and derive the effective action for *ζ*. If the mass of the heavy field *M* is much bigger than the Hubble scale, the loop corrections of *χ* only yields local terms in the effective action and then following the argument in the single field case, we can show the conservation of *ζ* at large scales. On the other hand, if the mass *M* is not large enough compared to the Hubble scale, the loop corrections of *χ* can give non-local contributions to the effective action. The presence of the non-local contribution can yield a qualitative difference from single field models.\nIn single field models of inflation, the conservation of the curvature perturbation at large scales is implemented by the scaling symmetry ${\\hbox{\\boldmath{$x$}}} \\to e^s {\\hbox{\\boldmath{$x$}}}$ with a constant parameter *s*, which changes $\\zeta(t, {\\hbox{\\boldmath{$x$}}})$ to $\\zeta(t,\\, e^{-s}{\\hbox{\\boldmath{$x$}}})-s$. The scaling symmetry is one of the gauge transformations and hence classically it should be preserved for a diffeomorphism invariant theory. However, when we quantize the system, the scaling symmetry is not always preserved, particularly when we allow an arbitrary initial quantum state . When the scaling symmetry is preserved, a part of the IR divergences is canceled out . (In order to eliminate all the IR divergences, we also need to preserve the invariance under other gauge transformations. For a detailed explanation, see, e.g., Ref. .) In Refs. , it was shown that when we choose the Euclidean vacuum, a.k.a., the adiabatic vacuum or the Bunch-Davies vacuum in the de Sitter limit, there exists a set of quantities which is free from the IR divergences.\nIn quantum field theory, a symmetry implies a corresponding identity, the so-called Ward-Takahashi (WT) identity. For one field model of inflation, the scaling symmetry yields the consistency relation, which relates the (*n* + 1)-point function of *ζ* with one soft external leg to the *n*-point function of *ζ* . The consistency relation is indeed the WT identity for the scaling symmetry . The consistency relation was first shown for the bi-spectrum in the squeezed limit by Maldacena in Ref. and it was extended to more general single field models in Ref. . The consistency relation for the arbitrary *n*-point function was derived in Ref. . In a single field inflation with diffeomorphism invariance, when the initial state is the Euclidean vacuum and the background trajectory is on attractor, the consistency relation generally holds. When one of these assumptions is not fulfilled, the consistency relation can be violated .\nVarious extensions of the consistency relation have been attempted so far. In Refs. , the squeezed bi-spectrum was computed in a non slow-roll setup and in Refs. , sub-leading contributions for the consistency relation were computed. (See also Refs. .) The consistency relation can also be obtained from the reparametrization invariance of the wave function of the universe . The use of the wave function is also motivated by the holographic description of inflation . In Refs. , the consistency relation was derived by solving the Callan-Symanzik equation in the dual boundary theory (see also Ref. ). A possible gauge issue for the consistency relation was discussed in Refs. .\nIn this paper, we derive the consistency relation for the heavy field *χ* from the requirement of the scaling symmetry. When the scaling symmetry is preserved at the quantum level, we obtain the corresponding WT identity. The WT identity for the scaling symmetry yields the consistency relation which relates the (*n* + 1)-point function of the *n* *χ*s and one soft curvature perturbation *ζ* to the *n*-point function of the *χ* field. The derivation of the consistency relation also applies in the presence of the loop corrections of the heavy field. Using the effective action for *ζ*, obtained by integrating out *χ*, we show that when the consistency relation for *χ* holds, the curvature perturbation *ζ* is conserved at the super Hubble scales.\nThis paper is organized as follows. In Sec. [Sec:single], we review the conservation of *ζ* in single field models of inflation, emphasizing the crucial role of the scaling symmetry. In Sec. [Sec:EA], after we describe our setup of the problem, we introduce the effective action for *ζ* by integrating out the heavy field in the in-in (or closed time path) formalism. In Sec. [Sec:WT], we derive the consistency relation for the heavy field from the Ward-Takahashi identity for the scaling symmetry. In Sec. [Sec:Conservation], using the consistency relation, derived in Sec. [Sec:WT], we show the conservation of *ζ* in the presence of the loop corrections of the heavy field. In Sec. [Renormalization], we briefly discuss the renormalization of the heavy field. Finally, in Sec. [Conclusion], we conclude.\nConservation of *ζ* and scaling symmetry in single field inflation\n==================================================================\nIn single field models of inflation, it is known that the curvature perturbation is conserved in the large scale limit. In this section, we show that the conservation of *ζ* is a direct consequence of the scaling symmetry.\nSingle field inflation\n----------------------\nFor illustrative purpose, we start our discussion by considering a single scalar field with the standard kinetic term, whose action is given by\n \n$$\\begin{aligned}\n S &= \\frac{1}{2} \\int \\sqrt{-g} \\left[ R - g^{\\mu \\nu}\n \\partial\\_\\mu \\phi \\partial\\_\\nu \\phi - 2V(\\phi) \\right] {{\\rm d}}^4 x\\,. \\label{Exp:actionS}\\end{aligned}$$ \nIn this paper, we set the gravitational constant *κ*2 ≡ 8*π**G* to 1. Using the ADM form of the line element:\n \n$$\\begin{aligned}\n {{\\rm d}}s^2 = - N^2 {{\\rm d}}t^2 + h\\_{ij} ({{\\rm d}}x^i + N^i {{\\rm d}}t) ({{\\rm d}}x^j + N^j {{\\rm d}}t)~,\n \\label{Exp:ADMmetric}\\end{aligned}$$ \nwhere we introduced the lapse function *N*, the shift vector *N**i*, and the spatial metric *h**i**j*, we can express the action ([Exp:actionS]) as\n \n$$\\begin{aligned}\n S &=\\frac{1}{2} \\int \\sqrt{h} \\Bigl[ N\\,{{^s\\!R}}- 2 N\n V(\\phi) + N (\\kappa\\_{ij} \\kappa^{ij} - \\kappa^2) \\cr\n & \\qquad \\qquad \\qquad \\qquad + \\frac{1}{N} ( \\dot{\\phi}\n - N^i \\partial\\_i \\phi )^2 - N h^{ij} \\partial\\_i \\phi \\partial\\_j \\phi\n \\Bigr] {{\\rm d}}^4x~, \\label{Eq:ADMaction}\\end{aligned}$$ \nwhere *s* *R* is the three-dimensional Ricci scalar, and *κ**i**j* and *κ* are the extrinsic curvature and its trace, defined by\n \n$$\\begin{aligned}\n \\kappa\\_{ij} = \\frac{1}{2 N} ( \\dot{h}\\_{ij} - D\\_i N\\_j\n - D\\_j N\\_i )~, \\qquad \n \\kappa = h^{ij} \\kappa\\_{ij} ~.\\end{aligned}$$ \nHere, the spatial indices *i*, *j*, ⋯ are raised or lowered by the spatial metric *h**i**j*, and *D**i* denotes the covariant differentiation associated with *h**i**j*. Taking the variation of the action with respect to *N* and *N**i*, which are the Lagrange multipliers, we obtain the Hamiltonian and momentum constraint equations as\n \n$$\\begin{aligned}\n & {{^s\\!R}}- 2 V - (\\kappa^{ij} \\kappa\\_{ij} - \\kappa^2 ) \n - N^{-2} ( \\dot{\\phi} - N^i \\partial\\_i \\phi)^2 - h^{ij}\n \\partial\\_i \\phi \\partial\\_j \\phi = 0~, \\\\\n & D\\_j ( \\kappa^j\\_{~i} - \\delta^j\\_{~i} \\kappa ) - N^{-1} \n \\partial\\_i \\phi~ ( \\dot{\\phi} - N^j \\partial\\_j \\phi) = 0~. \\end{aligned}$$ \nWe determine the time slicing, employing the uniform field gauge:\n \n$$\\begin{aligned}\n & \\delta \\phi=0\\,. \\label{GCphi}\\end{aligned}$$ \nWe express the spatial metric *h**i**j* as\n \n$$\\begin{aligned}\n & h\\_{ij}= e^{2(\\rho+\\zeta)} \\left[ e^{\\delta \\gamma} \\right]\\_{ij}\\,,\\end{aligned}$$ \nwhere the background scale factor is expressed as *a* ≡ *e**ρ* and *δ**γ**i**j* is set to traceless. As spatial gauge conditions, we impose\n \n$$\\begin{aligned}\n & \\partial^i \\delta \\gamma\\_{ij} =0\\,. \\label{GCT}\\end{aligned}$$ \nWith this gauge choice, the constraint equations are given by\n \n$$\\begin{aligned}\n & {{^s\\!R}}- 2 V - (\\kappa^{ij} \\kappa\\_{ij} - \\kappa^2 ) \n - N^{-2} \\dot{\\phi}^2 = 0\\,, \\\\ \n & D\\_j ( \\kappa^j\\_{~i} - \\delta^j\\_{~i} \\kappa ) = 0 ~.\\end{aligned}$$ \nInserting *N* and *N**i*, which are expressed in terms of *ζ* by solving these constraint equations, into the action ([Eq:ADMaction]), we can derive the action for *ζ* .\nScaling symmetry\n----------------\nThe transverse condition imposed on *δ**γ**i**j* is non-local and hence to determine the coordinates, we need to employ boundary conditions. For example, at linear order in perturbation, the tensor perturbation transforms under the spatial coordinate transformation *x**i* → *δ**x̃**i* = *x**i* + *δ**x**i* as\n \n$$\\begin{aligned}\n & \\delta \\tilde{\\gamma}\\_{ij} (x) = \\delta \\gamma\\_{ij}(x) - 2\n \\left(\\partial\\_{(i} \\delta x\\_{j)} - \\frac{1}{3} \\delta\\_{ij} \\partial^k\n \\delta x\\_k \\right) \\,. \\end{aligned}$$ \nThe transverse condition on *δ**γ**i**j* gives\n \n$$\\begin{aligned}\n & \\partial^2 \\delta x^i = - \\frac{1}{3} \\partial^i \\partial\\_j \\delta\n x^j\\,, \\label{Exp:condT}\\end{aligned}$$ \nwhich does not determine *δ**x**i* uniquely without specifying boundary conditions to solve Eq. ([Exp:condT]). For the scalar mode, all spatial coordinate transformations *δ**x**i* = ∂*i**δ**x* which satisfy ∂2∂*i**δ**x* = 0 still keep the transverse condition after the transformations. When we consider a compact support on each time slicing, we find an infinite way to impose the boundary conditions in solving ∂2∂*i**δ**x* = 0. We analyzed these *gauge* degrees of freedom in detail in Refs. , where we used the italic font for “gauge” to discriminate the *gauge* transformations defined within the compact support from those in the infinite spatial support. (See Ref. for a more recent work.) So far, we kept the tensor perturbation *δ**γ**i**j* for the illustrative purpose, but in the rest of this paper, we neglect it.\nAmong these transformations, the important one for *ζ* is the scale transformation *δ**x**i* = *s**x**i*, which is extended to *x**i* → *x**s**i* = *e**s**x**i* at non-linear orders. The parameter *s* can vary in time , but for our purpose, we do not need to consider the time dependence, and hence we set *s* to a constant parameter. Under this transformation, the spatial line element is recast into\n \n$$\\begin{aligned}\n & \\frac{{{\\rm d}}l^2\\_d}{a^2(t)} = e^{2 \\zeta(t,\\, {\\hbox{\\boldmath{\\scriptsize$x$}}})} {{\\rm d}}{\\hbox{\\boldmath{$x$}}}^2 = e^{2 \n \\zeta\\_s(t,\\, {\\hbox{\\boldmath{\\scriptsize$x$}}}\\_s)} {{\\rm d}}{\\hbox{\\boldmath{$x$}}}\\_s^2 = e^{2 \n \\{ \\zeta\\_s(t,\\, e^s {\\hbox{\\boldmath{\\scriptsize$x$}}}) +s \\}} {{\\rm d}}{\\hbox{\\boldmath{$x$}}}^2 \\,,\\end{aligned}$$ \nand then the curvature perturbation changes to\n \n$$\\begin{aligned}\n & \\zeta\\_s(t,\\, {\\hbox{\\boldmath{$x$}}}) = \\zeta(t,\\, e^{-s}{\\hbox{\\boldmath{$x$}}}) -s\\,. \\label{Exp:transzeta}\\end{aligned}$$ \nThis is a purely geometrical argument, and hence this transformation law also should apply to multi-field models.\nPreserving the scaling symmetry is crucial to ensure the infrared (IR) regularity of the curvature perturbation. In Refs. , we introduced the spatial Ricci scalar evaluated in the geodesic normal coordinates as a *gauge* invariant quantity. Since the contribution from the IR modes, which give rise to the IR divergence, can be eliminated by performing the corresponding *gauge* transformation, the IR divergence also can be removed from the *gauge* invariant quantity.\nBy construction, the spatial Ricci scalar evaluated in the spatial geodesic normal coordinates is *gauge* invariant and it serves a conceptually clear example of the *gauge* invariant quantity. Nevertheless, using the geodesic normal coordinates can alter the UV behaviour . For a practical use, we may use the smeared geodesic coordinates ${\\hbox{\\boldmath{$x$}}}\\_g(t)$ given by \n \n$$\\begin{aligned}\n & {\\hbox{\\boldmath{$x$}}}\\_g(t) \\equiv e^{{{^g\\!\\bar{\\zeta}}}(t)} {\\hbox{\\boldmath{$x$}}} \\,, \\label{Def:xg}\\end{aligned}$$ \nwhere *g* *ζ̄* is the averaged *ζ* at a compact support on each time slicing, given by\n \n$$\\begin{aligned}\n & {{^g\\!\\bar{\\zeta}}}(t) = \\frac{\\int {{\\rm d}}^3 {\\hbox{\\boldmath{$x$}}}\\_g W\\_t({\\hbox{\\boldmath{$x$}}}\\_g) \\zeta(t,\\,\n e^{- {{^g\\!\\bar{\\zeta}}}} {\\hbox{\\boldmath{$x$}}}\\_g)}{\\int {{\\rm d}}^3 {\\hbox{\\boldmath{$x$}}}\\_g W\\_t({\\hbox{\\boldmath{$x$}}}\\_g)} \\,, \\label{Def:gbz}\\end{aligned}$$ \nwhere $W\\_t({\\hbox{\\boldmath{$x$}}})$ is a window function which vanishes outside the compact support on the time constant slicing. The spatial Ricci scalar evaluated at ${\\hbox{\\boldmath{$x$}}}\\_g$ is not invariant under all the *gauge* transformations, but it is invariant under the scale transformation.\nConservation of *ζ* in single field inflation\n---------------------------------------------\nIn single field inflation, solving the Hamiltonian and momentum constraint equations, we can eliminate *N* and *N**i* and write down the action only in terms of *ζ*. Since the action for any diffeomorphism invariant theory remains invariant under the scale transformation, the action for *ζ* should take the following form:\n \n$$\\begin{aligned}\n & S [\\zeta] = \\int {{\\rm d}}t\\, {{\\rm d}}^3 {\\hbox{\\boldmath{$x$}}} e^{3(\\rho+\\zeta)} {\\cal\n L}\\_\\zeta\\! \\left[ \\partial\\_t\\zeta,\\, e^{-(\\rho+\\zeta)} \\partial\\_i \\zeta\n \\right]\\,, \\label{Exp:actionzeta}\\end{aligned}$$ \nwhere the Lagrangian density ${\\cal L}\\_\\zeta$ includes *ζ* only in the form ∂*t**ζ* or *e*− (*ρ* + *ζ*)∂*i**ζ*. (A detailed explanation can be found in Ref. .)\nTo address the conservation of *ζ* in the large scale limit, we neglect the terms which include *ζ* with the spatial derivative. Then, the action for *ζ*, written in the form ([Exp:actionzeta]), is given by\n \n$$\\begin{aligned}\n & S [\\zeta] \\approx \\int {{\\rm d}}t\\, {{\\rm d}}^3 {\\hbox{\\boldmath{$x$}}} e^{3(\\rho+\\zeta)}\n \\varepsilon \\left[ \\dot{\\zeta}^2 + \\sum\\_{n=3}^\\infty \\frac{2}{n} f\\_n(t)\n \\dot{\\zeta}^n \\right]\\,, \\label{Exp:actionlargescale}\\end{aligned}$$ \nwhere we schematically wrote the non-linear terms which include $\\dot{\\zeta}$. Here, the time dependent function *f**n*(*t*) is expressed only in terms of the background quantities such as $\\dot{\\rho}$ and the slow-roll parameters. Varying the action with respect to *ζ*, we obtain the equation of motion in the large scale limit as\n \n$$\\begin{aligned}\n & \\partial\\_t \\left[ e^{3(\\rho+\\zeta)} \\varepsilon \\left\\{ \\dot{\\zeta} +\n \\sum\\_{n=3}^\\infty f\\_n(t) \\dot{\\zeta}^{n-1} \\right\\} \\right] \\approx 0 \\,.\n \\label{Exp:eomlargescale} \\end{aligned}$$ \nThis equation motion has the anticipated constant solution in time as the non-decaying mode. I.e., if *ζ*(*x*) = *F*(*x*) is a solution of Eq. ([Exp:eomlargescale]), *F*(*x*) + *C* with a constant shift should also satisfy the equation. Then, after other solutions of the non-linear equation decay, *ζ* should be conserved at large scales. The relation between the shift symmetry and the conservation of *ζ* was pointed out also in Horndeski’s theory .\nEffective action for *ζ* with loop corrections of heavy field\n=============================================================\nNext, we consider a two-field model with one inflaton and one heavy field. The latter field does not contribute to the classical background evolution. Following the Feynman and Vernon’s method , in this section, we compute the effective action for the curvature perturbation with loop corrections of the massive field.\nTwo field model\n---------------\nIn this paper, we consider a light scalar field *ϕ* and a massive scalar field *χ* whose action is given by\n \n$$\\begin{aligned}\n S &= \\frac{1}{2} \\int \\sqrt{-g} \\left[ R - g^{\\mu \\nu}\n \\partial\\_\\mu \\phi \\partial\\_\\nu \\phi - g^{\\mu \\nu}\n \\partial\\_\\mu \\chi \\partial\\_\\nu \\chi - 2V(\\phi,\\, \\chi) \\right] {{\\rm d}}^{d+1} x\\,, \\label{Exp:action}\\end{aligned}$$ \nwhere *V*(*ϕ*, *χ*) is a potential for the scalar fields:\n \n$$\\begin{aligned}\n & V(\\phi,\\, \\chi) = V\\_{ph}(\\phi) + V\\_{ch}(\\phi,\\, \\chi) \\,, \\label{Exp:V}\\end{aligned}$$ \nwith\n \n$$\\begin{aligned}\n & V\\_{ch}(\\phi,\\, \\chi) \\equiv \\frac{1}{2} M^2(\\phi) \\chi^2 +\n \\frac{\\lambda}{4!}\\chi^4 \\,.\\end{aligned}$$ \nWe decomposed the potential *V*(*ϕ*, *χ*) into the *χ* independent part *V**p**h* and the rest *V**c**h*. When the mass of *χ* field, *M*, depends on the inflaton *ϕ*, this model includes the direct interaction between *ϕ* and *χ*, e.g., *ϕ*2*χ*2, which was addressed in Refs. . We assume that the mass of the inflaton *m* is much smaller than the Hubble parameter *H* as *m* ≪ *H*, while the mass of the field *χ* is bigger than *H* as *M* > *H*. Then, the classical background evolution is determined solely by the inflaton *ϕ* and (the linear perturbation of) *χ* becomes the pure isocurvature perturbation. In this paper, we only allow the renormalizable interactions for *χ*, while *W*(*ϕ*) may include non-renormalizable interactions. To perform the dimensional regularization, we consider a general (*d* + 1) dimensional spacetime. An extension to include non-renormalizable interactions for *χ* is straightforward as long as a finite order of loop corrections is concerned.\nAs in the single field case, we determine the time slicing, imposing\n \n$$\\begin{aligned}\n & \\delta \\phi=0\\,.\\end{aligned}$$ \nFor our later use, we discriminate the part which explicitly depends on *χ* from the rest as\n \n$$\\begin{aligned}\n S= S\\_{ad} [N,\\, N\\_i,\\, \\zeta] + S\\_\\chi[N,\\, N\\_i,\\, \\zeta,\\, \\chi]\\,,\\end{aligned}$$ \nwhere *S**a**d* agrees with the action in single field models, given in Eq. ([Eq:ADMaction]), and *S**χ* is given by\n \n$$\\begin{aligned}\n S\\_\\chi & = \\frac{1}{2} \\int e^{d(\\rho+ \\zeta)}\n \\biggl[ \\frac{1}{N} (\\dot{\\chi} - e^{-2(\\rho +\\zeta)} \\delta^{ij} N\\_i\n \\partial\\_j \\chi)^2 \\cr\n & \\qquad \\qquad \\qquad \\qquad \\quad - N e^{-2(\\rho+\n \\zeta)} (\\partial\\_i \\chi)^2 - 2 N V(\\phi,\\, \\chi) \\biggr] {{\\rm d}}t\\, {{\\rm d}}^d {\\hbox{\\boldmath{$x$}}}\\,.\\end{aligned}$$ \nEven if there is no direct interaction between *ϕ* and *χ*, gravity yields the non-linear interaction between *ζ* and the heavy field *χ*.\nEffective action\n----------------\nSince the mass of the field *χ* is much bigger than the Hubble scale *H*, it is natural to set the background value of *χ* to 0. Then, the field *χ* does not contribute to the classical background evolution. Meanwhile, because of the interaction between *ζ* and *χ*, the quantum fluctuation of *χ* can affect the evolution of the field *ζ*. In order to compute the evolution of *ζ* under the influence of *χ*, we compute the Feynman and Vernon’s influence functional , which can be obtained by integrating out the field *χ*, in the closed time path (or the in-in) formalism.\nAs in the single field case, the action *S* also includes the lapse function and the shift vector, which can be removed by solving the Hamiltonian and momentum constraint equations. These constraint equations are also modified due to the quantum fluctuation of the heavy field *χ*.\nUsing the closed time path, the *n*-point function of the curvature perturbation is given by\n \n$$\\begin{aligned}\n & \\langle \\Psi\\,| T\\zeta(x\\_1) \\cdots \\zeta(x\\_n) |\\, \\Psi \\rangle\\cr\n & = \\frac{\\int D\\zeta\\_+ \\int D\\chi\\_+ \\int D \\zeta\\_- \\int D\\chi\\_-\\,\n \\zeta\\_+(x\\_1) \\cdots \\zeta\\_+(x\\_n)\\, e^{i S[\\delta g\\_+,\\, \\chi\\_+]-\n i S[\\delta g\\_-,\\, \\chi\\_-]}}{\\int D\\zeta\\_+ \\int D\\chi\\_+ \\int D \\zeta\\_- \\int\n D\\chi\\_-\\, e^{i S[\\delta g\\_+,\\, \\chi\\_+]-\n i S[\\delta g\\_-,\\, \\chi\\_-]}}\\,, \\end{aligned}$$ \nwhere we used an abbreviation *δ**g* = (*δ**N*, *N**i*, *ζ*). In the closed time path, we double the fields: *δ**g*+ and *χ*+ denote the fields integrated from the past infinity to the time *t* and *δ**g*− and *χ*− denote the fields integrated from the time *t* to the past infinity. Here, *T* denotes the time ordering. Inserting *ζ*−(*x*) into the path integral in the numerator, we can compute the *n*-point function ordered in the anti-time ordering. Since *N* and *N**i* are not independent variables, we perform the path integral only regarding *ζ* and *χ*.\nIntroducing the effective action ${S\\_{\\rm eff}}$ as\n \n$$\\begin{aligned}\n &i {S\\_{\\rm eff}}[\\zeta\\_+,\\, \\zeta\\_-] \\equiv \\ln \\left[ \\int D \\chi\\_+ \\int D \\chi\\_- \\,\n e^{i S[\\delta g\\_+,\\, \\chi\\_+]- i S[\\delta g\\_-,\\, \\chi\\_-]}\\right]\\,, \\label{Def:Seff}\\end{aligned}$$ \nwe rewrite the *n*-point function for *ζ* as\n \n$$\\begin{aligned}\n & \\langle \\Psi\\,| T\\zeta(x\\_1) \\cdots \\zeta(x\\_n) |\\, \\Psi \\rangle\n = \\frac{\\int D\\zeta\\_+ \\int D \\zeta\\_- \\, \\zeta\\_+(x\\_1) \\cdots\n \\zeta\\_+(x\\_n)\\, e^{i {S\\_{\\rm eff}}[\\delta g\\_+,\\,\n \\delta g\\_-]}}{\\int D\\zeta\\_+ \\int D \\zeta\\_-\\,e^{i\n {S\\_{\\rm eff}}[\\delta g\\_+,\\,\\delta g\\_-]}} \\,. \\end{aligned}$$ \nBy inserting the action *S* into Eq. ([Def:Seff]), the effective action is recast into\n \n$$\\begin{aligned}\n & {S\\_{\\rm eff}}[\\delta g\\_+,\\, \\delta g\\_-] = S\\_{ad}[\\delta g\\_+] -\n S\\_{ad}[\\delta g\\_-] + {S\\_{\\rm eff}}'[\\delta g\\_+,\\, \\delta g\\_-]\\,, \\label{Exp:Seff}\\end{aligned}$$ \nwhere ${S\\_{\\rm eff}}'$ is the so-called influence functional, given by\n \n$$\\begin{aligned}\n &i {S\\_{\\rm eff}}'[\\delta g\\_+,\\, \\delta g\\_-] \n \\equiv \\ln \\left[ \\int D \\chi\\_+ \\int D \\chi\\_- \\,e^{i S\\_\\chi[\\delta g\\_+,\\,\n \\chi\\_+]- i S\\_\\chi[\\delta g\\_-,\\, \\chi\\_-]}\\right]\\,, \\label{Exp:Seffd}\\end{aligned}$$ \nwhere we factorized *S**a**d*[*δ**g*±] which commute with the path integral over *χ*±. The effective action ${S\\_{\\rm eff}}[\\delta g\\_+,\\, \\delta g\\_-]$ describes the evolution of the curvature perturbation affected by the quantum fluctuation of the heavy field *χ*.\nComputing the effective action\n------------------------------\nPerforming the path integrals about *χ*+ and *χ*−, we can compute the effective action ${S\\_{\\rm eff}}'[\\delta g\\_+,\\, \\delta g\\_-]$. Expanding ${S\\_{\\rm eff}}'$ in terms of *δ**g* = (*δ**N*, *N**i*, *ζ*), we obtain\n \n$$\\begin{aligned}\n & i {S\\_{\\rm eff}}'[\\delta g\\_+,\\, \\delta g\\_-] \\equiv \\sum\\_{n=0}^\\infty i\n {S'\\_{{\\rm eff}(n)}}[\\delta g\\_+,\\, \\delta g\\_-] \\,,\\end{aligned}$$ \nwhere ${S'\\_{{\\rm eff}(n)}}$ denotes the terms which include *n* *δ**g**α*s, given by\n \n$$\\begin{aligned}\n i {S'\\_{{\\rm eff}(n)}}[\\delta g\\_+,\\, \\delta g\\_-] \n &= \\frac{1}{n!}\n \\sum\\_{\\alpha\\_1 = \\pm} \\cdots \\sum\\_{\\alpha\\_n = \\pm}\n \\int {{\\rm d}}^{d+1} x\\_1 \\cdots \\int {{\\rm d}}^{d+1} x\\_n \\cr\n & \\qquad \\times \\delta g\\_{\\alpha\\_1}(x\\_1) \\cdots\n \\delta g\\_{\\alpha\\_n}(x\\_n)\\,W^{(n)}\\_{\\delta g\\_{\\alpha\\_1}\n \\cdots \\delta g\\_{\\alpha\\_n}}(x\\_1,\\, \\cdots,\\, x\\_n) \\,, \\label{Expn:Seffd}\\end{aligned}$$ \nwith\n \n$$\\begin{aligned}\n & W^{(n)}\\_{\\delta g\\_{\\alpha\\_1} \\cdots \\delta g\\_{\\alpha\\_n}}(x\\_1,\\, \\cdots,\\, x\\_n) \\equiv\n \\frac{\\delta^n i {S\\_{\\rm eff}}'[\\zeta\\_+,\\, \\zeta\\_-]}{\\delta g\\_{\\alpha\\_1}(x\\_1)\n \\cdots \\delta g\\_{\\alpha\\_n}(x\\_n)}\n \\bigg|\\_{\\delta g\\_{\\pm}=0} \\,. \\label{Def:tWn}\\end{aligned}$$ \nIn Eq. ([Expn:Seffd]), each *δ**g**α**m* with *m* = 1, ⋯, *n* should add up *δ**N**α**m*, *N**i*, *α**m*, and *ζ**α**m*. Here and hereafter, for notational brevity, we omit the summation symbol over *δ**g* unless necessary. Using Eq. ([Exp:Seffd]), we can express the variation of ${S\\_{\\rm eff}}'$ with respect to *δ**g*± by using the propagators for *χ*. Notice that the shift symmetry is not manifest in this series expansion.\nThe linear term in the effective action is given by\n \n$$\\begin{aligned}\n i {S'\\_{{\\rm eff}(1)}} &= \\sum\\_{\\alpha= \\pm} \\int {{\\rm d}}^{d+1} x\\,\n \\delta g\\_\\alpha(x) W^{(1)}\\_{\\delta g\\_\\alpha} (x)\n\\,, \\end{aligned}$$ \nwhere *W**δ**g**α*(1) is given by the expectation value as\n \n$$\\begin{aligned}\n & W^{(1)}\\_{\\delta g\\_+} (x) = - W^{(1)}\\_{\\delta g\\_-} (x) = \\left\\langle\n \\frac{\\delta i S\\_\\chi}{\\delta g(x)} \\bigg|\\_{\\delta g=0} \\right\\rangle\\,.\\end{aligned}$$ \nNext, we compute the quadratic terms in ${S\\_{\\rm eff}}'$. Taking the second variation of ${S\\_{\\rm eff}}'$ with respect to *δ**g*+, we obtain\n \n$$\\begin{aligned}\n W^{(2)}\\_{\\delta g\\_+ \\delta \\tilde{g}\\_+}(x\\_1,\\,x\\_2) & = i^2 \\left\\langle\n \\frac{\\delta S\\_\\chi[\\delta g\\_+,\\, \\chi\\_+]}{\\delta g\\_+(x\\_1)}\n \\bigg|\\_{\\delta g\\_+=0} \\frac{\\delta S\\_\\chi[\\delta g\\_+,\\, \\chi\\_+]}{\\delta\n \\tilde{g}\\_+(x\\_2)} \\bigg|\\_{\\delta g\\_+=0} \\right\\rangle\\_{\\pm} \\cr\n & \\qquad \\qquad \\qquad \\quad + i \\delta(x\\_1-x\\_2) \\left\\langle\n \\frac{\\delta^2 S\\_\\chi[\\zeta\\_+,\\, \\chi\\_+]}{\\delta g\\_+(x\\_1) \\delta \\tilde{g}\\_+(x\\_1)}\n \\bigg|\\_{\\delta g\\_+=0} \\right\\rangle\\_{\\pm}\\,, \\label{Exp:dS++}\\end{aligned}$$ \nwhere *δ**g* and *δ**g̃* are either *δ**N*, *N**i*, or *ζ*, and they can be different metric perturbations. Here, we introduced the expectation value:\n \n$$\\begin{aligned}\n & \\langle {\\cal O}[\\chi\\_+,\\, \\chi\\_-] \\rangle\\_{\\pm} \\equiv \\frac{\\int D \\chi\\_+\n \\int D \\chi\\_- \\,{\\cal O}[\\chi\\_+,\\, \\chi\\_-] e^{i S\\_\\chi[0,\\, \\chi\\_+]- i\n S\\_\\chi[0,\\,\\chi\\_-]}}{\\int D \\chi\\_+ \\int D \\chi\\_- \\,e^{i S\\_\\chi[0,\\,\\chi\\_+]- i\n S\\_\\chi[0,\\,\\chi\\_-]}} \\,.\\end{aligned}$$ \nSince the action *S**χ*[*δ**g*+, *χ*+] includes only local terms, the variation of *S**χ*[*δ**g*+, *χ*+] with respect to *δ**g*+(*x*1) and *δ**g̃*+(*x*2) yields the delta function *δ*(*x*1 − *x*2) in Eq. ([Exp:dS++]). Similarly, the second variation of ${S\\_{\\rm eff}}'$ with respect to *δ**g*− is given by\n \n$$\\begin{aligned}\n W^{(2)}\\_{\\delta g\\_- \\delta \\tilde{g}\\_-}(x\\_1,\\,x\\_2) & = i^2 \\left\\langle\n \\frac{\\delta S\\_\\chi[\\delta g\\_-,\\, \\chi\\_-]}{\\delta g\\_-(x\\_1)}\n \\bigg|\\_{\\delta g\\_-=0} \\frac{\\delta S\\_\\chi[\\delta g\\_-,\\, \\chi\\_-]}{\\delta\n \\tilde{g}\\_-(x\\_2)} \\bigg|\\_{\\delta g\\_-=0} \\right\\rangle\\_{\\pm} \\cr\n & \\qquad \\qquad \\qquad \\quad - i \\delta(x\\_1-x\\_2) \\left\\langle\n \\frac{\\delta^2 S\\_\\chi[\\delta g\\_-,\\, \\chi\\_-]}{\\delta g\\_-(x\\_1) \\delta \\tilde{g}\\_-(x\\_1)}\n \\bigg|\\_{\\delta g\\_-=0} \\right\\rangle\\_{\\pm}\\,. \\label{Exp:dS--}\\end{aligned}$$ \nTaking the derivative with respect to both *δ**g*+ and *δ**g*−, we obtain\n \n$$\\begin{aligned}\n W^{(2)}\\_{\\delta g\\_+ \\delta \\tilde{g}\\_-}(x\\_1,\\,x\\_2) & = - i^2 \\left\\langle\n \\frac{\\delta S\\_\\chi[\\delta g\\_+,\\, \\chi\\_+]}{\\delta g\\_+(x\\_1)}\n \\bigg|\\_{\\delta g\\_+=0} \\frac{\\delta S\\_\\chi[\\delta g\\_-,\\, \\chi\\_-]}{\\delta\n \\tilde{g}\\_-(x\\_2)} \\bigg|\\_{\\delta g\\_-=0} \\right\\rangle\\_{\\pm} \\,, \\label{Exp:dS+-}\\end{aligned}$$ \nand\n \n$$\\begin{aligned}\n W^{(2)}\\_{\\delta g\\_- \\delta \\tilde{g}\\_+}(x\\_1,\\,x\\_2)\n & = - i^2 \\left\\langle\n \\frac{\\delta S\\_\\chi[\\delta g\\_-,\\, \\chi\\_-]}{\\delta g\\_-(x\\_1)}\n \\bigg|\\_{\\delta g\\_-=0} \\frac{\\delta S\\_\\chi[\\delta g\\_+,\\, \\chi\\_+]}{\\delta\n \\tilde{g}\\_+(x\\_2)} \\bigg|\\_{\\delta g\\_+=0} \\right\\rangle\\_{\\pm} \\,. \\label{Exp:dS-+}\\end{aligned}$$ \nThese functions *W**δ**g**α*1*δ**g̃**α*2(2)(*x*1, *x*2) can be expanded by the propagators of *χ* for *λ* = 0, i.e., the time-ordered (Feynman) propagator:\n \n$$\\begin{aligned}\n & G\\_F(x\\_1,\\, x\\_2) \\equiv \\frac{\\int D \\chi\\_+\\, \\chi\\_+(x\\_1) \\chi\\_+(x\\_2)\\,e^{i\n {S\\_{\\chi,0}}[\\chi\\_+]}}{\\int D \\chi\\_+\\,e^{i {S\\_{\\chi,0}}[\\chi\\_+]}}\\,, \\end{aligned}$$ \nthe anti-time ordered (Dyson) propagator:\n \n$$\\begin{aligned}\n & G\\_D(x\\_1,\\, x\\_2) \\equiv \n \\frac{\\int D \\chi\\_-\\, \\chi\\_-(x\\_1) \\chi\\_-(x\\_2)\\,e^{- i\n {S\\_{\\chi,0}}[\\chi\\_-]}}{\\int D \\chi\\_-\\,e^{i {S\\_{\\chi,0}}[\\chi\\_-]}}\\,, \\end{aligned}$$ \nand the Wightman functions:\n \n$$\\begin{aligned}\n & G^+(x\\_1,\\, x\\_2) \\equiv \n \\frac{\\int D \\chi\\_+ \\int D \\chi\\_- \\,\\chi\\_-(x\\_1) \\chi\\_+(x\\_2) e^{i {S\\_{\\chi,0}}[ \\chi\\_+]- i\n {S\\_{\\chi,0}}[\\chi\\_-]}}{\\int D \\chi\\_+ \\int D \\chi\\_- \\,e^{i {S\\_{\\chi,0}}[\\chi\\_+]- i\n {S\\_{\\chi,0}}[\\chi\\_-]}} \\,, \\cr\n& G^-(x\\_1,\\, x\\_2) \\equiv \n \\frac{\\int D \\chi\\_+ \\int D \\chi\\_- \\,\\chi\\_+(x\\_1) \\chi\\_-(x\\_2) e^{i {S\\_{\\chi,0}}[ \\chi\\_+]- i\n {S\\_{\\chi,0}}[\\chi\\_-]}}{\\int D \\chi\\_+ \\int D \\chi\\_- \\,e^{i {S\\_{\\chi,0}}[\\chi\\_+]- i\n {S\\_{\\chi,0}}[\\chi\\_-]}} \\,. \\end{aligned}$$ \nHere, *S**χ*, 0[*χ*] denotes the action given by\n \n$$\\begin{aligned}\n {S\\_{\\chi,0}}[\\chi] = \\frac{1}{2} \\int e^{d\\rho} \\biggl[ \\dot{\\chi}^2 -\n e^{-2\\rho} (\\partial\\_i \\chi)^2 - M^2(\\phi) \\chi^2 \\biggr] {{\\rm d}}t \\, {{\\rm d}}^d {\\hbox{\\boldmath{$x$}}}\\,.\\end{aligned}$$ \nRecall that these propagators are mutually related as\n \n$$\\begin{aligned}\n & G^-(x\\_1,\\,x\\_2) = G^{+\\,\\*}(x\\_1,\\,x\\_2)\\,, \\label{Exp:Gpm} \\\\\n & G\\_F(x\\_1,\\, x\\_2) = \\theta(t\\_1 - t\\_2) G^+(x\\_1,\\, x\\_2) + \\theta(t\\_2-\n t\\_1) G^-(x\\_1,\\,x\\_2) \\,, \\label{Exp:GF} \\\\\n & G\\_D(x\\_1,\\, x\\_2) = \\theta(t\\_1 - t\\_2) G^-(x\\_1,\\, x\\_2) + \\theta(t\\_2-\n t\\_1) G^+(x\\_1,\\,x\\_2)\\,, \\label{Exp:GD}\\end{aligned}$$ \nwhere *θ* is the Heaviside function.\nPropagators for *χ*\n-------------------\nIn this subsection, solving the mode function for the heavy field *χ*, we compute the propagators introduced in the previous subsection. At the linear order of *χ*, the equation of motion is given by\n \n$$\\begin{aligned}\n & \\ddot{\\chi}\\_k + d \\dot{\\rho} \\dot{\\chi}\\_k + \\{ M^2(\\phi) + (k e^{-\\rho})^2 \\} \\chi\\_k\n =0\\,, \\label{Eq:modechi}\\end{aligned}$$ \nwhere *χ**k* is the Fourier mode of *χ*. Changing the variable from *χ**k* to $X\\_k(t)= e^{\\frac{d-1}{2} \\rho(t)} \\chi\\_k(t)$ and using the conformal time *η*, the mode equation ([Eq:modechi]) is recast into\n \n$$\\begin{aligned}\n & X\\_k'' + \\Omega\\_k^2(\\eta)\\, X\\_k =0\\,, \\label{Eq:modechit}\\end{aligned}$$ \nwhere the dash denotes the derivative with respect to the conformal time *η* and the time dependent frequency Ω*k* is given by\n \n$$\\begin{aligned}\n \\Omega\\_k^2(\\eta) = k^2 + (M e^\\rho)^2 - \\rho'' - {\\rho'}^2\\,.\\end{aligned}$$ \nUsing *W**k* which satisfies\n \n$$\\begin{aligned}\n & W\\_k^2 = \\Omega\\_k^2 + \\frac{3}{4} \\left( W\\_k' \\over W\\_k \\right)^2 -\n \\frac{1}{2} \\frac{W\\_k''}{W\\_k}\\,, \\label{Eq:Wk}\\end{aligned}$$ \nthe mode equation ([Eq:modechit]) can be solved as\n \n$$\\begin{aligned}\n & X\\_k(\\eta) = \\frac{1}{\\sqrt{2 W\\_k}} e^{- i \\int^\\eta {{\\rm d}}\\eta' W\\_k(\\eta')}\\,. \\label{Eq:WKB}\\end{aligned}$$ \nUsing the mode function *χ**k*, we quantize the non self-interacting heavy field as follows\n \n$$\\begin{aligned}\n & \\chi(x) = \\int \\frac{{{\\rm d}}^d {\\hbox{\\boldmath{$k$}}}}{(2\\pi)^{d/2}} e^{i {\\hbox{\\boldmath{\\scriptsize$k$}}} \\cdot\n {\\hbox{\\boldmath{\\scriptsize$x$}}}} a\\_{{\\hbox{\\boldmath{\\scriptsize$k$}}}} \\chi\\_k(\\eta) + ({\\rm h.c.})\\,, \\label{Exp:chif}\\end{aligned}$$ \nwhere $a\\_{{\\hbox{\\boldmath{\\scriptsize$k$}}}}$ denotes the annihilation operator. With this expansion, the Wightman function *G*+(*x*1, *x*2) is given by\n \n$$\\begin{aligned}\n & G^+(x\\_1,\\, x\\_2) = \\int \\frac{{{\\rm d}}^d {\\hbox{\\boldmath{$k$}}}}{(2\\pi)^d} e^{i {\\hbox{\\boldmath{\\scriptsize$k$}}}\n \\cdot ({\\hbox{\\boldmath{\\scriptsize$x$}}}\\_1- {\\hbox{\\boldmath{\\scriptsize$x$}}}\\_2)} \\chi\\_k(\\eta\\_1) \\chi\\_k^\\*(\\eta\\_2)\\,. \\label{Exp:G+}\\end{aligned}$$ \nOnce the mode function *χ**k*(*η*) is given, using Eqs. ([Exp:Gpm])-([Exp:GD]) and ([Exp:G+]), we can compute all the propagators which appear in the expansion ([Exp:dS++]).\nWard-Takahashi identity from the scaling symmetry\n=================================================\nAs was discussed in Sec. [SSec:scaling], the action for the single field model preserves the invariance under the scale transformation, which changes $\\zeta(t,\\, {\\hbox{\\boldmath{$x$}}})$ to $\\zeta(t,\\, e^{-s} {\\hbox{\\boldmath{$x$}}}) -s$. This symmetry is preserved classically also for multi-field models of inflation, since it is a part of the spatial diffeomorphism. In a quantum field theory, it is known that a symmetry leads to a corresponding Ward-Takahashi (WT) identity. When the scaling symmetry is also preserved at the quantum level, we obtain the WT identity. In Sec. [SSec:WT], we discuss the WT identity from the scaling symmetry of the correlators of *χ*. In single field models of inflation, it was shown that the WT identity of the scaling symmetry yields the consistency relation which relates the (*n* + 1)-point function of *ζ* with one soft external leg to *n*-point function of *ζ* . Likewise, in Sec. [SSec:CR], we find that the WT identity derived in Sec. [SSec:WT] gives the consistency relation which relates the (*n* + 1)-point cross-correlation with *n* *χ*s and one soft *ζ* to the *n*-point auto-correlation of *χ*.\nWard-Takahashi identity\n-----------------------\nIn single field models, an invariant quantity regarding the scale transformation was constructed in Refs. by using the smeared geodesic normal coordinates, defined in Eq. ([Def:xg]). Using ${\\hbox{\\boldmath{$x$}}}\\_g\\equiv {\\hbox{\\boldmath{$x$}}}\\_g(t\\_f)$, evaluated at the end of inflation *t* = *t**f*, we define\n \n$$\\begin{aligned}\n & {^g\\!\\chi}(t,\\, {\\hbox{\\boldmath{$x$}}}\\_g) = \\chi(t,\\, {\\hbox{\\boldmath{$x$}}})= \\chi(t,\\, e^{-\n {{^g\\!\\bar{\\zeta}}}} {\\hbox{\\boldmath{$x$}}}\\_g)\\,,\\end{aligned}$$ \nwhich is invariant under the scaling symmetry with the constant parameter *s*. Here, *g* *ζ̄* ≡ *g* *ζ̄*(*t**f*).\nWhen the scaling symmetry is also preserved for the quantum system, the correlation functions of ${^g\\!\\chi}(t,\\, {\\hbox{\\boldmath{$x$}}}\\_g)$ should be invariant under the scale transformation as\n \n$$\\begin{aligned}\n & \\langle \\chi\\_{\\alpha\\_1}(t\\_1,\\, e^{- {{^g\\!\\bar{\\zeta}}}}{\\hbox{\\boldmath{$x$}}}\\_{g1}) \\cdots \\chi\\_{\\alpha\\_n}(t\\_n,\\, e^{- {{^g\\!\\bar{\\zeta}}}}{\\hbox{\\boldmath{$x$}}}\\_{gn})\n \\rangle\\_{\\pm} \\big|\\_{\\delta g} \\cr\n &\\,\\,= \\langle \\chi\\_{\\alpha\\_1}(t\\_1,\\, e^{- {{^g\\!\\bar{\\zeta}}}+s} {\\hbox{\\boldmath{$x$}}}\\_{g1})\n \\cdots \\chi\\_{\\alpha\\_n}(t\\_n,\\, e^{- {{^g\\!\\bar{\\zeta}}}+s}{\\hbox{\\boldmath{$x$}}}\\_{gn}) \\rangle\\_{\\pm}\n \\big|\\_{\\delta g\\_s} \\label{WT} \\end{aligned}$$ \nwith *α**i* = ± . Here, *δ**g**s* denotes the metric perturbations after the scale transformation. Under the scale transformation, *δ**N* and *N**i* change as\n \n$$\\begin{aligned}\n & \\delta N\\_s(t,\\, {\\hbox{\\boldmath{$x$}}}) = \\delta N(t,\\, e^{-s}{\\hbox{\\boldmath{$x$}}}) \\,, \\label{Exp:transN} \\\\\n & N\\_{i, s} (t,\\, {\\hbox{\\boldmath{$x$}}}) = e^{-s} N\\_i(t, e^{-s} {\\hbox{\\boldmath{$x$}}})\\,, \\label{Exp:transNi}\\end{aligned}$$ \nand *ζ* changes as in Eq. ([Exp:transzeta]), and then *g* *ζ̄* changes to *g* *ζ̄**s* = *g* *ζ̄* − *s*. Equation ([WT]) holds only when the quantum state also preserves the scaling symmetry. This is the WT identity for the scaling symmetry. At ${\\cal O}(s)$, setting *δ**g* = 0, the WT identity yields\n \n$$\\begin{aligned}\n & \\sum\\_{i=1}^n {\\hbox{\\boldmath{$x$}}}\\_i \\cdot \\partial\\_{{\\hbox{\\boldmath{\\scriptsize$x$}}}\\_i} \\langle\n \\chi\\_{\\alpha\\_1}(x\\_1) \\cdots \\chi\\_{\\alpha\\_n}(x\\_n) \\rangle\\_{\\pm}\n - \\int {{\\rm d}}^{d+1} x \\left\\langle\n \\chi\\_{\\alpha\\_1}(x\\_1) \\cdots \\chi\\_{\\alpha\\_n}(x\\_n) \\frac{\\delta\n i S\\_\\chi[\\delta g\\_+,\\, \\chi\\_+]}{\\delta \\zeta\\_+(x)} \\bigg|\\_{\\delta g\\_+=0}\n \\right\\rangle\\_{\\!\\pm} \\nonumber \\\\\n & \\qquad \\qquad + \\int {{\\rm d}}^{d+1} x \\left\\langle\n \\chi\\_{\\alpha\\_1}(x\\_1) \\cdots \\chi\\_{\\alpha\\_n}(x\\_n) \\frac{\\delta\n i S\\_\\chi[\\delta g\\_-,\\, \\chi\\_-]}{\\delta \\zeta\\_-(x)} \\bigg|\\_{\\delta g\\_-=0}\n \\right\\rangle\\_{\\!\\pm}=0\\,. \\label{WTs1} \\end{aligned}$$ \nSince the changes of *δ**N* and *N**i* under the scale transformation are linear in *N* and *N**i* and their derivatives, they vanish after setting *δ**g* to 0.\nUsing the WT identity ([WTs1]) with *x*1 = ⋯ = *x**p* ≡ *x* and *x**p* + 1 = ⋯ = *x**n* ≡ *x*ʹ, we obtain\n \n$$\\begin{aligned}\n & ({\\hbox{\\boldmath{$x$}}} \\cdot \\partial\\_{{\\hbox{\\boldmath{\\scriptsize$x$}}}} + {\\hbox{\\boldmath{$x$}}}' \\cdot \\partial\\_{{\\hbox{\\boldmath{\\scriptsize$x$}}}'})\n \\langle \\chi\\_\\alpha^p(x) \\chi\\_\\alpha^{n-p}(x') \\rangle\\_{\\pm} \n - \\int {{\\rm d}}^{d+1} y \\left\\langle \\chi\\_\\alpha^p(x) \\chi\\_\\alpha^{n-p}(x') \\frac{\\delta\n i S\\_\\chi[\\delta g\\_+,\\, \\chi\\_+]}{\\delta \\zeta\\_+(y)} \\bigg|\\_{\\delta g\\_+=0}\n \\right\\rangle\\_{\\!\\pm} \\nonumber \\\\\n & \\qquad \\qquad + \\int {{\\rm d}}^{d+1} y \\left\\langle\n \\chi\\_\\alpha^p(x) \\chi\\_\\alpha^{n-p}(x') \\frac{\\delta\n i S\\_\\chi[\\delta g\\_-,\\, \\chi\\_-]}{\\delta \\zeta\\_-(y)} \\bigg|\\_{\\delta g\\_-=0}\n \\right\\rangle\\_{\\!\\pm}=0 \\,, \\label{Exp:WTpr}\\end{aligned}$$ \nwhere *α* = ± . In the next section, using these identities, we show the conservation of *ζ*, including the loop corrections of the heavy field.\nConsistency relation (Soft theorem)\n-----------------------------------\nIn single field models of inflation, it is known that the WT identity for the scaling symmetry gives the consistency relation. The consistency relation for *ζ* is an example of the soft theorem, which was first shown for the soft graviton scattering by Weinberg . Recently, Weinberg’s soft theorem was recaptured by Strominger *et al.* and was shown to be equivalent to a Ward-Takahashi identity in an asymptotically flat spacetime .\nHere, we show that the WT identity ([WTs1]) also gives a consistency relation in multi-field models. Performing the Fourier transformation of the WT identity ([WTs1]) evaluated at an equal time *t* with all *α**i*s chosen as + , we obtain\n \n$$\\begin{aligned}\n & \\left(\\sum\\_{i=1}^n {\\hbox{\\boldmath{$k$}}}\\_i \\cdot \\partial\\_{{\\hbox{\\boldmath{\\scriptsize$k$}}}\\_i} + nd \\right)\n \\langle \\chi\\_+({\\hbox{\\boldmath{$k$}}}\\_1) \\cdots \\chi\\_+({\\hbox{\\boldmath{$k$}}}\\_n) \\rangle\\_{\\pm} \\nonumber \\\\\n & \\qquad - i \\int {{\\rm d}}^{d+1} y \\left\\langle \\frac{\\delta\n S\\_{\\chi}[\\delta g\\_-,\\,\n \\chi\\_-]}{\\delta \\zeta\\_-(y)} \\bigg|\\_{\\delta g\\_-=0} \\chi\\_+({\\hbox{\\boldmath{$k$}}}\\_1)\n \\cdots \\chi\\_+({\\hbox{\\boldmath{$k$}}}\\_n) \\right\\rangle\\_{\\!\\pm} \\nonumber \\\\\n & \\qquad + i \\int {{\\rm d}}^{d+1} y \\left\\langle \\chi\\_+({\\hbox{\\boldmath{$k$}}}\\_1)\n \\cdots \\chi\\_+({\\hbox{\\boldmath{$k$}}}\\_n) \\frac{\\delta S\\_{\\chi}[\\delta g\\_+,\\,\n \\chi\\_+]}{\\delta \\zeta\\_+(y)} \\bigg|\\_{\\delta g\\_+=0} \\right\\rangle\\_{\\! \\pm}\n =0\\,, \\label{Eq:WTtf}\\end{aligned}$$ \nwhere we abbreviated *t* in the argument of *χ*s. The correlator in the first line is simply the in-in *n*-point function of $\\chi(t,\\,{\\hbox{\\boldmath{$k$}}})$. The correlator in the second line is given by the product of the Wightman function, the Feynman propagator, and the Dyson propagator, which appear by contracting *χ*± with *χ*∓, *χ*+ with *χ*+, and *χ*− with *χ*−, respectively. The correlator in the third line is given by the product of the Feynman propagator. The interaction vertices inserted at any time after *t* are canceled between the terms in the second and third lines. This ensures the causality in the closed time path formalism. Taking into account this cancellation, we can rewrite Eq. ([Eq:WTtf]) as\n \n$$\\begin{aligned}\n & \\left[ \\sum\\_{i=2}^n {\\hbox{\\boldmath{$k$}}}\\_i \\cdot \\partial\\_{{\\hbox{\\boldmath{\\scriptsize$k$}}}\\_i} + (n-1)d\n \\right] \\left\\langle \\chi\\left(- \\sum\\_{j=2}^n {\\hbox{\\boldmath{$k$}}}\\_j\\right) \\chi({\\hbox{\\boldmath{$k$}}}\\_2) \\cdots\n \\chi({\\hbox{\\boldmath{$k$}}}\\_n) \\right\\rangle' \\cr\n & = - i \\int^t {{\\rm d}}t\\_y \\int {{\\rm d}}^d {\\hbox{\\boldmath{$y$}}} \\left\\langle \\left[\n \\chi({\\hbox{\\boldmath{$k$}}}\\_1) \\cdots \\chi({\\hbox{\\boldmath{$k$}}}\\_n),\\, \\frac{\\delta S\\_\\chi}{\\delta\n \\zeta(y)} \\bigg|\\_{\\zeta=0} \\right] \\right\\rangle' \\,, \\label{Eq:CR} \\end{aligned}$$ \nwhere the correlation function with dash denotes the correlation function from which (2*π*)*d* and the delta function are removed, e.g.,\n \n$$\\begin{aligned}\n & \\langle \\chi({\\hbox{\\boldmath{$k$}}}\\_1) \\cdots \\chi({\\hbox{\\boldmath{$k$}}}\\_n) \\rangle \\equiv (2\\pi)^d\n \\delta({\\hbox{\\boldmath{$k$}}}\\_1 + \\cdots + {\\hbox{\\boldmath{$k$}}}\\_n) \\langle \\chi({\\hbox{\\boldmath{$k$}}}\\_1) \\cdots\n \\chi({\\hbox{\\boldmath{$k$}}}\\_n) \\rangle' \\,. \\end{aligned}$$ \nIn deriving Eq. ([Eq:CR]), we used\n \n$$\\begin{aligned}\n & \\sum\\_{i=1}^n {\\hbox{\\boldmath{$k$}}}\\_i \\cdot \\partial\\_{{\\hbox{\\boldmath{\\scriptsize$k$}}}\\_i} \\delta({\\hbox{\\boldmath{$k$}}}\\_1 +\n \\cdots {\\hbox{\\boldmath{$k$}}}\\_n) \\langle \\chi({\\hbox{\\boldmath{$k$}}}\\_1) \\cdots\n \\chi({\\hbox{\\boldmath{$k$}}}\\_n) \\rangle' \\cr\n & = \\delta({\\hbox{\\boldmath{$k$}}}\\_1 +\n \\cdots {\\hbox{\\boldmath{$k$}}}\\_n) \\left( \\sum\\_{i=2}^n {\\hbox{\\boldmath{$k$}}}\\_i \\cdot\n \\partial\\_{{\\hbox{\\boldmath{\\scriptsize$k$}}}\\_i} -d \\right) \\left\\langle \\chi\\left(- \\sum\\_{j=2}^n\n {\\hbox{\\boldmath{$k$}}}\\_j\\right) \\chi({\\hbox{\\boldmath{$k$}}}\\_2) \\cdots\n \\chi({\\hbox{\\boldmath{$k$}}}\\_n) \\right\\rangle' \\,. \\end{aligned}$$ \ncc \n[Fg:CR]\nThe correlation function in the second line of Eq. ([Eq:CR]) is the in-in *n*-point function of $\\chi({\\hbox{\\boldmath{$k$}}})$ where the gravitational interaction vertices with *n* heavy fields *χ* and one amputated soft curvature perturbation *ζ* are inserted. (See Fig. [Fg:CR].) Then, attaching the external soft propagator of *ζ* to this correlation function yields the (*n* + 1)-point function of *n* *χ*s and one soft *ζ*, i.e.,\n \n$$\\begin{aligned}\n & \\left[ \\sum\\_{i=2}^n {\\hbox{\\boldmath{$k$}}}\\_i \\cdot \\partial\\_{{\\hbox{\\boldmath{\\scriptsize$k$}}}\\_i} + (n-1)d\n \\right] \\left\\langle \\chi\\left(- \\sum\\_{j=2}^n {\\hbox{\\boldmath{$k$}}}\\_j\\right) \\chi({\\hbox{\\boldmath{$k$}}}\\_2) \\cdots\n \\chi({\\hbox{\\boldmath{$k$}}}\\_n) \\right\\rangle' \\cr\n & \\qquad = - \\lim\\_{k \\to 0} \\frac{ \\langle \\zeta({\\hbox{\\boldmath{$k$}}}) \\chi({\\hbox{\\boldmath{$k$}}}\\_1) \\cdots\n \\chi({\\hbox{\\boldmath{$k$}}}\\_n) \\rangle'}{P\\_\\zeta(k)} \\,,\\end{aligned}$$ \nwhere *P**ζ*(*k*) is the power spectrum of the free *ζ*. This is the consistency relation for the heavy field *χ*. The correlation function in the second line contains only one gravitational interaction vertex with *ζ*, but it can contain more than one self interaction vertexes for the heavy field *χ*. This is one example of the soft theorem.\nUsing the WT identity at ${\\cal O}(s)$, we derived the consistency relation for the correlation functions with one soft *ζ*. Using the WT identity at ${\\cal O}(s^p)$, we can derive the consistency relations with *p* soft *ζ*s.\nScaling symmetry and WKB solution\n---------------------------------\nIn this subsection, we explicitly analyze the consistency relation ([Eq:CR]) for the case with *λ* = 0. Inserting Eq. ([Exp:chif]) into Eq. ([Eq:CR]), we obtain\n \n$$\\begin{aligned}\n &({\\hbox{\\boldmath{$k$}}} \\cdot \\partial\\_{{\\hbox{\\boldmath{\\scriptsize$k$}}}} + d) |\\chi\\_k(t)|^2 \\cr\n & = 2 {\\rm Im} \\biggl[ \\int^t\n {{\\rm d}}t' e^{d\\rho(t')} \\chi\\_k^2(t) \\left\\{\n d(\\dot{\\chi}\\_k^\\*\\!^2(t') - M^2 \\chi\\_k^\\*\\!^2 (t') ) - (d-2)\n \\frac{k^2}{e^{2\\rho(t')}} \\chi\\_k^\\*\\!^2(t') \\right\\} \\biggr].\\end{aligned}$$ \nIntegrating by parts and using the mode equation, we obtain\n \n$$\\begin{aligned}\n & {\\hbox{\\boldmath{$k$}}} \\cdot \\partial\\_{{\\hbox{\\boldmath{\\scriptsize$k$}}}} |\\chi\\_k(t)|^2 = 4 \\, {\\rm Im} \\left[\n \\int^t {{\\rm d}}t' e^{d \\rho(t')} \\frac{k^2}{e^{2 \\rho(t')}} \\chi\\_k^2(t)\n \\chi\\_k^\\*\\!^2 (t') \\right]\\,, \\label{CRF}\\end{aligned}$$ \nwhere *d*∣*χ**k*(*t*)∣2 in the left hand side was canceled with the term which appears by operating the time derivative on the Heaviside function *θ*(*t* − *t*ʹ).\nWe can show that the WKB solution satisfies the WT identity ([CRF]). In order to show this statement, we rewrite Eq. ([CRF]) as\n \n$$\\begin{aligned}\n & \\chi\\_k(\\eta) L\\_k^\\*(\\eta) + \\chi\\_k^\\*(\\eta) L\\_k(\\eta) =0 \\,, \\label{CFT2}\\end{aligned}$$ \nintroducing\n \n$$\\begin{aligned}\n & L\\_k(\\eta) \\equiv k \\partial\\_k \\chi\\_k(\\eta) - 2 i \\chi\\_k^\\*(\\eta) \\int^\\eta {{\\rm d}}\\eta'\n e^{(d-1)\\rho(\\eta')} k^2 \\chi\\_k^2(\\eta') \\cr\n & \\qquad \\qquad \\quad + 2 i \\chi\\_k(\\eta) \\int^\\eta\\_{\\bar{\\eta}} {{\\rm d}}\\eta'\n e^{(d-1)\\rho(\\eta')} k^2 |\\chi\\_k(\\eta')|^2 + i k \\bar{\\eta} \\chi\\_k(\\eta)\\,. \\label{Def:Lkt}\\end{aligned}$$ \nThe last two terms in *L**k*(*η*) are canceled between the two terms in Eq. ([CFT2]). The time integral of the second term converges by rotating the time path as *η* → − ∞(1 + *i**ε*) where *ε* is a positive constant. For *L**k*\\*(*η*), the time integral of the second term should be rotated as *η* → − ∞(1 − *i**ε*). We choose *η̄* at a time in the distant past when the mode function can be well approximated by the leading order WKB solution with *W**k* = *k*. (To be precise, *η̄* differs for a different wavenumber *k*.) Since *L**k*(*η*) satisfies the mode equation for *χ**k*(*η*), i.e.,\n \n$$\\begin{aligned}\n & {L}''\\_k + (d-1) \\rho' L'\\_k + \\{ k^2 + M^2(\\phi) e^{2\\rho} \\} L\\_k\n =0\\,, \\end{aligned}$$ \nand the initial conditions *L**k*(*η̄*) = *L*ʹ*k*(*η̄*) = 0, *L**k*(*η*) vanishes all the time. Thus, we find that the WKB solution satisfies Eq. ([CFT2]).\nFor the exact de Sitter space, in the limit *k**e*− *ρ* ≪ *M* and *H* ≪ *M*, we can easily check that the WKB solution, given by\n \n$$\\begin{aligned}\n & \\chi\\_k (t) \\simeq \\frac{e^{\\frac{d}{2}\\rho(t)}}{\\sqrt{2 M}} \\left[ 1\n + \\frac{1}{4} \\left( \\frac{k}{M e^\\rho} \\right)^2 \\left( -1 + i\n \\frac{M}{H} \\right) \\right] e^{- i Mt}\\,,\\end{aligned}$$ \nsatisfies Eq. ([CRF]) as\n \n$$\\begin{aligned}\n & {\\hbox{\\boldmath{$k$}}} \\cdot \\partial\\_{{\\hbox{\\boldmath{\\scriptsize$k$}}}} |\\chi\\_k(t)|^2 = - e^{(d-1)\\rho(t)}\n \\frac{k^2}{2 \\omega\\_k^3(t)} \\simeq - \\frac{e^{-d \\rho(t)}}{2 M} \\left(\n \\frac{k}{M e^\\rho} \\right)^2 \\left\\{ 1 - \\frac{3}{2} \\left(\n \\frac{k}{M e^\\rho} \\right)^2 \\right\\} \\,. \\end{aligned}$$ \nConservation of *ζ* with loop corrections of heavy field\n========================================================\nIn this section, we show that when the scaling symmetry is preserved, the curvature perturbation *ζ* is conserved in time at super Hubble scales, including the loop correction of the heavy field *χ*. For this purpose, first we rewrite the effective action, using the WT identities derived in the previous section. Then, using the obtained effective action, we show the conservation of *ζ*.\nEffective action with scaling symmetry\n--------------------------------------\nAs discussed for the single field inflation in Sec. [SSec:single], the presence of the constant solution is implied by the scaling symmetry. In this subsection, using the WT identity, we rewrite the effective action in such a way that the scaling symmetry becomes manifest.\nTaking the variation of *S**χ* with respect to *δ**g*, we can compute *W**δ**g**α*1⋯*δ**g**α**n*(*n*) and the effective action. For instance, taking the *n*-th derivative of *S**χ* with respect to *ζ* and setting *δ**g* to 0, we obtain\n \n$$\\begin{aligned}\n &\\frac{\\delta^n S\\_\\chi[\\delta g,\\, \\chi]}{\\delta\n \\zeta(x\\_1) \\cdots \\delta \\zeta(x\\_n)} \\bigg|\\_{\\delta g=0} \\cr\n & = \\delta(x\\_1 - x\\_2) \\cdots \\delta(x\\_{n-1} - x\\_n) \\frac{e^{d \\rho(t\\_1)}}{2}\n \\cr & \\quad \\times \\left[ d^n \\! \\left( \\dot{\\chi}^2(x\\_1) - M^2\n \\chi^2(x\\_1)- \\frac{\\lambda}{12} \\chi^4(x\\_1) \\right) - (d-2)^n\n e^{-2\\rho(t\\_1)} (\\partial\\_{{\\hbox{\\boldmath{\\scriptsize$x$}}}\\_1} \\chi(x\\_1))^2\n \\right]. \\label{Exp:nS} \\end{aligned}$$ \nIn this subsection, we show that when the WT identity for *χ*, given in Eq. ([Exp:WTpr]), is fulfilled, the effective action for *ζ* preserves the scaling symmetry.\nTo show this, we further rewrite the WT identity ([Exp:WTpr]). Operating \n$$\\int {{\\rm d}}^d {\\hbox{\\boldmath{$x$}}} \\int {{\\rm d}}^{d+1} y\\, \\delta g\\_\\alpha(x) \\delta({\\hbox{\\boldmath{$x$}}} - {\\hbox{\\boldmath{$y$}}}) \\delta(t\\_x-t\\_y)$$ \n and performing the integration by parts, we obtain\n \n$$\\begin{aligned}\n & \\int {{\\rm d}}^d {\\hbox{\\boldmath{$x$}}} \\langle \\chi\\_\\alpha^n (x) \\rangle\n \\partial\\_{{\\hbox{\\boldmath{\\scriptsize$x$}}}} \\{ {\\hbox{\\boldmath{$x$}}} \\delta g\\_\\alpha (x) \\} + \\int {{\\rm d}}^d {\\hbox{\\boldmath{$x$}}}\n \\int {{\\rm d}}^{d+1}y\\, \\delta g\\_\\alpha(x) \\left\\langle \\chi\\_\\alpha^n(x) \\frac{\\delta\n i S\\_\\chi}{\\delta \\zeta\\_+(y)} \\bigg|\\_{\\delta g\\_+=0} \\right\\rangle \\cr\n & \\qquad \\qquad - \\int {{\\rm d}}^d {\\hbox{\\boldmath{$x$}}} \\int {{\\rm d}}^{d+1}y\\, \\delta g\\_\\alpha(x) \\left\\langle\n \\chi\\_\\alpha^n(x) \\frac{\\delta i S\\_\\chi}{\\delta \\zeta\\_-(y)} \\bigg|\\_{\\delta g\\_-=0}\n \\right\\rangle = 0\\,. \\label{Eq:WTnd}\\end{aligned}$$ \nHere, after rewriting $\\delta({\\hbox{\\boldmath{$x$}}}- {\\hbox{\\boldmath{$y$}}})({\\hbox{\\boldmath{$x$}}} \\cdot \\partial\\_{{\\hbox{\\boldmath{\\scriptsize$x$}}}} + {\\hbox{\\boldmath{$y$}}} \\cdot\n\\partial\\_{{\\hbox{\\boldmath{\\scriptsize$y$}}}})$ as $\\delta({\\hbox{\\boldmath{$x$}}}- {\\hbox{\\boldmath{$y$}}})({\\hbox{\\boldmath{$y$}}} \\cdot \\partial\\_{{\\hbox{\\boldmath{\\scriptsize$x$}}}} + {\\hbox{\\boldmath{$x$}}} \\cdot\n\\partial\\_{{\\hbox{\\boldmath{\\scriptsize$y$}}}})$, we performed the integration by parts and then we used \n$$({\\hbox{\\boldmath{$y$}}} \\cdot \\partial\\_{{\\hbox{\\boldmath{\\scriptsize$x$}}}} + {\\hbox{\\boldmath{$x$}}} \\cdot \\partial\\_{{\\hbox{\\boldmath{\\scriptsize$y$}}}})\n \\delta({\\hbox{\\boldmath{$x$}}}- {\\hbox{\\boldmath{$y$}}}) = ( \\partial\\_{{\\hbox{\\boldmath{\\scriptsize$x$}}}}\\, {\\hbox{\\boldmath{$x$}}} - {\\hbox{\\boldmath{$x$}}} \\cdot \\partial\\_{{\\hbox{\\boldmath{\\scriptsize$x$}}}})\n \\delta({\\hbox{\\boldmath{$x$}}}- {\\hbox{\\boldmath{$y$}}}) = d \\delta({\\hbox{\\boldmath{$x$}}}- {\\hbox{\\boldmath{$y$}}})\\,.$$ \n Similarly, operating \n$$\\int {{\\rm d}}^d {\\hbox{\\boldmath{$x$}}} \\int {{\\rm d}}^{d+1} y \\delta g\\_\\alpha(x) \\delta({\\hbox{\\boldmath{$x$}}} - {\\hbox{\\boldmath{$y$}}})\n\\delta(t\\_x-t\\_y) \\partial\\_{x^\\mu} \\partial\\_{y^\\nu} \\,,$$ \n on Eq. ([Def:tWn]) with *n* = 2 and *p* = 1, where *μ*, *ν* = 0, *i*, we obtain\n \n$$\\begin{aligned}\n & \\int {{\\rm d}}^d {\\hbox{\\boldmath{$x$}}} \\langle \\dot{\\chi}^2\\_\\alpha (x) \\rangle\n \\partial\\_{{\\hbox{\\boldmath{\\scriptsize$x$}}}} \\{ {\\hbox{\\boldmath{$x$}}} \\delta g\\_\\alpha (x) \\} + \\int {{\\rm d}}^d {\\hbox{\\boldmath{$x$}}}\n \\int {{\\rm d}}^{d+1}y\\, \\delta g\\_\\alpha(x) \\left\\langle \\dot{\\chi}\\_\\alpha^2(x) \\frac{\\delta\n i S\\_\\chi}{\\delta \\zeta\\_+(y)} \\bigg|\\_{\\delta g\\_+=0} \\right\\rangle \\cr\n & \\qquad \\qquad - \\int {{\\rm d}}^d {\\hbox{\\boldmath{$x$}}} \\int {{\\rm d}}^{d+1}y\\, \\delta g\\_\\alpha(x) \\left\\langle\n \\dot{\\chi}\\_\\alpha^2(x) \\frac{\\delta i S\\_\\chi}{\\delta \\zeta\\_-(y)} \\bigg|\\_{\\delta g\\_-=0}\n \\right\\rangle = 0\\,, \\label{Eq:WTtd} \\\\\n & \\int {{\\rm d}}^d {\\hbox{\\boldmath{$x$}}}\n \\int {{\\rm d}}^{d+1}y\\, \\delta g\\_\\alpha(x) \\left\\langle \\dot{\\chi}\\_\\alpha(x)\n \\partial\\_i \\chi\\_{\\alpha}(x) \\frac{\\delta\n i S\\_\\chi}{\\delta \\zeta\\_+(y)} \\bigg|\\_{\\delta g\\_+=0} \\right\\rangle \\cr\n & \\qquad \\qquad - \\int {{\\rm d}}^d {\\hbox{\\boldmath{$x$}}} \\int {{\\rm d}}^{d+1}y\\, \\delta g\\_\\alpha(x) \\left\\langle\n \\dot{\\chi}\\_\\alpha(x) \\partial\\_i \\chi\\_\\alpha(x) \\frac{\\delta i\n S\\_\\chi}{\\delta \\zeta\\_-(y)} \\bigg|\\_{\\delta g\\_-=0}\n \\right\\rangle = 0\\,,\\end{aligned}$$ \nand\n \n$$\\begin{aligned}\n & \\int {{\\rm d}}^d {\\hbox{\\boldmath{$x$}}} \\langle \\partial\\_i \\chi\\_\\alpha (x) \\partial^i \\chi\\_\\alpha (x) \\rangle\n \\{ {\\hbox{\\boldmath{$x$}}} \\cdot \\partial\\_{{\\hbox{\\boldmath{\\scriptsize$x$}}}} + (d-2) \\} \\delta g\\_\\alpha (x) \\cr\n & \\qquad \\qquad + \\int {{\\rm d}}^d {\\hbox{\\boldmath{$x$}}}\n \\int {{\\rm d}}^{d+1}y\\, \\delta g\\_\\alpha(x) \\left\\langle \\partial\\_i \\chi\\_\\alpha (x) \\partial^i \\chi\\_\\alpha (x) \\frac{\\delta\n i S\\_\\chi}{\\delta \\zeta\\_+(y)} \\bigg|\\_{\\delta g\\_+=0} \\right\\rangle \\cr\n & \\qquad \\qquad - \\int {{\\rm d}}^d {\\hbox{\\boldmath{$x$}}} \\int {{\\rm d}}^{d+1}y\\, \\delta g\\_\\alpha(x) \\left\\langle\n \\partial\\_i \\chi\\_\\alpha (x) \\partial^i \\chi\\_\\alpha (x) \\frac{\\delta\n i S\\_\\chi}{\\delta \\zeta\\_-(y)} \\bigg|\\_{\\delta g\\_-=0}\n \\right\\rangle = 0\\,, \\label{Eq:WTsd}\\end{aligned}$$ \nwhere we used $\\langle \\dot{\\chi}(x) \\partial\\_i \\chi(x) \\rangle=0$. Recalling the expressions of *W**δ**g**α*(1) and *W**δ**g**α*1*δ**g**α*2(2), given in Sec. [SSec:NaiveEA] and adding Eqs. ([Eq:WTnd]), ([Eq:WTtd]), and ([Eq:WTsd]) in such a way that their first terms give *W**δ**g**α*(1), we obtain\n \n$$\\begin{aligned}\n & \\int {{\\rm d}}^{d+1} x\\, \\{{\\hbox{\\boldmath{$x$}}} \\cdot \\partial\\_{{\\hbox{\\boldmath{\\scriptsize$x$}}}} \\delta g\\_{\\pm} (x) \\}\n W^{(1)}\\_{\\delta g\\_{\\pm}} (x) \\cr\n & \\qquad + \\int {{\\rm d}}^{d+1} x \\int {{\\rm d}}^{d+1} y\\, \\delta g\\_{\\pm}(x)\n \\left\\{ W^{(2)}\\_{\\delta g\\_{\\pm} \\zeta\\_{\\pm}}(x,\\, y) + W^{(2)}\\_{\\delta\n g\\_{\\pm} \\zeta\\_{\\mp}}(x,\\, y) \\right\\} =0 \\,. \\label{Eq:WTEA}\\end{aligned}$$ \nAs is clear from the derivation, Eq. ([Eq:WTEA]) also holds, even if we replace *δ**g*±(*x*) included in each term with an arbitrary function. Therefore, replacing *δ**g*±(*x*) with a constant nonzero number, we obtain\n \n$$\\begin{aligned}\n & \\int {{\\rm d}}^{d+1} x \\int {{\\rm d}}^{d+1} y \\left\\{ W^{(2)}\\_{\\delta g\\_{\\pm} \\zeta\\_{\\pm}}(x,\\, y) +\n W^{(2)}\\_{\\delta g\\_{\\pm} \\zeta\\_{\\mp}}(x,\\, y) \\right\\} = 0 \\,. \\label{Eq:WTEAct}\\end{aligned}$$ \nBy adding the left hand side of Eq. ([Eq:WTEA]) multiplied by a constant parameter − *s* and Eq. ([Eq:WTEAct]) with *δ**g*± = *ζ*± multiplied by − *s*2/2, the linear and the quadratic terms in the effective action can be given by\n \n$$\\begin{aligned}\n & i {S'\\_{{\\rm eff}(1)}}[\\delta g\\_+,\\, \\delta g\\_-] + i {S'\\_{{\\rm eff}(2)}}[\\delta g\\_+,\\,\n \\delta g\\_-]\n \\cr\n & = \\sum\\_{\\alpha = \\pm} \\int\n {{\\rm d}}^{d+1} x\\, \\delta g\\_{\\alpha,\\, s}(x) W^{(1)}\\_{\\delta g\\_\\alpha} (x) \\cr\n & \\qquad + \\frac{1}{2!} \\!\\sum\\_{\\alpha\\_1, \\alpha\\_2 = \\pm}\\! \\int {{\\rm d}}^{d+1} x\\_1 \\!\\int\\! {{\\rm d}}^{d+1} x\\_2 \\, \\delta g\\_{\\alpha\\_1,\\,\n s}(x\\_1) \\delta \\tilde{g}\\_{\\alpha\\_2,\\,s}(x\\_2) W^{(2)}\\_{\\delta g\\_{\\alpha\\_1}\n \\delta \\tilde{g}\\_{\\alpha\\_2}}(x\\_1,\\,x\\_2) \\cr\n & \\qquad + {\\cal O}(\\delta g^3) \\,, \\label{Expn:Seffdd}\\end{aligned}$$ \nwhere *δ**g**s* are related to *δ**g* as given in Eqs. ([Exp:transzeta]), ([Exp:transN]), and ([Exp:transNi]). Here, each *δ**g**i*, *α* (*i* = 1, 2) sums over *δ**N**α*, *s*, *N**i*, *α*, *s*, and *ζ**α*, *s*.\nWe can drop the term with one shift vector, because *W**N**i*, *α*(1), which is proportional to $\\langle \\dot{\\chi}\n\\partial\\_i \\chi \\rangle$, vanishes. In deriving Eq. ([Expn:Seffdd]), we used\n \n$$\\begin{aligned}\n & W^{(2)}\\_{\\delta g\\_{\\alpha\\_1} \\delta \\tilde{g}\\_{\\alpha\\_2}}(x\\_1,\\, x\\_2) =\n W^{(2)}\\_{\\delta \\tilde{g}\\_{\\alpha\\_2} \\delta g\\_{\\alpha\\_1}}(x\\_2,\\,\n x\\_1)\\,,\\end{aligned}$$ \nand *g* *ζ̄*+ = *g* *ζ̄*−, which holds since $\\zeta\\_+(t\\_f,\\,{\\hbox{\\boldmath{$x$}}})=\\zeta\\_-(t\\_f,\\, {\\hbox{\\boldmath{$x$}}})$. The first term in Eq. ([Eq:WTEA]) changes the argument of the metric perturbations in the linear term of Eq. ([Expn:Seffdd]). We also changed the arguments of the quadratic terms, since the modification appears only in higher orders of *δ**g*.\nEquation ([Expn:Seffdd]) shows that with the use of the WT identity, *δ**g**α*(*x*) in ${S\\_{\\rm eff}}'$ can be replaced with *δ**g**α*, *s*(*x*). Since the rest of the effective action, *S**a**d*, is simply the classical action for the single field model, it also should be invariant under this replacement. Therefore, when the WT identity ([WT]) holds, the total effective action $S\\_{\\rm eff}$ preserves the invariance under the change of *δ**g**α* to *δ**g**α*, *s*.\nThe effective action ([Expn:Seffdd]) includes the lapse function and the shift vector. By solving the Hamiltonian and momentum constraint equations, we can express *δ**N**s* and *N**i*, *s* in terms of *ζ**s*. Using these expressions, we can eliminate *δ**N**s* and *N**i*, *s* in the effective action as in the single field model . Since the constraint equations for *δ**g**s* are given by replacing *δ**g* with *δ**g**s* in the constraint equations for *δ**g*, the effective action for *ζ**s* obtained after eliminating *δ**N**s* and *N**i*, *s* should be given simply by replacing *ζ* with *ζ**s* in the effective action expressed only in terms of *ζ*.\nConservation of *ζ*\n-------------------\n### Tadpole contribution\nBefore we discuss the conservation, we show that the linear terms in the effective action ${S\\_{\\rm eff}}$, which is the tadpole terms, vanish all together. Taking the variation of the effective action with respect to *N* and *N**i*, we obtain the constraint equations. The Hamiltonian constraint for the FRW background gives\n \n$$\\begin{aligned}\n & d(d-1) \\dot{\\rho}^2 = \\dot{\\phi}^2 + \\langle \\dot{\\chi}^2 \\rangle +\n e^{-2\\rho} \\langle (\\partial\\_{{\\hbox{\\boldmath{\\scriptsize$x$}}}} \\chi)^2 \\rangle + 2 \\langle\n V(\\phi,\\, \\chi) \\rangle \\,, \\end{aligned}$$ \nand at the liner order\n \n$$\\begin{aligned}\n & (d-1) e^{-2 \\rho} \\dot{\\rho} \\partial^i N\\_i + \\delta N (2\n \\langle V \\rangle + e^{-2\\rho} \\langle (\\partial \\chi)^2 \\rangle )-\n d(d-1) \\dot{\\rho} \\dot{\\zeta} - \\zeta e^{-2 \\rho} \\langle (\\partial\n \\chi)^2 \\rangle = 0 \\label{HconstL}\\,,\\end{aligned}$$ \nwhere ∂*i* ≡ *δ**i**j*∂*j*. Here, we neglected the sub-leading contribution at large scales. The scalar part of the momentum constraint gives\n \n$$\\begin{aligned}\n & \\partial\\_i (\\dot{\\rho} \\delta N - \\dot{\\zeta})-\n \\frac{e^{-2\\rho}}{d-1} N\\_i \\langle (\\partial\\_i\\chi)^2 \\rangle =0\\,, \\label{MconstL}\\end{aligned}$$ \nwhere we used ⟨∂*i**χ*∂*j**χ*⟩ ∝ *δ**i**j*. The momentum constraint equation can be solved as\n \n$$\\begin{aligned}\n & \\delta N = \\frac{\\dot{\\zeta}}{\\dot{\\rho}} +\n \\frac{e^{-2\\rho}}{(d-1)\\dot{\\rho}} \\sum\\_{i=1}^3 \\langle (\\partial\\_i\\chi)^2 \\rangle\n \\partial^{-2} \\partial^i N\\_i \\,. \\label{Exp:deltaNwC}\\end{aligned}$$ \nWe can add a homogeneous solution of the Laplace equation on the right hand side.\nThe action *S**a**d* which is accurate at the linear order of *δ**g* is given by\n \n$$\\begin{aligned}\n S\\_{ad} &\\simeq \\frac{1}{2} \\int {{\\rm d}}^{d+1} x N e^{d(\\rho+\\zeta)}\n \\Bigl[- 2 W(\\phi) + \\frac{1}{N^2} \\bigl\\{ - d(d-1) \\dot{\\rho}^2 +\n \\dot{\\phi}^2 \\bigr\\} \\cr\n & \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad -2(d-1) \\dot{\\rho} \\delta N \\bigl\\{ d \\dot{\\zeta}\n - e^{-2(\\rho + \\zeta)} \\partial^i N\\_i\\bigr\\} \\Bigr] \\,,\\end{aligned}$$ \nwhere, for our purpose, we partially kept the higher order terms in the exponential form. The *n*-th order effective action ${S'\\_{{\\rm eff}(n)}}$ includes the local terms given by \n$$\\frac{1}{n!} \\int {{\\rm d}}^{d+1} x\\, \\delta g\\_{1,\\, \\alpha}(x) \\cdots \\delta g\\_{n,\\, \\alpha}(x) \\left\\langle\n \\frac{\\delta^n i S\\_\\chi[\\delta g\\_\\alpha,\\, \\chi\\_\\alpha]}{\\delta\n g\\_{1,\\alpha}(x) \\cdots \\delta g\\_{n,\\alpha}} \\right\\rangle$$ \n with *α* = ± . Adding up these local terms for all *n*, we obtain\n \n$$\\begin{aligned}\n & S\\_{{\\rm eff}, {\\rm local}}'[\\delta g\\_+,\\, \\delta g\\_-] =\n \\langle S\\_{\\chi}[\\delta g\\_+,\\, \\chi\\_+] \\rangle - \\langle\n S\\_{\\chi}[\\delta g\\_-,\\, \\chi\\_-] \\rangle \\,, \\end{aligned}$$ \nwhere the terms which do not depend on *δ**g* are canceled between the two terms on the right hand side. Adding these local terms to *S**a**d* and using the Hamiltonian constraint, we obtain a concise expression as\n \n$$\\begin{aligned}\n & S\\_{ad}[\\zeta\\_\\alpha] + \\langle S\\_{\\chi}[\\zeta\\_\\alpha,\\,\n \\chi\\_\\alpha] \\rangle \\cr\n & \\simeq - \\int {{\\rm d}}^{d+1} x\\, N e^{d(\\rho + \\zeta\\_\\alpha)}\n \\left[ 2 \\langle V(\\phi,\\, \\chi) \\rangle + e^{-2(\\rho + \\zeta\\_\\alpha)}\n \\langle (\\partial \\chi)^2 \\rangle \\right] \n. \\label{Exp:SzetaSeff}\\end{aligned}$$ \nAs in single field cases, solving the Hamiltonian and momentum constraint equations, we can express the lapse function and the shift vector in terms of *ζ*. Inserting Eq. ([Exp:deltaNwC]) into the Hamiltonian constraint ([HconstL]), we find that the degree of freedom to choose the solution of the Laplace equation changes the relation between *N**i* and *ζ*. Since the domain of integration extends to the spatial infinity, to regularize the spatial integral, the perturbed variables should approach to 0 in the spatial infinity. Therefore, we determine the constant degree of freedom in Eq. ([Exp:deltaNwC]), requesting the boundary condition:\n \n$$\\begin{aligned}\n & \\int {{\\rm d}}^d {\\hbox{\\boldmath{$x$}}} \\, \\partial^i N\\_i =0 \\,. \\label{BCNi}\\end{aligned}$$ \nWith this choice, the Hamiltonian constraint reads\n \n$$\\begin{aligned}\n & \\int {{\\rm d}}^d {\\hbox{\\boldmath{$x$}}}\\, \\delta N \\left\\{ 2 \\langle V \\rangle + e^{-2\\rho}\n \\langle (\\partial \\chi)^2 \\rangle \\right\\} = \\int {{\\rm d}}^d {\\hbox{\\boldmath{$x$}}} \\left\\{\n d(d-1) \\dot{\\rho} \\dot{\\zeta} + \\zeta e^{-2 \\rho} \\langle (\\partial\n \\chi)^2 \\rangle \\right\\}\\,. \\end{aligned}$$ \nUsing this relation, we can rewrite the action which is valid up to the linear order as\n \n$$\\begin{aligned}\n & S\\_{ad}[\\zeta\\_\\alpha] + \\langle S\\_{\\chi}[\\zeta\\_\\alpha,\\,\n \\chi\\_\\alpha] \\rangle \\cr\n & \\simeq - (d-1) \\int {{\\rm d}}^{d+1} x \\partial\\_t \\left( \\dot{\\rho}\n e^{d(\\rho+ \\zeta\\_\\alpha)} \\right) + \\int {{\\rm d}}^{d+1} x e^{d(\\rho+ \\zeta\\_\\alpha)} \\! \\left\\{ (d-1) \\ddot{\\rho}\n + \\dot{\\phi}^2 + \\langle \\dot{\\chi}^2 \\rangle + \\frac{e^{-2\\rho}}{d}\n \\langle (\\partial \\chi)^2 \\rangle \\right\\} \\cr\n & \\qquad - \\int {{\\rm d}}^{d+1} x e^{(d-2) \\rho} \\left\\{ e^{(d-2)\\zeta\\_\\alpha} - \\left(1\n - \\frac{1}{d}\\right) e^{d \\zeta\\_\\alpha} + \\zeta\\_\\alpha \\right\\} \\langle\n (\\partial \\chi)^2 \\rangle \\,. \\label{Exp:SzetaSeff2}\\end{aligned}$$ \nThe first term vanishes as a total derivative. The second term is proportional to\n \n$$\\begin{aligned}\n & (d-1) \\ddot{\\rho} = - \\dot{\\phi}^2 - \\langle \\dot{\\chi}^2 \\rangle - \\frac{1}{d}\n e^{-2\\rho} \\langle (\\partial \\chi)^2 \\rangle\\,,\\end{aligned}$$ \nwhich can be verified by using the time derivative of the Friedman equation and the field equations for *ϕ* and *χ*, given by\n \n$$\\begin{aligned}\n & \\ddot{\\phi} + d \\dot{\\rho} \\dot{\\phi}+ V\\_{ph}'(\\phi) + \\langle \\chi^2 \\rangle M\n M\\_\\phi =0\\,, \\label{Eq:H} \\\\\n & \\ddot{\\chi}\\_k + d \\dot{\\rho} \\dot{\\chi}\\_k + \\left( M^2 + \\frac{\\lambda}{2}\n \\langle \\chi^2 \\rangle + k^2 e^{- 2 \\rho} \\right) \\chi\\_k =0 \\,. \\label{Eq:dH} \\end{aligned}$$ \nThe tadpole terms contained in the last line cancel with each other and the term which does not depend on *ζ* is canceled between the action for + and the one for − . In this way, using the background equations and also choosing the boundary condition for *N**i* as in Eq. ([BCNi]), we can show that the tadpole contributions all disappear.\nAs we discussed in the previous subsection, the effective action $S\\_{\\rm eff}$ stays invariant under the replacement of *ζ**α*(*x*) with *ζ**α*, *s*(*x*). Therefore, the tadpole contribution for *ζ**s* should be given simply by replacing *ζ*(*x*) with *ζ**s*(*x*) in Eq. ([Exp:SzetaSeff]). When the background equations are satisfied and *N**i*, *s* is chosen to vanish at the spatial infinity (when *N**i* satisfies the boundary condition ([BCNi]), *N**i*, *s* also satisfies it), the terms in the second line of Eq. ([Expn:Seffdd]), which are linear in the metric perturbations, all vanish.\n### Existence of constant solution\nRemoving the tadpole contribution which vanishes with the use of the background equations, we only consider the quadratic terms about *ζ*. At the linear level, *ζ**α*, *s* simply gets the constant shift as\n \n$$\\begin{aligned}\n & \\zeta\\_{\\alpha,\\, s} (x) \\simeq \\zeta\\_\\alpha(x) -s\\,. \\end{aligned}$$ \nTherefore, the symmetry under the change of *ζ**α* into *ζ**α*, *s* immediately implies the existence of the constant solution also in the presence of the loop corrections of the heavy field.\nIn single filed cases, it is well known that only the constant solution survives while the other independent solution simply decays in the late time limit, as far as the background evolution is on an attractor (see, e.g., Ref. ). This fact explains why the curvature perturbation becomes time independent at super Hubble scales. When we add a quantum correction from a heavy field, in principle, the “decaying” mode can turn into a growing mode. Such a drastic change of the behaviour of perturbation can occur, in case the trajectory sizably deviates from the attractor solution, for instance, owing to an effect of an additional field. In the present context, the classical background evolution is determined only by the inflaton and we assume that the quantum effects of the heavy field always remain to be perturbative. In such cases, the effect of the heavy field does not drive the “decaying” mode to grow in time. Then, the presence of the constant mode implies the conservation of the curvature perturbation in time as well as in the presence of the loop corrections of the heavy field.\nRenormalization and scaling symmetry\n====================================\nAs is common in a non-linear quantum field theory, the effective action for *ζ* potentially diverges due to UV corrections. In our case, the bare coefficients of the effective action *W**δ**g**α*1⋯*δ**g**α**n*(*n*), which are expressed in terms of the correlators for *χ*, can diverge. The UV divergence should be renormalized by introducing counter terms. Depending on a way to introduce the counter terms, the scaling symmetry might be broken. If it were the case, the WT identity would not hold any more and the renormalized effective action does not preserve the scaling symmetry.\nWhen the counter terms are introduced in such a way that the scaling symmetry is preserved, the WT identity holds also after the UV renormalization. Then, inserting the WT identity into the effective action, which can be renormalized following the standard procedure since the theory (before the gauge fixing) is a local theory, and repeating the same argument as we did for the unrenormalized effective action, we can replace *ζ**α*(*x*) into *ζ**α*, *s*(*x*) in the renormalized effective action.\nSince only the heavy field is quantized in computing the effective action, the curvature perturbation *ζ* should be dealt with as a classical external field. We may set the arbitrary constant parameter *s* to a *c*-number variable *g* *ζ̄* in order to express the effective action in terms of the fluctuation in the local region. Then, the effective action includes the non-local contribution *g* *ζ̄*. Nonetheless, the renormalization should proceed in the standard way, because the inserted non-local contribution, which is schematically in the following form: \n$$0= ({\\rm WT~identity, which~identically~vanishes~and~is~local}) \\times\n {{^g\\!\\bar{\\zeta}}}^n \\qquad (n=1,\\,2)$$ \n is fictitious and does not introduce any non-local interactions.\nThis aspect may be instructive to speculate on the UV renormalization of an IR regular quantity. Preserving the scaling symmetry is crucial to cancel out the potentially IR divergent contribution. In Refs. , a quantity which preserves the scaling symmetry was proposed and it contains non-local contributions. Because of that, in Refs. , it was suggested that the quantity which preserves the scaling symmetry may not be able to be renormalized in the standard way by introducing local counter terms.\nIn this paper, we presented a handy example where the UV renormalization of the heavy field can be performed simply by introducing local counter terms as well as for a quantity which looks to include a non-local contribution. Here, we only considered the UV renormalization of the heavy field *χ*. It will be important to see if the UV renormalization of the curvature perturbation also can proceed by introducing local counter terms or not. We leave this issue for a future study.\nConcluding remarks\n==================\nString theory predicts the presence of a bunch of massive excitations after reduction to four dimensional spacetime, which may encode, for instance, the information on the structure of the compactification of the extra dimensions. It is important to explore a possible imprint of such massive modes on the curvature perturbation. In this paper, we considered an influence of a heavy scalar field on the curvature perturbation *ζ* at the super Hubble scales. When the mass of the heavy field *χ* is of ${\\cal O}(H)$, it can give non-local radiative corrections to the effective action of *ζ*, which may provide a distinctive imprint of the heavy field. We showed that the time evolution of *ζ* at the super Hubble scales is not affected by the loop corrections of the heavy field as far as the scaling symmetry, which is entailed in a covariant theory at the classical level, is preserved. The implies that the constant adiabatic mode exists as well as in the presence of the loop corrections of the heavy field.\nFor simplicity, we considered one heavy field with the standard canonical kinetic term. However, our argument can be extended in a straightforward manner to a more general model which contains more than one heavy fields with a non-canonical kinetic term.\nOur result indicates that in order to leave an imprint of massive fields well after the Hubble crossing, we need to break either of the following conditions [2](#fn2):\n* The massive fields do not alter the background evolution at the classical level.\n* The quantum system preserves the scaling symmetry, which yields the Ward-Takahashi identity.\n* The radiative corrections of the massive fields on the curvature perturbation *ζ* are perturbatively suppressed.\nIf the last condition does not hold, we need to perform a non-perturbative analysis to compute the radiative corrections of the massive fields.\nIn this paper, using the WT identity ([WT]) for the scaling symmetry at ${\\cal O}(s)$, we showed that the metric perturbation *δ**g*(*x*) in the effective action can be replaced with *δ**g**s*(*x*), given in Eqs. ([Exp:transzeta]), ([Exp:transN]), and ([Exp:transNi]), keeping up to the quadratic terms. This argument can be extended to higher orders in *δ**g*. Using the WT identity ([WT]), we can derive the WT identity which relates *W*(*n*) to *W*(*n*ʹ)s with *n*ʹ < *n*. Adding the WT identity for *W*(*m*) with *m* ≤ *n* (multiplied by some particular constant factors) to the effective action $S'\\_{\\rm eff}$, we can replace all *δ**g*(*x*) with *δ**g**s*(*x*) in $S'\\_{\\rm eff}$ up to the *n*-th order of perturbation. After removing the lapse function and the shift vector, we find that the effective action for the curvature perturbation is invariant under the replacement of *ζ*(*x*) with\n \n$$\\begin{aligned}\n & \\zeta\\_s(x)= \\zeta(x) - s - s {\\hbox{\\boldmath{$x$}}} \\cdot \\partial\\_{{\\hbox{\\boldmath{\\scriptsize$x$}}}} \\zeta(x) +\n\\frac{s^2}{2} ({\\hbox{\\boldmath{$x$}}} \\cdot \\partial\\_{{\\hbox{\\boldmath{\\scriptsize$x$}}}})^2 \\zeta(x) + {\\cal O}(s^3)\\,. \\label{Sol:zetas}\\end{aligned}$$ \nThis implies that the curvature perturbation includes a solution which is given by the *s*-dependent terms in Eq. ([Sol:zetas]), whose first term is the constant adiabatic mode. In order to keep the terms which explicitly depend on ${\\hbox{\\boldmath{$x$}}}$ perturbatively small, we need to confine the perturbation within a finite spatial region on each time slicing. For that, we will need to use other residual *gauge* degrees of freedom, which are addressed in Refs. .\nIn this paper, we studied a spin 0 scalar field as the heavy field. It will be interesting to extend the discussion to include a field with a more general spin . Our discussion does not rely on the explicit form of the interaction vertices nor the propagator. Therefore, we expect that this extension will be feasible. We leave this study for a future project .\nY. U. would like to thank N. Arkani-Hamed, M. Mirbabayi and M. Simonovi$\\acute{\\rm c}$ for interesting discussions about their works, which are relevant to this work. We are grateful to Y. Misonoh and S. Saga for their participations to the early state of this work. This work is supported by Grant-in-Aid for Scientific Research (B) No. 26287044. T. 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(See also Ref. .)[↩](#fnref2)\n"},"data_id":{"kind":"string","value":"arxiv_0000360"}}},{"rowIdx":97,"cells":{"content":{"kind":"string","value":"Unconventional Fusion and Braiding of Topological Defects in a Lattice Model\n============================================================================\nWe examine non-abelian topological defects in an abelian lattice model in two dimensions. We first construct an exact solvable lattice model that exhibits coexisting and intertwined topological and classical order. The anyon types of quasiparticle excitations are permuted by lattice symmetry operations like translations, rotations, and reflections. The global anyon permutation symmetry has a group structure of *S*3, the permutation group of three elements. Topological crystalline defects – dislocations and disclinations – change the anyon type of an orbiting quasiparticle. They exhibits multichannel order dependent fusion rules and projective braiding operations. Their braiding and exchange statistics breaks modular invariance and violates conventional spin-statistics theorem. We develop a framework to characterize these unconventional properties that originate from the semiclassical nature of defects.\nIntroduction\n============\nThe search for Majorana fermions has attracted tremendous attention in recent years due to their potential application in topological quantum computation . With the discovery of topological insulators , the quest has shifted from *p*-wave superconductors and quantum Hall states to superconductor-ferromagnet (SC-FM) heterostructures with quantum spin Hall insulators and strong spin-orbit coupled semiconductors . The more exotic fractional Majorana fermions that carry richer fusion and braiding characteristics are predicted at the SC-FM edge of fractional topological insulators and helical 1D Luttinger liquids .\nFractional Majorana fermions can be conceptually studied by *twist defects* in exact solvable lattice models. These include Ising-type dislocations in the Kitaev toric code , fractional Majorana-type dislocations in the Wen plaquette Z*k*-rotor model , and colored Majorana defects in a string net model . Twist defects even appear at dislocation line defects in 3D topological phases . Similar non-abelian defects can be constructed as dislocations in abelian *topological nematic states* such as a multiple Chern band with symmetry , described by *genons* in effective field theory and classified by Wilson structures of non-chiral gapped edges .\nThe underlying topological state of all systems mentioned above carries an ungauged symmetry, e.g. charge-flux duality in the toric code and Wen plaquette model , color permutation symmetry in the color code , and bilayer symmetry in a topological nematic state . Symmetries also appear in many strongly correlated systems such as electronic liquid-crystal phases of a doped Mott insulator and spin liquid . They intertwine with topology and offer a finer classification of topological phases . Even if the symmetry is not broken spontaneously by a Landau order parameter, it may still be *weakly broken* in the Kitaev sense by anyon labeling. Extrinsic twist defects further break the symmetry locally by winding the anyon labels. In this article, we demonstrate, using twist defects in an exact solvable lattice model, some fundamental distinctions in fusion and braiding that separate semiclassical symmetry defects from quantum deconfined anyons in true topological phases such as the Kitaev honeycomb Ising phase or quasi-topological phases such as the physical Pfaffian quantum Hall state .\nTopological defects involve windings of certain non-dynamical extensive order parameters such as pairing phase, Dirac mass, spin polarization, etc. The semiclassical order parameter forbids quantum superposition of different defect configurations. Although defect excitations in SC-FM heterostructures have exponentially localized wavefunctions and twist defects in integrable models only violate symmetry locally at a point, they should be treated as quasi-extensive objects because of the bulk order parameter associated with each of them. This extensiveness provides the means to circumvent locality restrictions and gives rise to non-abelian statistics in (3 + 1)-dimensions .\nIn this article, we address three major consequences of the quasi-extensive nature of twist defects in an exact solvable topologically ordered abelian system with symmetry.\n1. **Non-commutative fusion** Contrary to a non-abelian discrete gauge theory where fluxons are labeled by conjugacy classes, twist defects are instead labeled by symmetry group elements and the fusion of defects depends on their order.\n2. **Modified spin statistics** Topological spin for a defect *λ* can only be robustly defined through a 2*π* × ord(*λ*) rotation, where ord(*λ*) is the order of the defect, so that the initial and rotated systems are classically indistinguishable. The exchange phase is then identical to this new definition of spin.\n3. **Modified modular invariance** Unlike modular functors in conformal field theory , fractional quantum Hall states or topological quantum field theories , defect exchange and braiding does not obey the full modular group *S**L*(2; Z), but is restricted to a congruent subgroup.\nOutline and Summary\n-------------------\nWe begin in section [sec:honeycombmodel] by presenting an abelian Z*k* rotor lattice model on a honeycomb lattice or in general bipartite trivalent planar graph. Its Z2 version has been studied by Bombin and Martin-Delgado and is called the color code. The model possesses a hidden non-abelian symmetry $S\\_3=\\mathbb{Z}\\_2\\ltimes\\mathbb{Z}\\_3$, the permutation of group on three elements. There is a *k*4 ground state degeneracy on a torus. The *k*4 abelian anyon excitations can be labelled and bipartitioned into ${\\bf a}=({\\bf a}\\_\\bullet,{\\bf a}\\_\\circ)$, where each spinless component ${\\bf a}\\_{\\bullet/\\circ}$ lives on a two dimensional triangular Z*k*-lattice (see figure [fig:abeliananyonlattice]). Threefold *cyclic color permutations* Λ3, Λ3− 1 in *S*3 act as rotations on the anyon lattice, leaving the bipartite • , ∘ -label untouched. Twofold *color sublattice transpositions* Λ*Y*, Λ*R*, Λ*B* in *S*3 interchange • ↔ ∘ and act as three mirror planes on the anyon lattice. This model has fusion and braiding properties identical to two copies of the Z*k* toric code. However the symmetry group is non-trivially extended from charge-flux Z2 duality to *S*3 by a threefold cyclic color permutation.\nWe construct non-abelian twist defects in section [sec:twistdefect]. A defect is classified by a group element Λ in *S*3 that characterizes the change of the label ${\\bf a}\\to\\Lambda\\cdot{\\bf a}$ of an encirling abelian anyon (see figure [fig:anyontwist]). There are two threefold defects [1/3] and its anti-particle $[\\overline{1/3}]$, and three twofold defects [1/2]*χ* labeled by color *χ* = *Y*, *R*, *B*. Primitive threefold defects are constructed in the lattice level by ± 120∘ disclinations at tetravalent/bivalent vertex, and twofold defects are associated with ± 60∘ disclinations at pentagons and heptagons (see figure [fig:twistdefects]). Quantum dimensions can be deduced either by counting plaquette and vertex degrees of freedom (section [sec:latticedefecthamiltonian]) or evaluated by ground state degeneracy corresponding to the non-local Wilson loop algebra (section [sec:nonlocalWilsonalgebra]). They are given by\n \n$$\\begin{aligned}\nd\\_{[1/3]}=d\\_{[\\overline{1/3}]}=\\left\\{\\begin{array}{\\*{20}c}k^2,&\\mbox{if $3\\centernot\\mid k$}\\\\k^2/3,&\\mbox{if $3\\mid k$}\\end{array}\\right.,\\quad d\\_{[1/2]}=k\\end{aligned}$$ \nNote that the dimension for the twofold defect matches that of two copies of Z*k*-fractional Majorana fermion. A defect contains a phase parameter that determines the value of a local Wilson observable (see figure [fig:defectlocalloop]). It subdivides twofold defects into *k*2 species ${\\bf l}$ and threefold defects into 9 species ${\\bf s}$ when *k* is divisible by 3. Species labels can mutate by absorbing or releasing abelian anyons, a process driven by continuously tuning the local defect phase parameter. This novel species characterization of defect-anyon composites is essential in a complete description of fusion and braiding. The Wilson loop algebra of an arbitrary multi-defect system is studied using word presentation consisting of open Wilson paths in section [sec:AlphabeticpresentationofWilsonalgebra], where the *S*3-transformation of defects and symmetry structure of the Wilson algebra are discussed.\nIn section [sec:defectfusion], we investigate the non-commutative defect fusion category. The objects consist of abelian anyons and twist defects labeled with species. Due to the semi-classical nature of the non-dynamical *S*3-symmetry, defects are not grouped into conjugacy classes of *S*3-fluxes and anyons are not projected into *S*3-orbifold superselection sectors. Loosely speaking the fusion rules originate from the non-abelian group structure of *S*3 and take the following multi-channel form\n \n$$\\begin{aligned}\n{\\sum\\_{\\bf a}}'[{\\bf a}]&\\simeq[1/2]\\_\\chi\\times[1/2]\\_\\chi\\nonumber\\\\&\\simeq[1/3]\\times[1/3]\\times[1/3]\\nonumber\\\\&\\simeq[1/2]\\_\\chi\\times[1/2]\\_{\\chi+1}\\times[\\overline{1/3}]\\nonumber\\\\&\\simeq[1/2]\\_\\chi\\times[1/2]\\_{\\chi-1}\\times[1/3]\\label{fusionrulesummary}\\end{aligned}$$ \nwhere *χ* = 0, 1, 2 mod 3 represents the three colors *Y*, *R*, *B*, the equation is unaffected by cyclic permutation of defect order on the right, and the sum of abelian anyons ${\\bf a}$ on the left is restricted by defect species so that the particle-antiparticle duality requires a change of species label $[1/2]\\_{\\chi,{\\bf l}\\_1}\\times[{\\bf a}]=[1/2]\\_{\\chi,{\\bf l}\\_2}$ when absorbing or releasing an abelian anyon in general. The second equality requires fusion degeneracy $[1/3]\\times[1/3]=d\\_{[1/3]}[\\overline{1/3}]$, and the third and forth equalities exhibit fusion non-commutativity. A more precise set of fusion rules can be found in section [sec:fusionrules]. The choice of a particular set of splitting states (or Wilson string configurations) shown in figure [fig:splittingspaces] in section [sec:splittingspaces] fixes the gauge for a consistent set of basis transformations between different maximally commuting sets of observables, called *F*-symbols, that characterizes the fusion category. A few calculated examples are illustrated in section [sec:Fmoves] and a complete list of *F*-matrices (up to *S*3-symmetry) is included in table [tab:Fsymbols] in appendix [sec:Fsymbols].\nWe describe exchange and braiding between defects of the same type in section [sec:defectexchangebraiding]. Although the non-commutative fusion category cannot be fully braided, a subset of 180∘ rotation operations or *R*-symbols between commuting objects can be defined and are evaluated in section [sec:statisticsZ3] and [sec:statisticsZ2] by rotating Wilson strings. While threefold defects do not carry spin, twofold defects have non-trivial species dependent statistics as shown in section [sec:statisticsZ2] and they are identified with the spin phase of 2 × (360∘) rotation. The braiding *S*-matrices are defined among defects of the same *S*3-type with entries labeled by species. The *S* and *T* matrices for twofold defects are identified in section [sec:defectmodulartransformation] with Dehn twist *t**y* and double Dehn twist *t**x*2 respectively on the ground states on a torus decorated with a branch cut along the *y*-cycle. In general, they form a unitary group structure of the congruent subgroup Γ0(2) rather than the full modular group *S**L*(2; Z). Physical unitary braiding operations or *B*-matrices between defects of the same type are computed in section [sec:Bmatrices] and they demonstrate the non-abelian nature of twist defects. A certain compactification braiding identity of the sphere braid group that is expected to hold in a closed anyon system is now only *projectively* satisfied for defects.\nThe rotor lattice model\n=======================\nWe consider a Z*k* rotor model (or spin-1/2 model for *k* = 2) on vertices of a honeycomb lattice, or in general a bipartite trivalent planar graph with the following properties.\n1. **Tricoloring** Each plaquette may be colored with one of three colors, say yellow (Y), red (R) and blue (B), such that adjacent plaquettes never carry the same color.\n2. **Bipartite** There is a sublattice structure so that vertices can be labeled by black ( • ) or white ( ∘ ) and adjacent vertices are always of differing types.\nIn addition, in the absence of twist defects and branch cuts which will be explained later, we require the graph to satisfy the above constraints globally. On a torus, they will be globally preserved with compatible periodic boundary conditions, when the length of the two primitive cycles are multiples of three. The model can be put on a closed surface with arbitrary genus by adding an appropriate number of defect squares or octagons (figure [fig:squareoctagon]) on the regular honeycomb lattice. These are trivial *twistless* defects in the sense that they do not violate the tricoloring and bipartite order.\n[fig:squareoctagon]\nThe Hamiltonian\n---------------\nThe degrees of freedom (the “spins” for *k* = 2) of the Z*k*-lattice model live on vertices. Z*k* rotors are operators *σ* and *τ* that take eigenvalues in {1, *e*2*π**i*/*k*, …, *e*2*π**i*(*k* − 1)/*k*} and commute up to a Z*k* phase\n \n*τ**σ* = *w**σ**τ*, *σ**k* = *τ**k* = 1 \nwhere *w* is a *k**t**h* root of unity. We will assume *w* = *e*2*π**i*/*k* hereafter. They can be represented by *k*-dimensional matrices\n \n$$\\tau=\\left(\\begin{array}{\\*{20}c}0&1&\\cdots&0\\\\\\vdots&\\vdots&\\ddots&\\vdots\\\\0&0&\\cdots&1\\\\1&0&\\cdots&0\\end{array}\\right),\\quad\\sigma=\\left(\\begin{array}{\\*{20}c}1&0&\\cdots&0\\\\0&w&\\cdots&0\\\\\\vdots&\\vdots&\\ddots&\\vdots\\\\0&0&\\cdots&w^{k-1}\\end{array}\\right)\\label{sigmataurep}$$ \nEach vertex carries a *k*-dimensional Hilbert space and a set of rotors. The total Hilbert space is a tensor product over vertices, and rotors at different vertices commute.\n[fig:stabilzers]\nGiven a bipartite • , ∘ assignment of vertices, each plaquette *P* carries two stablilizer operators\n \n*P̂*• = ∏*v*• ∈ *P**σ**v*•∏*v*∘ ∈ *P**τ**v*∘, *P̂*∘ = ∏*v*• ∈ *P**τ**v*•∏*v*∘ ∈ *P**σ**v*∘ \nwhere *σ**v*, *τ**v* are rotors at vertices *v* around the plaquette and tensor products have been suppressed. The bipartite structure ensures all plaquettes have the same number of • and ∘ vertices and two neighboring plaquettes share exactly one • and one ∘ vertex. Then ensures mutual commutativity of the plaquettes operators, which form a set of good quantum numbers referred to as Z*k* stabilizers or fluxes. The Hamiltonian is defined by the sum of stablilizers,\n \n*H* = − *J*∑*P*(*P̂*• + *P̂*•†) − *J*∑*P*(*P̂*∘ + *P̂*∘†) \nGround states are trivial flux configurations where *P̂*• = *P̂*∘ = 1 for all stabilizers.\nIt is indicative to notice that the model describes the topological phase of two copies of Z*k* version of Kitaev toric code. However, Hamiltonian realizes a much richer *S*3 *geometric symmetry* that extends the original charge-flux Z2 duality in the quantum double model. The *S*3 symmetry can be understood in the microscopic geometric level by the action of space group on the tricoloring and bipartite pattern of the honeycomb lattice. These correspond the threefold and twofold generators of the permutation group *S*3 and will be discussed in detail in section [sec:compositelatticedefects]. This generalizes of the *e*-*m* duality in the Kitaev’s toric code that involves geometrically switching vertices and plaquettes.\nThe ground state degeneracy of the Hamiltonian on a trivalent graph on any closed orientable genus *g* surface can be evaluated by counting vertices and independent plaquette stabilizers. Denote the total number of vertices by #*V* and the number of plaquettes by #*P*. For a regular honeycomb lattice on a torus, #*V* = 2 × #*P*, which is commensurate with the two stabilizer operators eq. located at each plaquette. However there is an overcounting since certain products of stabilizers are identical to the identity. The number of these relations is independent of system size and therefore gives rise to a *topological* ground state degeneracy (*k*4 on the torus if the honeycomb is globally tricolorable).\nAs mentioned earlier, the model can be put on any closed orientable surface without violating *local* tricolorability and the • , ∘ -sublattice structure, by adding square or octagon defects as showen in figure [fig:squareoctagon]. The Gauss-Bonnet theorem (or Euler characteristic) requires\n \n#octagons − #squares = 6(*g* − 1). \nSince an octagon/square carries two greater/fewer vertices than a hexagon,\n \n#*V* = 2 × #*P* + 4(*g* − 1). \non a genus *g* surface.\nWe will first investigate this overcounting in the case when the trivalent graph is *globally* tricolorable. This is an additional topological constraint that ensures that the tricoloration remains unchanged around a non-trivial cycle. In particular, the total number of plaquettes must be a multiple of three and appropriate twisted periodic boundary conditions must be applied if individual cycles have lengths not divisible by three.\nGiven a global *Y**R**B*-coloration of the plaquettes *P**Y*, *P**R* and *P**B*, stabilizer operators are overcounted by the following four cocycle relations,\n \n$$\\begin{aligned}\n1&=\\prod\\_{P^Y}\\hat{P}^Y\\_\\bullet\\left(\\prod\\_{P^R}\\hat{P}\\_\\bullet^R\\right)^\\dagger=\\prod\\_{P^R}\\hat{P}^R\\_\\bullet\\left(\\prod\\_{P^B}\\hat{P}\\_\\bullet^B\\right)^\\dagger\\nonumber\\\\&=\\prod\\_{P^Y}\\hat{P}^Y\\_\\circ\\left(\\prod\\_{P^R}\\hat{P}\\_\\circ^R\\right)^\\dagger=\\prod\\_{P^R}\\hat{P}^R\\_\\circ\\left(\\prod\\_{P^B}\\hat{P}\\_\\circ^B\\right)^\\dagger\\label{stabilizerovercounting}\\end{aligned}$$ \nThese can be understood by observing that the product of all vertex rotors can be given by product of plaquettes of a particular color, i.e. ∏*v*•*σ**v*•∏*v*∘*τ**v*∘ = ∏*P**Y**P̂*•*Y* = ∏*P**R**P̂*•*R* = ∏*P**B**P̂*•*B* and similarly for the other set, *P̂*∘. Thus the number of indepedent stabilizers is,\n \n#stabilizers = 2 × #*P* − 4 \nand the ground state degeneracy (G.S.D.) of a globally tricolorable graph on a genus *g* closed surface is\n \n*G*.*S*.*D*. = *k*#*V* − #\\small stabilizers = *k*4*g* \nWilson algebra and ground state degeneracy\n------------------------------------------\n[fig:trivialloop]\n[fig:Wilsonstring]\nGround states can be written down as an equally weighted sum of plaquette operators acting on a suitably chosen state. One example is,\n \n$$|0\\rangle\\_\\bullet=\\frac{1}{\\sqrt{\\mathcal{N}}}\\prod\\_{P}\\left(\\sum\\_{r=0}^{k-1}\\hat{P}\\_\\circ^r\\right)|\\sigma\\_{v\\_\\bullet}=\\tau\\_{v\\_\\circ}=+1\\rangle\\label{GS0}$$ \nwhere ∣*σ**v*• = *τ**v*∘ = + 1⟩ is the tensor product eigenstate state of *σ* for each • -vertex and *τ* for ∘ -vertex, and N = *k*#*P* − 2 is a normalization factor. It is a simultaneous eigenstate for all plaquette operators *P̂*•∣0⟩ = *P̂*∘∣0⟩ = ∣0⟩. The ground state can be interpreted as a condensate of of trivial *Wilson loops*. A Wilson loop is a string of rotors that commutes with each stabilizer, and trivial if it can be written as a product of stabilizers. They are labeled by the • , ∘ -sublattice type and colors *Y**R**B*, so that a Wilson loop along the boundary of a domain Ω is of the form\n \n$$\\begin{aligned}\nW(\\partial\\Omega)^\\chi\\_{\\bullet}&=\\prod\\_{P\\subseteq\\Omega}\\hat{P}^{\\chi+1}\\_{\\bullet}(\\hat{P}^{\\chi-1}\\_{\\bullet})^{-1}\\\\W(\\partial\\Omega)^\\chi\\_{\\circ}&=\\prod\\_{P\\subseteq\\Omega}\\hat{P}^{\\chi-1}\\_{\\circ}(\\hat{P}^{\\chi+1}\\_{\\circ})^{-1}\\end{aligned}$$ \nfor *χ* ∈ {0, 1, 2} labels the colors {*Y*, *R*, *B*} mod 3, and the product is taken over either • - or ∘ -plaquettes of two complementing colors *χ* ± 1 inside the open domain Ω. Operators in the interior of Ω cancel each other and leave a string of rotors along the boundary ∂Ω connecting a strand of plaquettes of the same color *χ*. This is illustrated by figure [fig:trivialloop] for *χ* = *B*.\n[fig:colorspliting]\nThere is an important redundancy in the color labels. A blue string can split into a yellow and red pair, both propagating in the opposite direction. This can be done by multiplying plaquettes on the string, and results in a pair of tricolor, trivalent, sources and drains as shown in figure [fig:colorspliting]. Similarly, a parallel triplet of co-propagating Wilson strings of three different colors can be locally cancelled by plaquette operators. The color redundancy can be summarized by the fusion\n \n*Y*• × *R*• × *B*• = *Y*∘ × *R*∘ × *B*∘ = 1 \nfor 1 being the vacuum.\nIn general a closed Wilson loop is built by joining colored, directed paths emanating from sources and drains. Each colored path is a string of *σ*± 1 and *τ*± 1 rotors that connects plaquettes of that color as illustrated in figure [fig:Wilsonstring].\n \n*W*•*χ* = ∏*v*•*σ**v*•± 1∏*v*∘*τ**v*∘± 1, *W*∘*χ* = ∏*v*•*τ**v*•± 1∏*v*∘*σ**v*∘± 1 \nThe signs of the rotors are dictated by the arrow directions in figure [fig:trivialloop](b) and [fig:Wilsonstring]. For instance, rotors along a string switch signs about a ± 60∘ or 180∘ corner as shown in figure [fig:trivialloop]. It is straightforward to check that closed Wilson strings commute with all stabilizers through phase cancellation. Evidently, any product of plaquette operators is a closed Wilson string, which can be pulled out of the condensate ground state without an energy cost. However, the converse is not true — a closed Wilson string is not necessarily the product of plaquettes. There are non-trivial cycles on the genus *g* ≥ 1 surface that do not bound an open domain.\n[fig:genusgcycles]\n[fig:intersection]\nOn a genus *g* surface, there are 2*g* non-trivial primitive cycles C*i*, where C2*l* − 1, C2*l* are the two cycles associated with the *l**t**h* handle (see figure [fig:genusgcycles]). Each one can be labeled by sublattice type • , ∘ and color *Y**R**B*. Together, the Wilson loops form a non-commutative algebra with the algebraic relations\n \n$$\\begin{aligned}\n\\left(W(\\mathcal{C}\\_i)^{\\chi}\\_{\\bullet/\\circ}\\right)^k&=1\\label{wilsoncomm1}\\\\\\left[W(\\mathcal{C}\\_{i})^{\\chi\\_1}\\_\\bullet,W(\\mathcal{C}\\_{j})^{\\chi\\_2}\\_\\bullet\\right]&=\\left[W(\\mathcal{C}\\_{j})^{\\chi\\_1}\\_\\circ,W(\\mathcal{C}\\_{j})^{\\chi\\_2}\\_\\circ\\right]=0\\label{wilsoncomm2}\\\\ W(\\mathcal{C}\\_i)^{\\chi\\_1}\\_\\bullet W(\\mathcal{C}\\_{j})^{\\chi\\_2}\\_\\circ&=e^{i\\frac{2\\pi}{k}\\langle\\mathcal{C}\\_i^{\\chi\\_1},\\mathcal{C}\\_j^{\\chi\\_2}\\rangle}W(\\mathcal{C}\\_{j})^{\\chi\\_2}\\_\\circ W(\\mathcal{C}\\_{i})^{\\chi\\_1}\\_\\bullet\\label{wilsoncomm3}\\end{aligned}$$ \nwhere the *χ*’s run over the colors *Y**R**B* and the pairing ⟨C*i**χ*1, C*j**χ*2⟩ is determined by intersection number between Wilson strings, summarized in figure [fig:intersection]. The intersection form ⟨ \\* , \\* ⟩ is bilinear and symmetric. It counts the total Z*k*-phase accumulated by interchanging rotor operators *σ*, *τ* at overlapping vertices according to eq.. Intersection is invariant under cyclic permutation of colors, and changes sign if the direction of one of the Wilson string is reversed. The color fusion rule eq. forces null intersection between Wilson cycles of the same color. Since primitive cycles only intersect when they correspond to the same handle, the intersection ⟨C*i**χ*1, C*j**χ*2⟩ is zero unless *i* = 2*l* − 1, *j* = 2*l* or vice versa, in which the number according to figure [fig:genusgcycles] and [fig:intersection](b) would be\n \n$$\\langle\\mathcal{C}\\_{2l-1}^{\\chi\\_1},\\mathcal{C}\\_{2l}^{\\chi\\_2}\\rangle=\\left\\{\\begin{array}{\\*{20}c}0&\\mbox{for $\\chi\\_1=\\chi\\_2$}\\\\-1&\\mbox{for $\\chi\\_1=\\chi\\_2-1$}\\\\1&\\mbox{for $\\chi\\_1=\\chi\\_2+1$}\\end{array}\\right.\\label{wilsoncomm4}$$ \nfor *χ* = *Y*, *R*, *B* = 0, 1, 2 modulo 3.\nThe three colors are not independent due to color fusion. For example, the products of the parallel triplets\n \n*W*(C*i*)•*Y**W*(C*i*)•*R**W*(C*i*)•*B* = *W*(C*i*)∘*Y**W*(C*i*)∘*R**W*(C*i*)∘*B* = 1 \nsince these can be written as products of plaquette operators and act as the identity on the ground state. Therefore, there are *k*8*g* independent Wilson loops generated by the primitive cycles *W*(C*i*)•*Y*, *W*(C*i*)•*R*, *W*(C*i*)∘*Y* and *W*(C*i*)∘*R*, for *i* = 1, …, 2*g*,\nAn orthonormal basis for the ground state Hilbert space can be written down using the Wilson algebra starting with the particular ground state ∣0⟩• in eq.. Define the normalized ground state\n \n$$|{\\bf m},{\\bf n}\\rangle\\_\\bullet=\\left[\\prod\\_{i=1,\\ldots,2g}\\left(W(\\mathcal{C}\\_i)^Y\\_\\circ\\right)^{m\\_i}\\left(W(\\mathcal{C}\\_i)^R\\_\\circ\\right)^{n\\_i}\\right]|0\\rangle\\_\\bullet\\label{GSbasis}$$ \nwhere ${\\bf m}=(m\\_i)$, ${\\bf n}=(n\\_i)$ have integers modulo *k* entries, for *i* = 1, …, 2*g*. The • -Wilson loops *W*(C*i*)•*χ* form a maximal set of commuting generators and share simultaneous eigenstates. The eigenvalues can be evaluated using the intersection relation and.\n \n$$W(\\mathcal{C}\\_j)^\\chi\\_\\bullet|{\\bf m},{\\bf n}\\rangle\\_\\bullet=e^{i\\frac{2\\pi}{k}\\varphi\\_j^\\chi({\\bf m},{\\bf n})}|{\\bf m},{\\bf n}\\rangle\\_\\bullet\\label{GSeigenvalues}$$ \nwhere the Z*k* phase is given by\n \n$$\\begin{aligned}\n\\varphi\\_j^\\chi({\\bf m},{\\bf n})&=(-1)^j\\left[n\\_{j-(-1)^j}(\\delta^\\chi\\_0-\\delta^\\chi\\_2)\\right.\\nonumber\\\\&\\quad\\quad\\left.+m\\_{j-(-1)^j}(\\delta^\\chi\\_2-\\delta^\\chi\\_1)\\right]\\end{aligned}$$ \nmodulo *k*, where *χ* = 0, 1, 2 index the colors *Y*, *R*, *B*. As different ground states in eq. are distinguished by their eigenvalues with respect to the • -Wilson loops, they must be mutually orthogonal. The *k*4*g* dimensional space of degenerate ground states forms an irreducible unitary representation of the Wilson algebra.\n### Obstruction to global tricolorability\nWe will spend the remaining of the subsection on the Wilson algebra and ground state degeneracy when the trivalent graph is *not* globally tricolorable. The topological obstruction is characterized by closed branch cuts where same color plaquettes share edges and vertices (see figure [fig:torusbranchcut](a)). Branch cuts are not physical domain walls as the Hamiltonian does not depend on an explicit plaquette color definition. A closed branch cut that runs along a trivial loop can be removed by cyclic permuting the colors inside the area bounded by the loop. A branch cut going along a non-contractible cycle is however irremovable (unless canceled by another branch cut). This topological color inconsistency has a reducing effect on the Wilson algebra and consequently ground state degeneracy. Similar issue of branch cuts also arise for Z*k* toric code over a checkerboard lattice on a torus , where the charge-flux duality is realized as the bicolor structure of checkerboard plaquettes. The situation in the tricolored model is qualitative different as the Wilson algebra and ground state degeneracy in the presence of branch cuts depend on the divisibility of *k* by 3.\n[fig:torusbranchcut]\nFor simplicity, we will only demonstrate the case when the model is put on a torus and there is a single branch cut along a non-trivial cycle, say the meridian direction. This can be achieved by adapting a twisted boundary condition on a regular honeycomb lattice through introducing a lattice displacement along an zig-zag edge illustrated in figure [fig:torusbranchcut](a). We will see in the section [sec:twistdefect] that an open color branch cut ends at a conjugate pair of non-abelian threefold *twist defects*. And therefore the twisted boundary condition can be constructed by draging a threefold defect around a cycle. Note that this is fundamentally different from threading a non-abelian quantum flux in a true topological phase as the underlying semiclassical configuration explicitly breaks *S**L*(2; Z) modular invariance. The branch cut picks out a particular non-trivial cycle on the torus and the Wilson algebra does not close under the *S**L*(2; Z) action. Modular transformations will be discussed in more detail in section [sec:defectmodulartransformation].\nA Wilson string will change color through cyclic permutation across the branch cut as shown in figure [fig:torusbranchcut](b). As a result the longitudinal cycle no longer corresponds a Wilson loop as the string will not close back onto itself after passing across the branch cut. Wilson loops along the meridian direction on the other hand are still closed as they do not intersect the parallel branch cut. However, they will change color if the entire loop is dragged around the torus. This gives rise to a color ambiguity since meridian Wilson loops of different colors are now interchangable through plaquette stabilizers, and they are indistinguishable on the ground states. The color fusion rule *Y* × *R* × *B* = 1 then implies all meridian Wilson loops to be of order 3, *W*(C2)3 = 1. Being built by Z*k* rotors, a Wilson loop is automatically of order *k* as seen in. And therefore unless *k* is divisible by 3, the meridian Wilson loop is trivial and is a product of plaquette operators\n \n*W*(C2)• / ∘ = ∏*P*(*P̂*• / ∘*Y*)2*s*(*P̂*• / ∘*R*)− *s*(*P̂*• / ∘*B*)− *s* \nwhere *s* is an integer so that 3*s* ≡ 1 mod *k*.\nWhen *k* is a multiple of 3, the meridian Wilson loop *W*(C2)• / ∘ is not a contractible boundary and it intersects with a closed Wilson string *W*(Σ1)• / ∘ in the longitudinal direction consists of *k*/3 tricolor trivalent sources and a unicolor *k*-valent drain depicted in figure [fig:torusbranchcut](c). It commutes with all plaquette stabilizers because of the color fusion *Y* × *R* × *B* = 1 at the tricolor sources and Z*k* fusion *Y**k* = 1 at the unicolor drain. *W*(Σ1)• / ∘ is equivalent to dragging the abelian anyon $\\boldsymbol\\kappa\\_{\\bullet/\\circ}$ (figure [fig:abeliananyonlattice]), a non-trivial anyon invariant under cyclic color permutation only when *k* divisible by 3, around the longitudinal cycle. The Wilson algebra is then generated by *W*(Σ1)•, *W*(C2)•, *W*(Σ1)∘ and *W*(C2)∘ that satisfy the following algebraic relations.\n \n$$\\begin{aligned}\n\\left(W(\\Sigma\\_1)\\_{\\bullet/\\circ}\\right)^3&=\\left(W(\\mathcal{C}\\_2)\\_{\\bullet/\\circ}\\right)^3=1\\\\\\left[W(\\Sigma\\_1)\\_\\bullet,W(\\mathcal{C}\\_2)\\_\\bullet\\right]&=\\left[W(\\Sigma\\_1)\\_\\circ,W(\\mathcal{C}\\_2)\\_\\circ\\right]=0\\\\W(\\Sigma\\_1)\\_\\bullet W(\\mathcal{C}\\_2)\\_\\circ&=e^{2\\pi i/3}W(\\mathcal{C}\\_2)\\_\\circ W(\\Sigma\\_2)\\_\\bullet\\\\W(\\mathcal{C}\\_2)\\_\\bullet W(\\Sigma\\_1)\\_\\circ&=e^{2\\pi i/3}W(\\Sigma\\_1)\\_\\circ W(\\mathcal{C}\\_2)\\_\\bullet\\end{aligned}$$ \nThis gives rise to a ground state degeneracy of 9 = 32 on a torus.\nThe ground state degeneracy on a torus is identical to the total number of deconfined anyon types . On a globally tricolorable graph, abelian anyons can be uniquely labeled by the particle numbers mod *k* of fundamental constituents *Y*•, *R*•, *Y*∘, *R*∘, and thus the ground state degeneracy is *k*4. When there is a color ambiguity from non-contractible branch cut, there will be less particle types which are now referred as species because *Y*• = *R*• = *B*• and *Y*∘ = *R*∘ = *B*∘. The three colors and Z*k* fusion implies *B*• / ∘3 = *B*• / ∘*k* = 1. And hence there will not be non-trivial species unless *k* is a multiple of 3, in which case they will be labeled by the particle number of the two fundamental generators *B*•, *B*∘ modulo 3. This gives rise to a 32-fold degeneracy on a torus and corresponds 32 species of threefold defects distinguishable by Σ• / ∘ discussed in more detail in section [sec:twistdefect].\nAbelian anyon excitations, effective field theory and -symmetry\n---------------------------------------------------------------\nExcitations of the Hamiltonian are eigenstates of plaquette stabilizers with non-unit eigenvalues. They can be constructed by letting open Wilson (also called Jordan-Wigner) string operators *W*(S)∘ / • act on a ground state ∣*G**S*⟩.\n \n∣∂S⟩• / ∘ = *W*(S)∘ / •∣*G**S*⟩ \nOpen Wilson strings ( • or ∘ ) do not commute with local Wilson loops (*W*(L)∘ and *W*(L)• resp.) surrounding the end points in ∂S (see figure [fig:excitations] and [fig:coloranyons]). Since trivial closed Wilson loops condense in the ground state, the excitation state ∣∂S⟩ in depends only on its plaquette eigenvalues at the end points of the string S rather than the path itself as long as it does not wrap an extra non-trivial cycle. The excited state (or in general a collection of states due to ground state degeneracy) can therefore be labeled by local *abelian anyon* configurations,, measured by the eigenvalues of plaquette stabilizers\n \n*P̂*• / ∘∣∂S⟩• / ∘± = *e*± 2*π**i*/*k*∣∂S⟩• / ∘ \nfor ∂S = ∑*P*+ − ∑*P*− are directed plaquettes where the open string S ends (see figure [fig:excitations] and [fig:coloranyons] for sign definition).\n[fig:excitations]\n[fig:coloranyons]\nAnyons are in general detected by local Wilson loops that encircle the quasi-particle excitation (see figure [fig:coloranyons]) with eigenvalues given by Wilson strings intersection (see figure [fig:intersection]). Primitive anyons are labelled by color, *Y*, *R*, *B*, and sublattice type, • , ∘ with fusion relation\n \n$$\\underbrace{\\chi\\_{\\bullet/\\circ}\\times\\ldots\\times\\chi\\_{\\bullet/\\circ}}\\_{k}=Y\\_{\\bullet/\\circ}\\times R\\_{\\bullet/\\circ}\\times B\\_{\\bullet/\\circ}=1\\label{orderkcolorfusion}$$ \nfor *χ* = *Y*, *R*, *B*, so that no Wilson loop measurement can tell apart these combination from the trivial vacuum 1 without enclosing a proper subset. Composite anyons $[{\\bf a}]$ are labeled by particle numbers (modulo *k*) of independent primitive anyons.\n \n$$\\begin{aligned}\n&=(Y\\_\\bullet)^{y\\_1}(R\\_\\bullet)^{r\\_1}(Y\\_\\circ)^{y\\_2}(R\\_\\circ)^{r\\_2}\\label{abeliananyonlabel1}\\\\{\\bf a}&=({\\bf a}\\_\\bullet,{\\bf a}\\_\\circ),\\quad\\left\\{\\begin{array}{\\*{20}c}{\\bf a}\\_\\bullet=y\\_1{\\bf e}\\_Y+r\\_1{\\bf e}\\_R\\\\{\\bf a}\\_\\circ=y\\_2{\\bf f}^Y+r\\_2{\\bf f}^R\\hfill\\end{array}\\right.\\label{abeliananyonlabel2}\\end{aligned}$$ \nfor *y*1, *y*2, *r*1, *r*2 in Z*k*. The *k*4 anyon are mutually distinguishable by Wilson loops. They are represented as a pair of Z*k*-valued two dimensional vectors, ${\\bf a}\\_\\bullet$ and ${\\bf a}\\_\\circ$, on two triangular lattices (see figure [fig:abeliananyonlattice]), one for • -anyons, another for ∘ -anyons.\n[fig:abeliananyonlattice]\nAbelian anyons support single channel fusion\n \n$$\\times[{\\bf b}\\_\\bullet,{\\bf b}\\_\\circ]=[{\\bf a}\\_\\bullet+{\\bf b}\\_\\bullet,{\\bf a}\\_\\circ+{\\bf b}\\_\\circ]\\label{abeliananyonfusion}$$ \nand carry unit quantum dimension $d\\_{[\\bf a]}=1$. A basis of the one dimensional splitting space $V^{[{\\bf a}][{\\bf b}]}\\_{[{\\bf a}+{\\bf b}]}$ is given by letting the Jordan-Wigner string in figure [fig:abeliananyonsplittingspace](a) act on the ground state (projected locally inside the dotted line with fixed boundary condition). We adopt the time-ordering convention that a • -string always acts on the ground state before a ∘ -one.\n \n$$\\begin{aligned}\n\\left|V^{[{\\bf a}][{\\bf b}]}\\_{[{\\bf a}+{\\bf b}]}\\right\\rangle=W\\left(\\vcenter{\\hbox{\\includegraphics[width=0.2in]{Y2string.pdf}}}\\right)\\_\\circ W\\left(\\vcenter{\\hbox{\\includegraphics[width=0.2in]{Y1string.pdf}}}\\right)\\_\\bullet|GS\\rangle\\label{anyonsplittingbasis}\\end{aligned}$$ \nThe string ordering is a gauge choice for splitting state. This will be generalized to twist defects in figure [fig:splittingspaces] in section [sec:defectfusion].\n[fig:abeliananyonsplittingspace]\nExchange and braiding operations are represented by abelian Z*k* phases. They stem from intersection between the anyon worldlines and Jordan-Wigner strings, which can be shown to be identical to the linking number between worldlines of anyons such as those shown in figure [fig:abelianspin]. The *R*-symbol of exchanging abelian anyon $[{\\bf a}]$ and $[{\\bf b}]$ under the basis choice of splitting space $V^{[{\\bf a}][{\\bf b}]}\\_{[{\\bf a}+{\\bf b}]}$ in eq. and figure [fig:abeliananyonsplittingspace](a) is given by\n \n$$R^{[{\\bf a}][{\\bf b}]}\\_{[{\\bf a}+{\\bf b}]}=e^{i\\frac{2\\pi}{k}{\\bf a}\\_\\circ^Ti\\sigma\\_y{\\bf b}\\_\\bullet}=e^{i\\frac{2\\pi}{k}(y\\_2r'\\_1-r\\_2y'\\_1)}\\label{Rabeliananyon}$$ \nand is illustrated in figure [fig:abeliananyonsplittingspace](b), for ${\\bf a}\\_\\circ=y\\_2{\\bf f}^Y+r\\_2{\\bf f}^R$ and ${\\bf b}\\_\\bullet=y'\\_1{\\bf e}\\_Y+r'\\_1{\\bf e}\\_R$. The topological spin of an anyon can be evaluated by a 360∘ rotation (figure [fig:abelianspin]) or exchange, giving the spin-statisticcal phase\n \n$$\\theta\\_{[{\\bf a}]}=R^{[{\\bf a}][{\\bf a}]}\\_{[2{\\bf a}]}=\\vcenter{\\hbox{\\includegraphics[width=0.7in]{anyonexchange.pdf}}}=e^{i\\frac{2\\pi}{k}{\\bf a}\\_\\circ^Ti\\sigma\\_y{\\bf a}\\_\\bullet}\\label{anyonexchangespin}$$ \nThe full braiding between anyon $[{\\bf a}]$ and $[{\\bf b}]$ is given by the symmetric\n \n$$S\\_{[{\\bf a}][{\\bf b}]}=\\frac{1}{k^2}\\vcenter{\\hbox{\\includegraphics[width=0.5in]{anyonfullbraid.pdf}}}=\\frac{1}{k^2}e^{i\\frac{2\\pi}{k}({\\bf a}\\_\\circ^Ti\\sigma\\_y{\\bf b}\\_\\bullet+{\\bf b}\\_\\circ^Ti\\sigma\\_y{\\bf a}\\_\\bullet)}\\label{anyonfullbraiding}$$ \nwhere the normalization $\\mathcal{D}=\\sqrt{\\sum\\_{\\bf a}d^2\\_{\\bf a}}=k^2$ equals the total quantum dimension of the abelian topological phase that is responsible for its topological entanglement entropy , and is added so that the *S*-matrix is unitary. Both spin *θ* and braiding *S* are gauge invariant.\n[fig:abelianspin]\n### Low energy effective field theory\nThe model Hamiltonian can be described by the low energy effective Chern-Simons theory \n \n$$\\mathcal{L}=\\frac{1}{4\\pi}\\int K\\_{IJ}a\\_I\\wedge da\\_J\\label{CS}$$ \nwith the 4 × 4 *K*-matrix\n \n$$K=k\\left(\\begin{array}{\\*{20}c}0&0&0&-1\\\\0&0&1&0\\\\0&1&0&0\\\\-1&0&0&0\\end{array}\\right)\\label{Kmatrix}$$ \nwhere *Y*•, *R*•, *Y*∘, *R*∘ are anyon charges for the *U*(1)-gauge fields *a*1, *a*2, *a*3, *a*4 respectively. This Chern-Simons theory has the same *k*4 anyon types with identical fusion and braiding. It also supports an identical Wilson algebra eq.([wilsoncomm1]-[wilsoncomm4]) on a genus *g* surface. The Hamiltonian therefore has the same low energy description as two copies of the Z*k* version of Kitaev’s toric code . The novelty of is the apparent *S*3-symmetry relating the tricoloring and bipartite structure of the lattice, which enriches the charge-flux (or plaquette-vertex) Z2-duality in Kitaev’s toric code. Dislocations or twist defects are topological defects that violate certain duality or symmetry and carry non-abelian signature. The lattice structure in Hamiltonian facilitates *S*3-twist defects naturally through lattice dislocations and disclinations. Similar topological defects are more obscure and less motivated in the low energy Chern-Simons theory or double Z*k*-plaquette model. Their field theoretical constructions rely on an explicit branch cut in real space where the gauge fields are discontinuous. The cut can be *gauged away* only when the *K*-matrix is symmetric under symmetry transformation. In a lattice description of twist defects, we will see that branch cuts are absent completely.\nAlthough the Chern-Simons theory is not our fundamental tool in this article, we expect a *genon* description of the *S*3-twist defects similar to ref.[]. Here we identify the *S*3-symmetry action on the multicomponent *U*(1)-gauge fields. The permutation group *S*3 of three elements is generated by a non-commutative threefold and twofold symmetries that act respectively as cyclic color permutation\n \n$$\\Lambda\\_3:\\left(\\begin{array}{\\*{20}c}Y\\_\\bullet&R\\_\\bullet&B\\_\\bullet\\\\Y\\_\\circ&R\\_\\circ&B\\_\\circ\\end{array}\\right)\\to\\left(\\begin{array}{\\*{20}c}R\\_\\bullet&B\\_\\bullet&Y\\_\\bullet\\\\R\\_\\circ&B\\_\\circ&Y\\_\\circ\\end{array}\\right)\\label{ST}$$ \nand transposition for color and rotor types\n \n$$\\Lambda\\_{B}:\\left(\\begin{array}{\\*{20}c}Y\\_\\bullet&R\\_\\bullet&B\\_\\bullet\\\\Y\\_\\circ&R\\_\\circ&B\\_\\circ\\end{array}\\right)\\to\\left(\\begin{array}{\\*{20}c}R\\_\\circ&Y\\_\\circ&B\\_\\circ\\\\R\\_\\bullet&Y\\_\\bullet&B\\_\\bullet\\end{array}\\right)\\label{S}$$ \nΛ*B* represents the charge-flux duality within each copy of Z*k*-Kitaev toric code pair, and Λ3 corresponds an extra Z3 symmetry that intertwines the two copies. These will arise naturally as space group operators of the lattice model described in section [sec:compositelatticedefects]. They are represented by the 4 × 4 matrices\n \n$$\\Lambda\\_3=\\left(\\begin{array}{\\*{20}c}0&-1&0&0\\\\1&-1&0&0\\\\0&0&0&-1\\\\0&0&1&-1\\end{array}\\right),\\quad\\Lambda\\_{B}=\\left(\\begin{array}{\\*{20}c}0&0&0&1\\\\0&0&1&0\\\\0&1&0&0\\\\1&0&0&0\\end{array}\\right)\\label{STCS}$$ \nacting on the gauge fields (*a*1, *a*2, *a*3, *a*4). The *K*-matrix is invariant under symmetric transformation, *K* = Λ*T**K*Λ for Λ ∈ *S*3, and therefore the theory is symmetric under the *S*3-transformation *a**i* → Λ*i**j**a**j*.\nConsequently fusion and braiding are also invariant under the symmetry transformation $[{\\bf a}\\_\\bullet,{\\bf a}\\_\\circ]\\to\\Lambda\\cdot[{\\bf a}\\_\\bullet,{\\bf a}\\_\\circ]$ according to and. These transformations rename the color and sublattice labels for abelian anyons. Cyclic color permutation Λ3 corresponds a threefold rotation of the triangular anyon lattice in figure [fig:abeliananyonlattice] while keeping the sublattice label • , ∘ fixes, and transposition Λ*B* corresponds a mirror operation while flipping between a • ↔ ∘ . Notice that for *k* divisible by 3, there are non-trivial anyons $\\boldsymbol\\kappa\\_\\bullet=\\overline{\\boldsymbol\\kappa}\\_\\bullet^\\dagger=(Y\\_\\bullet)^{k/3}(R\\_\\bullet)^{-k/3}$ and $\\boldsymbol\\kappa\\_\\circ=\\overline{\\boldsymbol\\kappa}\\_\\circ^\\dagger=(Y\\_\\circ)^{k/3}(R\\_\\circ)^{-k/3}$ that are invariant under cyclic color permutation $(ST)\\cdot\\boldsymbol\\kappa\\_{\\bullet/\\circ}=\\boldsymbol\\kappa\\_{\\bullet/\\circ}$. Furthermore, unlike over complex coefficients where the finite group *S*3 has only 2-dimenional faithful irreducible representation, the two 4-dimensional matrices cannot be simultaneously further block diagonalized with discrete coefficients. This means that the anyon Hilbert space cannot be decomposed into tensor product without violating *S*3-symmetry.\nIn mathematical terms, the symmetry group *S*3 is a subgroup of Γ1, the group of relabeling of anyons, or precisely the group of invertible functors A → A of the unitary braided fusion category A. In the lattice rotor model with *K*-matrix, Γ1 is given by the group of outer automorphisms\n \n$$\\Gamma\\_1=\\mbox{Outer}(K)\\equiv\\frac{\\mbox{Aut}(K)}{\\mbox{Inner}(K)}$$ \nwhere the automorphism group\n \n$$\\begin{aligned}\n\\mbox{Aut}(K)=O(K;\\mathbb{Z})\\equiv\\left\\{g\\in GL(4;\\mathbb{Z}):g^TKg=K\\right\\}\\end{aligned}$$ \nis given by orthogonal transformations that leave the *K*-matrix invariant, and the subgroup of inner automorphism\n \n$$\\begin{aligned}\n\\mbox{Inner}(K)\\equiv\\left\\{g\\in\\mbox{Aut}(K):g\\cdot[{\\bf a}\\_\\bullet,{\\bf a}\\_\\circ]\\equiv[{\\bf a}\\_\\bullet,{\\bf a}\\_\\circ]\\;\\mbox{mod $K$}\\right\\}\\end{aligned}$$ \ncontain orthogonal transformations that fix the anyon lattice Z4/*K*Z4. As shown later in eq., the symmetry subgroup *S*3 is inherited from and identical to the symmetry of the underlying trivalent bipartite planar graph. For example the color permutation Λ3 is induced by a lattice translation on the honeycomb and color sublattice transposition Λ*B* is induced by a lattice inversion. The correspondence between symmetry of the microscopic Hamiltonian and anyon relabeling symmetry\n \n*ω*1 : *S*3 → Γ1 = Outer(*K*) \nis a *first level weak symmetry breaking* according to Kitaev. Twist defects are explicit local violations of the underlying symmetry, and because of they are also symmetry defects that alter anyon sectors.\nNon-Abelian Twist Defects\n=========================\nThe *Y**R**B*-plaquette coloration and • , ∘ -sublattice types give rise to the four fundamental abelian anyon excitations *Y*•, *Y*∘, *R*•, *R*∘ in the lattice rotor model (recall the color redundancy *Y* × *R* × *B* = 1). The arbitrariness of color and sublattice labeling of abelian anyons is summarized by the *S*3-symmetry generated by cyclic color permutation Λ3 and transposition Λ*B* in eq. and. A twist defect is a topological defect that locally violates the symmetry by altering, or *twisting*, the color and rotor label of an anyon that goes around it (see figure [fig:anyontwist]). In other words, a Wilson string that circles around a twist defect does not close back to itself, and therefore unlike abelian anyons which can be locally detected by small Wilson loops such as plaquette stabilizers, there are no local observables measuring a twist defect state. This non-locality is a central theme of many non-abelian anyons, such as vortex-bound Majorana fermions in chiral *p* + *i**p* superconductors , Ising anyon in the Kitaev’s honeycomb model and Pfaffian fractional quantum Hall state . The non-abelian anyons associated with twist defects considered in this article, however, are not fundamental deconfined excitations of a true topological phase. They are qualitatively more similar to (fractional) Majorana excitations at SC-FM heterostructures with (fractional) topological insulators or strongly spin-orbit coupled quantum wires . Their existence rely on the topological winding of certain classical non-dynamical *order parameter field*, such as pairing and spin/charge gap . The tricoloring and bipartite structure of the lattice Hamiltonian can be regarded as a discrete *order parameter* of the condensate, and its winding around a point defect supports the color and sublattice twisting.\n[fig:anyontwist]\nA twist defect in our lattice model is labeled by an element Λ in the symmetry group *S*3 according to its action on the anyon label so that when an anyon $[{\\bf a}]=Y\\_\\bullet^{y\\_1}R\\_\\bullet^{r\\_1}Y\\_\\circ^{y\\_2}R\\_\\circ^{r\\_2}$ passes counter-clockwise around the twist defect, it changes into $\\Lambda\\cdot[{\\bf a}]$, where Λ is some product combinations of Λ*B* and Λ3 in and. Threefold cyclic permutation and twofold transposition are the two conjugacy classes of *S*3 and correspond to two threefold twist defects $[1/3],[\\overline{1/3}]$ and three twofold ones [1/2]*Y*, [1/2]*R*, [1/2]*B* respectively. Their twisting actions on abelian anyons going around them are summarized in figure [fig:anyontwist]. The fraction label is chosen to match with the *fractionalization* of abelian anyons so that the denominator shows the minimal number of identical defect copies required to fuse into an abelian channel (see ).\n[fig:twistdefects]\n[fig:Z3dislocation]\nCrystalline defects are predicted to carry topologically protected excitations in topological insulators and superconductors . They are expected to hold fractional quantum vortices in FFLO states . Here we realize twist defects in the lattice model as crystalline defects, such as the disclinations and dislocations illustrated in figure [fig:twistdefects] and [fig:Z3dislocation], that center at non trivalent vertices for threefold twist defects, or odd sided plaquettes for twofold twist defects. These are topological lattice defects that carry curvature or torsion singularities and locally breaks lattice rotation and translation symmetry. Through local violation of tricolorability and/or bipartite structure of the Hamiltonian, they change the color and/or sublattice type of abelian anyons that go around them. This gives rise to additional non-contractible Wilson loops, ground state degeneracies and non-trivial quantum dimensions (*d* > 1) associate with the twist defects.\nWe begin this section by writing down the local lattice Hamiltonian for the two kinds of twist defects. Their quantum dimensions (or ground state degeneracies) can be deduced by counting plaquette stabilizers and vertices. Next we show the topological degeneracies are inherited from non-trivial Wilson string operators, each surrounds multiple defects. These form a set of non-commuting physical observables with Z*k*-valued measurements. As a quantum state is labeled by its simultaneous eigenvalues of a maximal set of commuting Wilson operators, it cannot be *accidentally* measured by local observation since the Wilson strings are non-local operators passing through spatially separated twist defects. This non-local storage of quantum information between non-abelian anyons provides topological protection against decoherence and forms the basis for fault tolerant topological quantum computation . Similar to the algebraic relation eq., the non-commutativity of Wilson operators are characterized by an intersection form ⟨C*i**χ*1, C*j**χ*2⟩ between Wilson strings. We compute these pairings explicitly in this section and show their covariant behavior under *S*3-transformation that differs from the invariant one from the previous section. The basis of Wilson strings and their intersection properties will be useful for characterizing defect fusion and braiding in the following section.\nLattice defect Hamiltonian\n--------------------------\nWe describe the lattice model modifications at primitive ± 120∘-disclinations centered at tetravalent or bivalent vertices and ± 60∘-disclinations at heptagon or pentagon plaquettes (see figure [fig:twistdefects]) corresponding to threefold and twofold twist defects respectively. Unlike the square or octagon disclinations in figure [fig:squareoctagon], these non-trivial lattice defects require additional sets of vertex rotors or allow less plaquette operators in order for stabilizers to remain mutually commutative. The extra rotor degree of freedom increases the ground state degeneracy and associates non-trivial quantum dimensions to twist defects. The *S*3-classification of primitive disclinations are summarized in table [tab:twistdefects].\n[ht]\ncl *S*3-label & \n & + 120∘ disclination at a tetravalent ∘ -vertex \n& − 120∘ disclination at a bivalent • -vertex \n & + 120∘ disclination at a tetravalent • -vertex \n& − 120∘ disclination at a bivalent ∘ -vertex \n & + 60∘ disclination at a *χ*-colored heptagon \n& − 60∘ disclination at a *χ*-colored pentagon \n[tab:twistdefects]\n### Threefold twist defects\n[fig:4valentvertex]\nInstead of accommodating a single set of rotors *σ*, *τ* and a *k*-dimensional Hilbert space like an ordinary trivalent vertex, a tetravalent vertex hosts four sets of rotors *σ*(*i*), *τ*(*i*) and a *k*4-dimensional tensor product space. We define the rotor operators at a tetravalent vertex by the following tensor products\n \n$$\\begin{array}{\\*{20}c}\\sigma(1)=\\sigma\\otimes\\openone\\_k\\otimes\\openone\\_k\\otimes\\openone\\_k\\\\\\sigma(2)=\\openone\\_k\\otimes\\sigma\\otimes\\openone\\_k\\otimes\\openone\\_k\\\\\\sigma(3)=\\openone\\_k\\otimes\\openone\\_k\\otimes\\sigma\\otimes\\openone\\_k\\\\\\sigma(4)=\\openone\\_k\\otimes\\openone\\_k\\otimes\\openone\\_k\\otimes\\sigma\\end{array}\\quad\\begin{array}{\\*{20}c}\\tau(1)=\\tau\\otimes\\tau\\otimes\\openone\\_k\\otimes\\tau\\\\\\tau(2)=\\tau\\otimes\\tau\\otimes\\tau\\otimes\\openone\\_k\\\\\\tau(3)=\\openone\\_k\\otimes\\tau\\otimes\\tau\\otimes\\tau\\\\\\tau(4)=\\tau\\otimes\\openone\\_k\\otimes\\tau\\otimes\\tau\\end{array}\\label{4valentrotorrep}$$ \nwhere $\\openone\\_k$ is the *k* × *k* identity matrix and *σ*, *τ* are the usual rotors with matrix representations. They satisfy a modified algebraic relation\n \n$$\\begin{aligned}\n\\tau(i)\\sigma(j)\\tau(i)^{-1}\\sigma(j)^{-1}=\\left\\{\\begin{array}{\\*{20}c}w,&\\mbox{if $|i-j|\\neq2$}\\\\1,&\\mbox{if $|i-j|=2$}\\end{array}\\right.\\label{4valentrotorcomm1}\\\\\\sigma(i)^k=\\tau(i)^k=1,\\quad[\\sigma(i),\\sigma(j)]=[\\tau(i),\\tau(j)]=0\\label{4valentrotorcomm2}\\end{aligned}$$ \nwhere *w* = *e*2*π**i*/*k*. The four adjacent plaquette stabilizers around the tetravalent vertex *u*4 in figure [fig:4valentvertex] are defined with the new rotor operators (suppressing tensor products)\n \n$$\\begin{aligned}\n\\begin{array}{\\*{20}c}\\hat{P}\\_\\bullet(1)=\\tau\\_{u\\_4}(1)\\sigma\\_{v\\_1}\\tau\\_{v\\_2}\\sigma\\_{v\\_3}\\tau\\_{v\\_4}\\sigma\\_{v\\_5}\\hfill\\\\\\hat{P}\\_\\bullet(2)=\\tau\\_{u\\_4}(2)\\sigma\\_{v\\_5}\\tau\\_{v\\_6}\\sigma\\_{v\\_7}\\tau\\_{v\\_8}\\sigma\\_{v\\_9}\\hfill\\\\\\hat{P}\\_\\bullet(3)=\\tau\\_{u\\_4}(3)\\sigma\\_{v\\_9}\\tau\\_{v\\_{10}}\\sigma\\_{v\\_{11}}\\tau\\_{v\\_{12}}\\sigma\\_{v\\_{13}}\\\\\\hat{P}\\_\\bullet(4)=\\tau\\_{u\\_4}(4)\\sigma\\_{v\\_{13}}\\tau\\_{v\\_{14}}\\sigma\\_{v\\_{15}}\\tau\\_{v\\_{16}}\\sigma\\_{v\\_1}\\end{array}\\end{aligned}$$ \nand similarly the ∘ -plaquette operators *P̂*∘(*i*) are defined by interchanging *σ* ↔ *τ*.\nA bivalent vertex on the other hand is hosting two sets of rotor operators and a *k*2-dimensional tensor product space. The rotors are defined by\n \n$$\\begin{array}{\\*{20}c}\\sigma(1)=\\sigma\\otimes\\openone\\_k\\\\\\sigma(2)=\\openone\\_k\\otimes\\sigma\\end{array}\\quad\\begin{array}{\\*{20}c}\\tau(1)=\\tau\\otimes\\tau^2\\\\\\tau(2)=\\tau^2\\otimes\\tau\\end{array}\\label{2valentrotorrep}$$ \nand satisfy the algebraic relations as well as\n \n$$\\tau(i)\\sigma(j)\\tau(i)^{-1}\\sigma(j)^{-1}=\\left\\{\\begin{array}{\\*{20}c}w,&\\mbox{if $i=j$}\\\\w^2,&\\mbox{if $i\\neq j$}\\end{array}\\right.\\label{2valentrotorcomm1}$$ \nThe two adjacent plaquette stabilizers of the bivalent vertex *u*2 in figure [fig:4valentvertex] are defined by\n \n$$\\begin{array}{\\*{20}c}\\hat{P}\\_\\bullet(1)=\\tau\\_{u\\_2}(1)\\sigma\\_{v\\_1}\\tau\\_{v\\_2}\\sigma\\_{v\\_3}\\tau\\_{v\\_4}\\sigma\\_{v\\_5}\\\\\\hat{P}\\_\\bullet(2)=\\tau\\_{u\\_2}(2)\\sigma\\_{v\\_5}\\tau\\_{v\\_6}\\sigma\\_{v\\_7}\\tau\\_{v\\_8}\\sigma\\_{v\\_1}\\end{array}$$ \nand similarly for the ∘ -plaquette operators.\nThanks to the modified algebraic relations and, all plaquette operators around a tetravalent or bivalent vertex mutually commute. The Hamiltonian is then defined just as in eq. by summing over all plaquette stabilizers. In fact and are not just sufficient but also necessary for the stabilizers to form good quantum numbers. The *k*4 or *k*2 dimensional tensor product rotor representations or are not accidental. Their dimensionality are *topologically protected* by non-contractible Wilson strings around them and are directly related to the quantum dimension of threefold twist defect. These will be explained in detail in the following subsection.\nThere is however a caveat when *k* is a multiple of 3. By examining the rotor representation at a tetravalent vertex or that at a bivalent one, we have the additional torsion relation\n \n(∏*i* = 14*τ**u*4(*i*))*k*/3 = (∏*i* = 12*τ**u*2(*i*))*k*/3 = 1 \nand the order three center operators that commute with all rotors\n \nΣ*u*4 = (∏*i* = 14*σ**u*4(*i*))*k*/3, Σ*u*2 = (∏*i* = 12*σ**u*2(*i*))*k*/3 \nThis means that the *k*4 (or *k*2) dimensional tensor product space at the tetravalent (resp. bivalent) vertex is no longer irreducible for the rotor algebra eq. (or eq.) as it can be decomposed according to the Z3-valued good quantum number according to Σ*u*4 (resp. Σ*u*2). In this case, we restrict the tensor product space at any tetravalent (or bivalent) vertices to one of the *k*4/3 (resp. *k*2/3) dimensional sector by specifying the Z3-phase for the central observables Σ*u*4 (resp. Σ*u*2). Or equivalently the *k*4 (resp. *k*2) dimensional rotor space can be broken down by the local defect Hamiltonian\n \n*H**u*4/*u*2 = − *J*\\*(*e*− *i**ϕ*Σ*u*4/*u*2 + *e**i**ϕ*Σ*u*4/*u*2†) \nwhere the phase variable *ϕ* controls the Z3-value of Σ*u*4/*u*2 in the ground state except at *ϕ* = *π*, ± *π*/3 where level crossings occur. Consequently, when *k* is divisible by 3, threefold twist defects are further subdivided into nine different species distinguished by the eigenvalues of the Σ’s, which are local measurements described by certain order 3 Wilson strings described in the upcoming subsection.\nUsing the Euler characteristics 2 − 2*g* = #*P* − #*E* + #*V*, eq. is modified into\n \n#*V* = 2 × #*P* + 4(*g* − 1) − #*u*4 + #*u*2 \nwhere #*V*, #*P* are the total numbers of vertices, plaquettes and #*u*4, #*u*2 are the numbers of tetravalent, bivalent vertices respectively. We are interested in how the ground state degeneracy (G.S.D.) scales with #*u*4 and #*u*2 in the thermodynamic limit. For this, we will ignore the genus *g* and the overcounting of plaquettes such as that will contribute to the G.S.D. by a proportionality constant independent from the twist defect number.\nThe total Hilbert space is a tensor product of rotor spaces of dimension *k*#*V* + 3#*u*4 + #*u*2 for *k* not divisible by 3, or *k*#*V* + 3#*u*4 + #*u*2/3#*u*4 + #*u*2 otherwise. Stabilized by the two operators *P̂*•, *P̂*∘ per plaquette, the ground state degeneracy scales as\n \n$$G.S.D\\sim\\left\\{\\begin{array}{\\*{20}c}(k^2)^{\\#u\\_4+\\#u\\_2},&\\mbox{if $3\\centernot\\mid k$}\\\\(k^2/3)^{\\#u\\_4+\\#u\\_2},&\\mbox{if $3\\mid k$}\\end{array}\\right.$$ \nwhere #*u*4 + #*u*2 is the total number of defects. Or equivalently the quantum dimensions of a threefold twist defect is given by\n \n$$d\\_{[1/3]}=d\\_{[\\overline{1/3}]}=\\left\\{\\begin{array}{\\*{20}c}k^2,&\\mbox{if $3\\centernot\\mid k$}\\\\k^2/3,&\\mbox{if $3\\mid k$}\\end{array}\\right.\\label{Z3dimension}$$ \n### Twofold twist defects\n[fig:pentagonheptagon]\nHeptagon and pentagon are odd-sided plaquettes and therefore the two operators will not commute with one another. There are instead only one plaquette stabilizer at a heptagon and pentagon (or any odd-sided plaquettes)\n \n*P̂*7 = ∏*p* = *i*7*σ**v**i**τ**v**i*, *P̂*5 = ∏*p* = *i*5*σ**v**i**τ**v**i* \nAll faces around the heptagon and pentagon are ordinary even-sided plaquettes. Each carries two usual stabilizers and commute with the heptagon and pentagon. Similar to dislocation operators in Kitaev’s toric code or Z*k*-Wen plaquette model , the problem with *P̂*7 and *P̂*5 is that they are not necessarily *k**t**h* roots of unity since (*σ**τ*)*k* = *w**k*(*k* − 1)/2 = ( − 1)*k* − 1. And in general the defect Hamiltonian can depend on a phase\n \n*H*5, 7(*ϕ*) = − *J*\\*(*e*− *i**ϕ**P̂*5, 7 + *e**i**ϕ**P̂*5, 7†) \nIts ground state is labeled by the stabilizer eigenvalues *P̂*5, 7 = *e*2*π**i**p*(*ϕ*)/*k* for *p*(*ϕ*) being the integer for *k* odd (or half-integer for *k* even) between\n \n$$\\frac{k\\phi}{2\\pi}-\\frac{1}{2}<p(\\phi)<\\frac{k\\phi}{2\\pi}+\\frac{1}{2}$$ \nThere are level crossings when *k**ϕ*/2*π* is a half-integer (or integer) when *k* is odd (resp even) where two eigenstates for *P̂*5, 7 have the same energy and the system becomes gapless (see figure [fig:levelcrossing]).\n[fig:levelcrossing]\nWe treat this phase variable *ϕ* as a non-dynamic classical parameter associate to a twofold twist defect. This is a fundamental feature differing the twist defect from usual deconfined non-abelian anyons in topologically ordered system. *p*(*ϕ*) modulo *k* is a locally measurable quantity by the pentagon or heptagon plaquette operator *P̂*5, 7. In general twofold twist defects are labeled by small Wilson loops circling them (to be discussed in the following subsection) and are divided into *k*2 species apart from their colors *χ*. Each species has distinct fusion and braiding characteristics such as pair measurement restrictions and topological spin as will be seen in section [sec:defectfusion] and [sec:defectexchangebraiding]. Slow evolution of the phase variable *ϕ* → *ϕ* + 2*π*/*k* drives a ground state adiabatically to an excited state due to a level crossing shown in figure [fig:levelcrossing]. The excited state can relax to a different ground state by absorbing or emitting an abelian anyon and thereby driving a *species mutation* of the twofold twist defect. A successive phase winding of a series of twofold twist defects can lead to non-local transportation of abelian anyons without actually moving the defects. This pumping process is a Z*k* analogue of the *U*(1) Thouless charge pump or the Z2 fermion parity pump .\nThe ground state degeneracy of multiple twofold twist defects can be estimated in the thermodynamic limit by counting degree of freedom. Assume the number heptagons and pentagons are identical so that they do not contribute to the net curvature in the Gauss Bonnet theorem. Since there are only one stabilizer per pentagon or heptagon, the ground state degeneracy scales as *k**N* where *N* is the total number of twofold twist defects. And therefore its quantum dimension is given by\n \n*d*[1/2] = *k* \nThis matches the $\\sqrt{k}$ quantum dimension for dislocation twist defects in Z*k*-rotor model or $\\sqrt{2}$ for Kitaev’s toric code since our Hamiltonian is identical to two copies of quantum double Z*k*-model. The difference here is that there are three types of twofold twist defects [1/2]*χ* labeled by colors *χ*, which is the plaquette color for primitive pentagon or heptagon defects. Two identical twofold defects fuse to *k*2 measurable abelian channels while a pair of twofold defects of different colors fuse into a threefold one, [1/2]*χ* × [1/2]*χ* ± 1 = [1/3] or $[\\overline{1/3}]$ as will be discussed further in section [sec:defectfusion].\n### Composite lattice defects\nClassical lattice defects are topologically classified by the holonomy around it such as the Burgers’ vector ${\\bf B}$ of a dislocation or the Frank angle Ω of a disclination. The holonomy is an element in the space group counting the net amount of discrete rotations *r*(Ω) and translations ${\\bf t}$ along a loop around a point defect or a combination of them on a lattice. The space group in our model on a honeycomb lattice is the semi-direct product $P6=C\\_6\\ltimes\\mathcal{L}$, where *C*6 is the 6-fold rotation group and L = Z2 is the translation group of a triangular lattice. The holonomy $(r(\\Omega),{\\bf t})$ is path independent as long as the loop encloses the same defects but it may change according to conjugacy transformation into $(r(\\Omega),{\\bf t}+r(\\Omega)\\cdot{\\bf d}-{\\bf d})$ if the starting and ending point of the loop is displaced by ${\\bf d}$. Thus lattice defects are precisely characterized by the conjugacy classes of holonomy $(r(\\Omega),[{\\bf t}]\\_\\Omega)$, where *r*(Ω) = *e**i*Ω*σ**y* is rotation with Frank angle Ω, a multiple of *π*/3, and $[{\\bf t}]\\_\\Omega$ is the conjugacy class of translation in the quotient group\n \n$$\\frac{\\mathcal{L}}{(r(\\Omega)-1)\\mathcal{L}}=\\left\\{\\begin{array}{\\*{20}c}\\mathcal{L},\\hfill&\\mbox{for $\\Omega=0$}\\hfill\\\\0,\\hfill&\\mbox{for $\\Omega=\\pm60^\\circ$}\\hfill\\\\\\mathbb{Z}\\_3,\\hfill&\\mbox{for $\\Omega=\\pm120^\\circ$}\\hfill\\\\\\mathbb{Z}\\_2\\oplus\\mathbb{Z}\\_2,&\\mbox{for $\\Omega=180^\\circ$}\\hfill\\end{array}\\right.\\label{disclinationclassification}$$ \nIn particular $[{\\bf t}]\\_\\Omega$ differentiates disclinations with the same Frank angle and curvature singularity. The Z3 classification distinguishes 120∘ disclinations centered at octagons such as figure [fig:squareoctagon](a), tetravalent ∘ -vertices such as figure [fig:twistdefects](a) and a tetravalent • -vertices. And therefore tricolorability is violated when $[{\\bf t}]\\_{120^\\circ}$ is non-trivial in Z3. This remains true for any composite defect with an overall ± 120∘ Frank angle. Since the classification for ± 60∘-disclinations is trivial, all such composite defect is holonomically equivalent to a pentagon or heptagon which breaks both tricolorability and bipartite structure. Dislocations are composite defects consist of a collection of disclinations with canceling Frank angles and curvature. The torsional singularity of a dislocation, characterized holonomically by a Burgers’ vector ${\\bf B}$ in L, originates from the spacial separation of its constituent disclinations. For example, each dislocation in figure [fig:Z3dislocation] is a disclination dipole with a tetravalent vertex (Ω = + 120∘) separated from a square (Ω = − 120∘) by half a lattice spacing. And figure [fig:twistdefects](c) shows an overall dislocation from the ± 60-disclination pair. All dislocation preserves the • , ∘ bipartite structure since it always consists of the same number of pentagons and heptagons. It violates tricolorability when its Burgers’ vector ${\\bf B}$ sends a hexagon plaquette to a different color one. Examples of dislocation twist defects include ± 120∘-disclination dipole with unequal translation pieces $[{\\bf t}]\\_{+120^\\circ}\\neq[{\\bf t}]\\_{+120^\\circ}\\in\\mathbb{Z}\\_3$ (i.e. inequivalent defect centers, • , ∘ -vertex or plaquette) such as those in figure [fig:Z3dislocation], and ± 60∘ disclination dipole with different color pentagon and heptagon.\nLet Lʹ be the translation subgroup of L that preserves tricoloration and that L/Lʹ = Z3 classified ± 120∘-disclinations in. Any dislocation with Burgers’ vector in Lʹ and any ± 120∘-disclination with trivial Z3 translation piece $[{\\bf t}]\\_{120^\\circ}$ are *twistless* defects that do not violate tricolorability and bipartite strucutre such as those in figure [fig:squareoctagon]. The *P*6 honeycomb space group symmetry collapses when there is are twistless dislocations and ± 120∘-disclinations in the lattice that break discrete Lʹ translation and *C*3 rotation symmetry respectively. Certain *discreteness* in the space group that represents tricoloration and bipartite structure is however left over. The presence of twistless lattice defects breaks the space group symmetry into the following residue\n \n$$\\frac{C\\_6\\ltimes\\mathcal{L}}{C\\_3\\ltimes\\mathcal{L}'}=\\mathbb{Z}\\_2\\ltimes\\mathbb{Z}\\_3=S\\_3\\label{spacegroupquotient}$$ \nwhich is not surprisingly identical to the group of symmetry between abelian anyons and or gauge fields in. In a trivalent bipartite graph, defects are holonomically classified by the *residue* group *S*3. For instance, a pentagon defect is indistinguishable from a heptagon one as ± 60∘-disclinations are interchangeable by absorbing or releasing a square or octagon disclination. A primitive dislocation is equivalent to a tetravalent vertex by releasing a square disclination and in the long length scale indistinguishable from a pentagon-heptagon dipole separated by one lattice spacing.\nNon-local Wilson algebra\n------------------------\nFigure [fig:anyontwist] shows how an abelian anyon changes type around a twist defect and its Wilson string does not close back to itself after one cycle. A closed Wilson string is either non-local so that it encloses multiple twist defects or wraps around a twist defect multiple times. The former contributes to a ground state degeneracy as it intersects and therefore does not commute with other non-local Wilson strings. The latter can be represented by an abelian phase as it can be shrunk to the point defect and will not intersect with other Wilson strings.\n[fig:defectWilsonloop]\n[fig:defectlocalloop]\nAll non-local Wilson loops are generated by primitive ones shown in figure [fig:defectWilsonloop] (and another one in figure [fig:splittingspaces](g) which will not be used in this section), each encircles two twist defects. A particular arbitrary choice of branch cut is assigned in the figure to keep track of color and sublattice transformation according to figure [fig:anyontwist] and [fig:twistdefects]. Twist defects are further subdivided into species according to the abelian phase eigenvalues of local Wilson loops, Θ*χ* around twofold twist defect and Σ•, Σ∘ around threefold twist defects when *k* is divisible by 3. Each circles a single defect multiple times as shown in figure [fig:defectlocalloop]. Θ*χ* are continuum versions of defect pentagon and heptagon operators, and Σ•, Σ∘ are related to the good quantum numbers associate with each tetravalent and bivalent vertex.\n### Threefold twist defects\nA threefold twist defect is characterized by its 3-fold anyon color twisting described in figure [fig:anyontwist]. It can be generated by primitive dislocation, tetravalent or bivalent lattice disclination, or any composite defect with the same overall color permutation character.\n[fig:Z3loopdefinition]\nWe consider a system of *N* threefold twist defects [1/3] for *N* is some multiple of three so that the system can be closed on a sphere, i.e. the *N* defects fuse to the trivial abelian anyon channel. All non-local Wilson loops are combinations of the one in figure [fig:defectWilsonloop](a) between neighboring defects and we label the one enclosing the *i**t**h* and (*i* + 1)*t**h* defect by A*i*, *i* + 1*χ*, • and A*i*, *i* + 1*χ*, ∘ (see figure [fig:Z3loopdefinition]), where *χ* is the color of the string at the fixed view point (grey square). Due to color redundancy *Y* × *R* × *B* = 1, it is enough to take *χ* = *Y*, *R*. And the final loop A*N* − 1, *N* can be expressed in terms of the previous ones because the system is closed and the Wilson loop enclosing all *N* defects is trivial.\n \n$$\\begin{aligned}\n\\prod\\_{i=1}^{N/3}\\mathcal{A}\\_{3i-2,3i-1}^{\\chi,\\bullet/\\circ}\\left(\\mathcal{A}\\_{3i-1,3i}^{\\chi-1,\\bullet/\\circ}\\right)^\\dagger=\\vcenter{\\hbox{\\includegraphics[width=0.8in]{outsideloop.pdf}}}=1\\label{Z3closeness}\\end{aligned}$$ \nfor [*i*]3 ≡ *i* modulo 3 cyclically permutes the color *χ*.\n[fig:Z3intersection]\nSince • -Wilson loops mutually commute, the 2(*N* − 2) • -loops form a maximal commuting set of Wilson operators. The Wilson algebra\n \n$$\\mathcal{A}\\_{i,i+1}^{\\chi,\\bullet}\\mathcal{A}\\_{j,j+1}^{\\chi',\\circ}=e^{i\\frac{2\\pi}{k}\\langle\\mathcal{A}\\_{i,i+1}^\\chi,\\mathcal{A}\\_{j,j+1}^{\\chi'}\\rangle}\\mathcal{A}\\_{j,j+1}^{\\chi',\\circ}\\mathcal{A}\\_{i,i+1}^{\\chi,\\bullet}\\label{Z3Wilsonalgebra}$$ \nis determined by the symmetric pairing matrix I = ⟨ \\* , \\* ⟩ between colored strings. It is non-zero only when ∣*i* − *j*∣ ≤ 1 and can be evaluated by local intersection number according to figure [fig:intersection]. All non-trivial intersection are shown in figure [fig:Z3intersection]. Note that the result is independent from the arbitrary choice of branch cut. The intersection form for *N* threefold twist defects is given by the 2(*N* − 2) × 2(*N* − 2) symmetric matrix I\n \n$$\\langle\\mathcal{A}\\_{i,i+1},\\mathcal{A}\\_{j,j+1}\\rangle=\\left(\\begin{array}{\\*{20}c}I\\_0&I\\_1&0&0&\\ldots&0\\\\I\\_1^T&I\\_0&I\\_1&0&\\ldots&0\\\\0&I\\_1^T&I\\_0&I\\_1&\\ldots&0\\\\0&0&I\\_1^T&I\\_0&\\ldots&0\\\\\\vdots&\\vdots&\\vdots&\\ddots&\\ddots&\\vdots\\\\0&0&0&0&\\ldots&I\\_0\\end{array}\\right)\\label{Z3intersectionmatrix}$$ \n \n$$I\\_0=\\left(\\begin{array}{\\*{20}c}-2&1\\\\1&-2\\end{array}\\right),\\quad I\\_1=I\\_0\\Lambda\\_3=\\left(\\begin{array}{\\*{20}c}1&1\\\\-2&1\\end{array}\\right)\\label{I0I1}$$ \nwhere the (*N* − 2) rows and columns of the intersection matrix is enumerated according to the loops {A*i*, *i* + 1 : *i* = 1, …, *N* − 2}, its 2 × 2 matrix entries act on color degree of freedom *χ* = *Y*, *R*, and $\\Lambda\\_3=\\left(\\begin{array}{\\*{20}c}0&-1\\\\1&-1\\end{array}\\right)$ is cyclic color permutation. *I*0 comes from the intersection between Wilson loops about the same defect pair (first two diagrams in figure [fig:Z3intersection]) while *I*1 comes from the intersection between Wilson loops about adjacent defect pairs (last three diagrams in figure [fig:Z3intersection]).\nWe notice that the Wilson algebra is symmetric under cyclic color permutation Λ3± 1 : *χ* → *χ* ± 1 but is not invariant under color and sublattice transposition Λ*B* : *Y*• / ∘ ↔ *R*∘ / •. This can be understood by observing that the Wilson loop in figure [fig:defectWilsonloop](a) violates transposition explicitly. Color label *χ* of the Wilson operator A*i*, *i* + 1*χ*, • / ∘ in figure [fig:Z3loopdefinition] is defined with respect to a base point, and a change of base point does not commute, hence inconsistent, with all transpositions Λ*Y*, Λ*R*, Λ*B*. In fact color and sublattice transposition switches a threefold defect into its anti-partner\n \n$$\\Lambda\\_B:[1/3]\\leftrightarrow[\\overline{1/3}]$$ \nand the intersection form is *covariant* under Λ*B*\n \n$$\\Lambda\\_B^T\\cdot\\mathbb{I}\\_{[1/3]}\\cdot\\Lambda\\_B=-\\mathbb{I}\\_{[\\overline{1/3}]}$$ \nwhere Λ*B* = *σ**x* acts on the colors *χ* = *Y*, *R* and $\\mathbb{I}\\_{[\\overline{1/3}]}$ is the intersection matrix for *N* anti-threefold defects. In the microscopic lattice level, transposition Λ*B* originates from inversion which interchanges the • , ∘ sublattice type and switches $[1/3]\\leftrightarrow[\\overline{1/3}]$ according to table [tab:twistdefects] for primitive disclination twist defects. This can also be understood in the continuum by seeing transposition interchanges the abelian anyon twisting between [1/3] and $[\\overline{1/3}]$ in figure [fig:anyontwist](a) and (c).\nFor later convenience, we adapt the multi-exponents notation ${\\bf m}=(m\\_1,m\\_2,\\ldots,m\\_{2N-5},m\\_{2N-4})$ for Wilson operator product\n \n$$\\left(\\mathcal{A}^\\bullet\\right)^{\\bf m}=\\prod\\_{i=1}^{N-2}\\left(\\mathcal{A}\\_{i,i+1}^{Y,\\bullet}\\right)^{m\\_{2i-1}}\\left(\\mathcal{A}\\_{i,i+1}^{R,\\bullet}\\right)^{m\\_{2i}}$$ \nand similarly for the dual ones $\\left(\\mathcal{A}^\\circ\\right)^{\\bf m}$, where *m**j* are integers modulo *k*.\nWhen *k* is *not* divisible by 3, starting with the particular ground state ∣0⟩• in eq. fixed by the • -Wilson operators $\\left(\\mathcal{A}^\\bullet\\right)^{\\bf m}|0\\rangle\\_\\bullet=|0\\rangle\\_\\bullet$, the dual ∘ -Wilson operators generate all ground states by product combinations\n \n$$|{\\bf m}\\rangle\\_\\bullet=\\left(\\mathcal{A}^\\circ\\right)^{\\bf m}|0\\rangle\\_\\bullet\\label{GSZ3}$$ \nThese are simultaneous eigenstates for the maximally commuting set of • -Wilson operators. The matrix elements of A• and A∘ with respect to this basis are given by\n \n$$\\begin{aligned}\n\\langle{\\bf m}'|\\left(\\mathcal{A}^\\bullet\\right)^{\\bf n}|{\\bf m}\\rangle&=e^{i\\frac{2\\pi}{k}{\\bf n}^T\\mathbb{I}{\\bf m}}\\delta\\_{{\\bf m}',{\\bf m}}\\label{Z3Wilsoneigenvalues}\\\\\\langle{\\bf m}'|\\left(\\mathcal{A}^\\circ\\right)^{\\bf n}|{\\bf m}\\rangle&=\\delta\\_{{\\bf m}',{\\bf m}+{\\bf n}}\\end{aligned}$$ \nwhere I is the intersection form in.\n[fig:3foldsublattice]\nWe notice that 3I− 1 has integer entries, and therefore the intersection matrix I is invertible (with Z*k* entries) when there is an integer *s* with 3*s* ≡ 1 mod *k*, in other words *k* is not divisible by 3. This means different ground states $|{\\bf m}\\rangle\\_\\bullet$ in can be distinguished by their set of eigenvalues for the • -Wilson operators, and thus forms a complete orthonormal basis for the ground state. Alternatively one can label the ground states using • -Wilson operator eigenvalues ${\\boldsymbol\\alpha}=(y\\_1,r\\_1,\\ldots,y\\_{N-2},r\\_{N-2})$ by\n \n$$|{\\boldsymbol\\alpha}\\rangle\\_\\bullet=\\left(\\mathcal{A}^\\circ\\right)^{\\bf m}|0\\rangle\\_\\bullet,\\quad{\\bf m}=\\mathbb{I}^{-1}{\\boldsymbol\\alpha}\\label{Z3groundstatealpha}$$ \nwhere *y**i*, *r**i* are integers modulo *k*, so that the matrix elements of Wilson operators are\n \n$$\\begin{aligned}\n\\langle{\\boldsymbol\\alpha}'|\\left(\\mathcal{A}^{\\bullet}\\right)^{\\bf n}|{\\boldsymbol\\alpha}\\rangle&=e^{i\\frac{2\\pi}{k}{\\bf n}^T{\\boldsymbol\\alpha}}\\delta\\_{{\\boldsymbol\\alpha}',{\\boldsymbol\\alpha}}\\label{Z3Wilsonmatix1}\\\\\\langle{\\boldsymbol\\alpha}'|\\left(\\mathcal{A}^{\\circ}\\right)^{\\bf n}|{\\boldsymbol\\alpha}\\rangle&=\\delta\\_{{\\boldsymbol\\alpha}',\\mathbb{I}{\\bf n}+{\\boldsymbol\\alpha}}\\label{Z3Wilsonmatix2}\\end{aligned}$$ \nOne can put the ground states $|{\\boldsymbol\\alpha}\\rangle\\_\\bullet$ on the Cartesian product of *N* − 2 periodic 2D lattices shown in figure [fig:3foldsublattice](a). Each lattice point ${\\boldsymbol\\alpha}\\_i=y\\_i{\\bf e}\\_1+r\\_i{\\bf e}\\_2$ on the integer mod *k* triangular lattice represents the eigenvalues *e*2*π**i**y**i*/*k*, *e*2*π**i**r**i*/*k* for A*i*, *i* + 1*Y*, •, A*i*, *i* + 1*R*, •. The ground state degeneracy (G.S.D.) for a closed spherical system with *N* threefold twist defects is given by\n \n*G*.*S*.*D*. = *k*2(*N* − 2) \nwhich matches the quantum dimension *d*[1/3] = *k*2 predicted by counting lattice degree of freedom in eq..\nWhen *k* is divisible by 3, there is a non-trivial center that commute with all Wilson operators. It is generated by\n \n$$\\begin{aligned}\n\\Sigma\\_{i,i+1}^{\\bullet/\\circ}&=\\left(\\mathcal{A}\\_{i,i+1}^{Y,\\bullet/\\circ}\\right)^{-k/3}\\left(\\mathcal{A}\\_{i,i+1}^{R,\\bullet/\\circ}\\right)^{k/3}\\nonumber\\\\&=\\Sigma\\_i^{\\bullet/\\circ}\\left(\\Sigma\\_{i+1}^{\\bullet/\\circ}\\right)^{-1}=e^{i\\frac{2\\pi}{3}\\left(s^{\\bullet/\\circ}\\_i-s^{\\bullet/\\circ}\\_{i+1}\\right)}\\in\\mathbb{Z}\\_3\\label{Sigmaii+1}\\end{aligned}$$ \nwhere the eigenvalues of the local Wilson operator Σ• = *e*2*π**i**s*•/3 and Σ∘ = *e*2*π**i**s*∘/3 in figure [fig:defectlocalloop](b) distinguishes the nine species ${\\bf s}=(s\\_\\bullet,s\\_\\circ)\\in\\mathbb{Z}\\_3\\oplus\\mathbb{Z}\\_3$ for each threefold defects $[1/3]\\_{\\bf s}$. It can be shown, up to plaquette stabilizers, that the local Wilson loop Σ*i*• / ∘ around a tetravalent (or bivalent) • / ∘ -vertex is Σ*u*4− 1 (resp. Σ*u*2) in eq. or 1 around a ∘ / • -vertex. And therefore the Z3-phases for the small loops Σ*i*• / ∘ are determined by local defect Hamiltonian. The species are interchangeable by tuning the defect phase *ϕ* → *ϕ* ± 2*π*/3 in that drive the system into an excited state and locally relax back to a ground state by emitting or absorbing an abelian anyon. This process changes the Z3-values for Σ• and Σ∘ and thus switches the species. This extra phase degree of freedom provides a possibility for non-local transport of abelian anyon along a series of coupled [1/3] defects that is a unique feature only when *k* is a multiple of 3.\nThe ground states $|{\\bf m}\\rangle\\_\\bullet$ in do not form an orthonormal set as they overcount the ground state degeneracy. In fact the allowed eigenvalues $e^{i\\frac{2\\pi}{k}{\\bf n}^T\\boldsymbol\\alpha}$ for $(\\mathcal{A}^\\bullet)^{\\bf n}$ are restricted by the species *s*• / ∘ in. The ground states $|\\boldsymbol\\alpha\\rangle$ now form a sublattice in figure [fig:3foldsublattice](b), which contains a third of lattice points in the origin triangular lattices. The ground state degeneracy (G.S.D.) is thus reduced by 3 for each defects.\n \n$$G.S.D.=\\left(\\frac{k^2}{3}\\right)^{N-2}\\label{Z33dimensionGSD}$$ \nThis matches the quantum dimension *d*[1/3] = *k*2/3 in predicted by microscopic lattice derivation. A complete description for the Wilson structure of threefold defects when 3 divides *k* can be found in appendix [sec:Z33Wilsonalgebraappendix].\n### Twofold twist defects\nA twofold twist defect is characterized by its twofold twisting of abelian anyons circling around as shown in figure [fig:anyontwist]. It can be realized as pentagon or heptagon defects in the lattice, or any composite defects that violate tricolorability and bipartite structure. Since there are three transpositions Λ*Y*, Λ*R*, Λ*B* in the permutation group *S*3, twofold defects are classified into three types according to colors, [1/2]*Y*, [1/2]*R* and [1/2]*B*. We consider a closed collection of *N* twofold twist defects for some even *N*. We assume for simplicity that all defects are of the same color type, say blue (*B*), and they fuse to the overall vacuum channel and the system is compactified on a sphere. A more general discussion on coexisting multi-type defects will be given in the upcomming subsection.\nThe twofold defect [1/2]*B* is further subdivided into *k*2 species according to eigenvalues of two *local* measurements at the defect, the small Wilson loops Θ*χ* (see figure [fig:defectlocalloop](a) and [fig:Z2loopdefinition]) that circles the defect twice for *χ* = *Y*, *R*, *B*. For primitive pentagon or heptagon defects, they are identical (up to a Z*k* phase) to the defect plaquette operator *P̂*5, 7 in eq., and therefore their eigenvalues *w**l**χ*, *w* = *e*2*π**i*/*k*, is fixed by the local defect Hamiltonian. Similar to *P̂*5, 7, the self-intersection of the small Wilson loop Θ*χ* causes a minus sign for even *k* so that (Θ*χ*)*k* = ( − 1)*k* − 1 for *χ* not equal *B*, the color of the twofold defect. This means that *l**χ* is an integer modulo *k* for *k* odd but a half-integer modulo *k* when *k* is even and *χ* ≠ *B*. Color redundancy *Y* × *R* × *B* = 1 requires *l**Y* + *l**R* + *l**B* = 0. The two independent phases ${\\bf l}=(l\\_Y,l\\_R)$ in Z*k* ⊕ Z*k* for *k* odd or in (Z + 1/2)*k* ⊕ (Z + 1/2)*k* for *k* even form the species label for $[1/2]\\_{B,{\\bf l}}$.\n[fig:Z2loopdefinition]\n[fig:doublelooppair]\nThe prototype of a Wilson loop is depicted in figure [fig:defectWilsonloop](c) and [fig:Z2loopdefinition]. We denote the Wilson loop enclosing the *i**t**h* and (*i* + 1)*t**h* defect by Z*i*, *i* + 1*χ* according to the string color *χ* = *Y*, *R*, *B* observed at the base point (grey square). We can assume the loop always originates as a • -sublattice type string from the base point because different sublattice types are interchangeable up to a phase as shown in figure [fig:defectWilsonloop](c). The phase is fixed precisely by the local Wilson loops Θ*i**χ* = *w**l**i**χ* and Θ*i* + 1*χ* = *w**l**i* + 1*χ* and an unlinking procedure illustrated in figure [fig:doublelooppair] and is shown in the right side of the equation below.\n \n$$\\mathcal{Z}\\_{i,i+1}^{\\chi',\\circ}\\mathcal{Z}\\_{i,i+1}^{\\chi,\\bullet}=e^{i\\frac{2\\pi}{k}\\left(\\delta^\\chi\\_{\\chi'+1}-\\delta^\\chi\\_{\\chi'-1}+l^\\chi\\_i+l^\\chi\\_{i+1}\\right)}\\label{Zloopinsideout}$$ \nwhere the color *χ*, *χ*ʹ = *Y*, *R*, *B* are indexed by 0, 1, 2 mod 3, and (*χ*ʹ, ∘ ) and (*χ*, • ) are related by the transposition that characterizes the twisting of twofold defects defined in figure [fig:anyontwist].\n[fig:Z2intersection]\nAll Wilson operators can be generated by the prototype Z*i*, *i* + 1*χ* for *χ* = *Y*, *R* (recall *B* = *Y*− 1*R*− 1) and *i* = 1, …, *N* − 2 since Z*N* − 1, *N**χ* can be generated by the compactification relation\n \n$$\\begin{aligned}\n\\prod\\_{j=1}^{N/2}\\mathcal{Z}\\_{2j-1,2j}^\\chi=\\vcenter{\\hbox{\\includegraphics[width=1in]{outsideloopp.pdf}}}=1\\label{Z2closeness}\\end{aligned}$$ \n{Z2*j* − 1, 2*j**χ*}*j* = 1, …, *N*/2 − 1 and {Z2*j*, 2*j* + 1*χ*}*j* = 1, …, *N*/2 − 1 forms two maximal commuting set of Wilson operators because loops in the same set do not intersect. The Wilson algebra is characterized by the intersection relation between the two sets.\n \n$$\\mathcal{Z}\\_{2i-1,2i}^\\chi\\mathcal{Z}\\_{2j,2j+1}^{\\chi'}=e^{i\\frac{2\\pi}{k}\\langle\\mathcal{Z}\\_{2i-1,2i}^\\chi,\\mathcal{Z}\\_{2j,2j+1}^{\\chi'}\\rangle}\\mathcal{Z}\\_{2j,2j+1}^{\\chi'}\\mathcal{Z}\\_{2i-1,2i}^\\chi$$ \nwhere the pairing I = ⟨ \\* , \\* ⟩ can be deduced by counting intersections between adjacent loops in figure [fig:Z2intersection] according to the rules set by figure [fig:intersection], and is given by the following (*N* − 2) × (*N* − 2) invertible matrix I.\n \nI = ⟨Z2*i* − 1, 2*i**χ*, Z2*j*, 2*j* + 1*χ*ʹ⟩ = *J* ⊗ J0 \n \n$$J\\_Y=\\left(\\begin{array}{\\*{20}c}0&1\\\\1&-1\\end{array}\\right),\\quad J\\_R=\\left(\\begin{array}{\\*{20}c}1&-1\\\\-1&0\\end{array}\\right),\\quad J\\_B=\\left(\\begin{array}{\\*{20}c}-1&0\\\\0&1\\end{array}\\right)\\label{Z2JYRB}$$ \n \n$$\\mathbb{J}\\_0=\\left(\\begin{array}{\\*{20}c}1&0&0&\\ldots&0\\\\-1&1&0&\\ldots&0\\\\0&-1&1&\\ldots&0\\\\\\vdots&\\vdots&\\vdots&\\ddots&\\vdots\\\\0&0&0&\\ldots&1\\\\\\end{array}\\right)\\label{Z2J0}$$ \nwhere the (*N* − 2)/2 rows and columns of J0 in run over the maximal commuting sets of Wilson loops {Z2*i* − 1, 2*i*}*i* = 1, …, *N*/2 − 1 and {Z2*j*, 2*j* + 1}*i* = 1, …, *N*/2 − 1 respectively, the 2 × 2 entries *J* = *J**Y*, *J**R*, *J**B* are the corresponding matrices for the three types of twofold defects [1/2]*Y*, [1/2]*R*, [1/2]*B* that act on *χ*-degree of freedom of Z*χ*, for *χ* = *Y*, *R*. Contrary to threefold defects, the Wilson algebra specified by the intersection form preserves transposition but not symmetric under color permutation. It is evident from table [tab:twistdefects] and figure [fig:anyontwist] that twofold defects change type upon color permutation\n \nΛ3 : [1/2]*χ* → [1/2]*χ* + 1 \nwhere *χ* = 0, 1, 2 mod 3 index the colors *Y*, *R*, *B*, and the intersection matrix is Λ3-*covariant*\n \nΛ3*T* ⋅ I[1/2]*χ* ⋅ Λ3 = I[1/2]*χ* − 1 \nwhere $\\Lambda\\_3=\\left(\\begin{array}{\\*{20}c}0&-1\\\\1&-1\\end{array}\\right)$ acts on the color degree of freedom *χ* = *Y*, *R*.\nGiven an arbitrary choice of branch cut that specified the sublattice type • , ∘ of all vertices, one can write down a particular ground state (up to a normalization constant)\n \n∣0⟩ ∝ ∏*i*[∑*r* = 0*k* − 1*w*− *r**p**i**D̂**i**r*]∏*P*[∑*r* = 0*k* − 1*P̂**r*]∣*σ**v*• = *τ**v*∘ = 1⟩ \nwhere *D̂**i* = *P̂*5, 7 are the pentagon or heptagon plaquette operators in eq. at the *i**t**h* defect, and *P̂* = *P̂*•, *P̂*∘ runs over all other even sided plaquettes. Here *p**i* are integer (half-integer) mod *k* for *k* odd (even) that specify the ground state eigenvalues *D̂**i*∣0⟩ = *w**p**i*∣0⟩ determined by local defect Hamiltonian. All ground states can be generated by Wilson operators\n \n$$|{\\bf m}\\rangle=\\left(\\mathcal{Z}\\_{even}\\right)^{\\bf m}|0\\rangle$$ \n \n$$\\begin{aligned}\n\\left(\\mathcal{Z}\\_{even}\\right)^{\\bf m}&=\\prod\\_{j=1}^{N/2-1}\\left(\\mathcal{Z}\\_{2j,2j+1}^Y\\right)^{m\\_{2j-1}}\\left(\\mathcal{Z}\\_{2j,2j+1}^R\\right)^{m\\_{2j}}\\\\\\left(\\mathcal{Z}\\_{odd}\\right)^{\\bf m}&=\\prod\\_{j=1}^{N/2-1}\\left(\\mathcal{Z}\\_{2j-1,2j}^Y\\right)^{m\\_{2j-1}}\\left(\\mathcal{Z}\\_{2j-1,2j}^R\\right)^{m\\_{2j}}\\end{aligned}$$ \nwhere ${\\bf m}=(m\\_1,\\ldots,m\\_{N-2})$ are the multi-exponents for Wilson loops. Assuming the branch cuts are chosen to avoid cutting across the odd Wilson loops Z2*j* − 1, 2*j* such as those in figure [fig:Z2intersection], the odd Wilson operators fix the particular ground state $\\left(\\mathcal{Z}\\_{odd}\\right)^{\\bf m}|0\\rangle=|0\\rangle$. The matrix elements of observables are given by\n \n$$\\begin{aligned}\n\\langle{\\bf m}'|\\left(\\mathcal{Z}\\_{odd}\\right)^{\\bf n}|{\\bf m}\\rangle&=e^{i\\frac{2\\pi}{k}{\\bf n}^T\\mathbb{I}{\\bf m}}\\delta\\_{{\\bf m}',{\\bf m}}\\label{Z2Zoddeigenvalues}\\\\\\langle{\\bf m}'|\\left(\\mathcal{Z}\\_{even}\\right)^{\\bf n}|{\\bf m}\\rangle&=\\delta\\_{{\\bf m}',{\\bf m}+{\\bf n}}\\end{aligned}$$ \nSince I is invertible, ground states $|{\\bf m}\\rangle$ are completely distinguished by its eigenvalues with respect to Z*o**d**d* and therefore form an orthonormal basis for the ground state Hilbert space. The ground state degeneracy (G.S.D.) is given by\n \n*G*.*S*.*D*. = *k**N* − 2 \nand matches the quantum dimension *d*[1/2] = *k* predicted in by counting lattice degree of freedom.\n[fig:Z2fusion]\nIt would be more convenient for later considerations in fusion and braiding if the ground states are labeled by the abelian anyon fusion channels $[{\\bf a}\\_j]=[{\\bf a}\\_\\bullet^j,{\\bf a}\\_\\circ^j]$ of the *j**t**h* pair of twofold defects (see figure [fig:Z2fusion]), where ${\\bf a}\\_\\bullet^j=y\\_1^j{\\bf e}\\_Y+r\\_1^j{\\bf e}\\_R$ and ${\\bf a}\\_\\circ^j=y\\_2^j{\\bf f}^Y+r\\_2^j{\\bf f}^R$ are discrete vectors in the abelian anyon lattice (figure [fig:abeliananyonlattice]). The • -part of the anyon channels ${\\bf a}\\_\\bullet=(y\\_1^1,r\\_1^1,\\ldots,y\\_1^{N/2-1},r\\_1^{N/2-1})$ can be read off from for the ground state $|{\\bf m}\\rangle$.\n \n$${\\bf a}\\_\\bullet=-i\\sigma\\_yJ\\otimes\\mathbb{J}\\_0{\\bf m}$$ \nEq. ensures a constraint on the ∘ -part of the abelian fusion channels ${\\bf a}\\_\\circ=(y\\_2^1,r\\_2^1,\\ldots,y\\_2^{N/2-1},r\\_2^{N/2-1})$ so that\n \n$${\\bf a}\\_\\circ=\\mathbb{J}\\_0{\\bf m}+{\\boldsymbol\\theta}\\_\\circ$$ \nwhere ${\\boldsymbol\\theta}\\_\\circ=(\\boldsymbol\\theta\\_1,\\ldots,\\boldsymbol\\theta\\_{N/2-1})$, $\\boldsymbol\\theta\\_j=(\\theta\\_j^Y,\\theta\\_j^R)$, depends only on defect species ${\\bf l}\\_i=(l^Y\\_i,l^R\\_i)$, for Θ*i**χ* = *w**l**i**χ*.\n \n$$\\boldsymbol\\theta\\_j={\\bf f}-J^{-1}({\\bf l}\\_{2j-1}+{\\bf l}\\_{2j})$$ \nwhere ${\\bf f}={\\bf f}^Y,{\\bf f}^R,{\\bf f}^B=(1,0),(0,1),(-1,-1)$ are basis vectors in the ∘ -anyon lattice in figure [fig:abeliananyonlattice] and *J* = *J**Y*, *J**R*, *J**B* are matrices in determined by the color of twofold defects [1/2]*Y*, [1/2]*R*, [1/2]*B*.\nUnder the fusion channel basis, the matrix elements for the Wilson operators are\n \n$$\\begin{aligned}\n\\langle{\\bf a}'|\\left(\\mathcal{Z}\\_{odd}\\right)^{\\bf n}|{\\bf a}\\rangle&=e^{i\\frac{2\\pi}{k}{\\bf n}^T i\\sigma\\_y{\\bf a}\\_\\bullet}\\delta\\_{{\\bf m}',{\\bf m}}\\label{Z2groundstatea}\\\\\\langle{\\bf a}'|\\left(\\mathcal{Z}\\_{even}\\right)^{\\bf n}|{\\bf a}\\rangle&=\\delta\\_{{\\bf a}'\\_\\bullet,{\\bf a}\\_\\bullet-i\\sigma\\_y\\mathbb{I}{\\bf n}}\\end{aligned}$$ \nwhere the abelian fusion channels ${\\bf a}=({\\bf a}\\_\\bullet,{\\bf a}\\_\\circ)$ are constrained by\n \n$${\\bf a}\\_\\circ=J^{-1}i\\sigma\\_y{\\bf a}\\_\\bullet+\\boldsymbol\\theta\\_\\circ\\label{Z2fusionconstraint}$$ \nNotice that the vacuum fusion channel may not be allowed by. This is because the (2*j* − 1)*t**h* defect is not the anti-partner of the (2*j*)*t**h* in general unless the right hand side of eq. or ${\\boldsymbol\\theta}\\_\\circ$ is trivial, i.e. the species labels satisfy\n \n$${\\bf l}\\_{2j-1}+{\\bf l}\\_{2j}=J{\\bf f}$$ \nWe end this subsection with a cautionary remark on the phase variables of local defect Hamiltonians and in a closed system. Similar to Z2-fermion parity in a closed electronic system, there is a Z3 ⊕ Z3-*anyon parity* in a closed system of threefold defects for *k* divisible by 3 and a Z*k* ⊕ Z*k*-*anyon parity* in a closed system of twofold defects. They are good quantum numbers that imply global restrictions. The closeness condition requires the species labels *s**i* = (*s**i*•, *s**i*∘) of the threefold twist defects to satisfy\n \n$$\\prod\\_{i=1}^N\\Sigma^{\\bullet/\\circ}\\_i=e^{i\\frac{2\\pi}{3}\\sum\\_{i=1}^Ns\\_i^{\\bullet/\\circ}}=1\\label{Z3speciescloseness}$$ \nfor *k* divisible by 3. While for twofold defects, similarly requires the species labels ${\\bf l}\\_i=(l^Y\\_i,l^R\\_i)$ to obey\n \n$$\\sum\\_{i=1}^N{\\bf l}\\_i=\\frac{N}{2}J{\\bf f}\\label{Z2speciescloseness}$$ \nwhere ${\\bf f}={\\bf f}^Y,{\\bf f}^R,{\\bf f}^B$ depending on the color type of [1/2]*Y*, [1/2]*R*, [1/2]*B*. The species ${\\bf s}\\_i$ and ${\\bf l}\\_i$ are however completely determined by phase variables *ϕ**i* of *local* defect Hamiltonians and in the lattice level, and there are no reasons for the local phases to be restricted by the global closeness conditions or. When the local defect phase variables *ϕ**i* are incompatible with or, there is a topological obstruction in obtaining the absolute lowest energy state. For instance the state in eq. would be identically zero, ∣0⟩ = 0. The system would be forced to have local excitations where energy is not locally minimized. There will be two phases depending on the relative magnitude between *J*\\*, the energy scale of defect Hamiltonians and, and *J*, the underlying energy scale of the original model. For *J*\\* < < *J*, excitations would be bounded and localized at defect centers, and eq., would be effectively restored. And therefore the Wilson algebras and ground state degeneracies described before would still persist. If *J*\\* > > *J*, there will be delocalized abelian anyon excitations causing an infinite number of ground states. We consider only the former scenario.\nWord presentation of Wilson algebra\n-----------------------------------\n[fig:multidefects]\nWe describe a presentation of the group of Wilson algebra in a system with multi-type twist defects. A defect type is defined by its *S*3-twisting action on abelian anyon labels (see figure [fig:anyontwist]) when an abelian anyon is dragged along a path around the defect. Examples of these paths are shown by the viewing curves *λ̂**i* in figure [fig:multidefects], each encloses one and only one defect. An abelian anyon $[{\\bf a}]=(Y\\_\\bullet)^{y\\_1}(R\\_\\bullet)^{r\\_1}(Y\\_\\circ)^{y\\_2}(R\\_\\circ)^{r\\_2}$ will be twisted into $\\Lambda\\_i\\cdot[{\\bf a}]$ according to and if it moves along *λ̂**i*, where Λ*i* is an element in *S*3 that distinguishes the type of the *i**t**h* defect by [1/3], $[\\overline{1/3}]$ or [1/2]*χ*. The defect type or Λ*i* may depend on the viewing curve *λ̂**i* for a multi-type defect system that contains twofold defects. Defects can be arranged and ordered in series as shown in figure [fig:multidefects]. A re-ordering of viewing curves will in general alter the defect type. Upon switching the viewing order of the *i**t**h* and (*i* + 1)*t**h* defects, their *S*3-classifications change according to conjugation\n \n(Λ*i*, Λ*i* + 1) ↦ (Λʹ*i*, Λʹ*i* + 1) = (Λ*i*− 1Λ*i* + 1Λ*i*, Λ*i*) \nwhere this is demonstrated by redefining the viewing curves for the 6*t**h* and 7*t**h* defects in figure [fig:multidefects]. This can be understood in the microscopic lattice level by observing that the • , ∘ -sublattice type of a threefold defect center or the *Y**R**B*-plaquette color of a twofold defect center depends on the viewing path in the presence of other defects. Twist defects are therefore coarsely classified by the conjugacy classes of *S*3, which include the class of threefold defects $[1/3],[\\overline{1/3}]$ and the class of twofold defects [1/2]*χ*. However we do not consider these conjugacy classes as superselection sectors as the *S*3-symmetry is not gauged, unlike in Ref .\nWe denote the *alphabet* $[\\hat{\\lambda}\\_i]\\_{\\bf a}$ to be an open Wilson string of dragging an abelian anyon ${\\bf a}=[{\\bf a}\\_\\bullet,{\\bf a}\\_\\circ]$ along the viewing curve *λ̂**i* starting from the base point (grey square in figure [fig:multidefects]). The ordering of ${\\bf a}\\_\\bullet$ and ${\\bf a}\\_\\circ$ will only affect the Wilson operator by an over phase due to self-intersection, and if necessary, we will adapt the convention that the ${\\bf a}\\_\\bullet$-string acts before and sits below the ${\\bf a}\\_\\circ$ one. The Wilson algebra of the defect system consists of closed Wilson loops presented in series of the alphabets\n \n$$\\begin{aligned}\nW(\\mathcal{C})=\\left[\\hat{\\lambda}\\_{i\\_m}\\ldots\\hat{\\lambda}\\_{i\\_2}\\hat{\\lambda}\\_{i\\_1}\\right]\\_{\\bf a}=\\prod\\_{r=1}^m\\left[\\hat{\\lambda}\\_{i\\_r}\\right]\\_{\\prod\\_{l=1}^{r-1}\\Lambda\\_{i\\_l}\\cdot{\\bf a}}\\label{alphabetpresentation}\\end{aligned}$$ \nwhere C is the loop of the ordered composition *λ̂**i**m* \\* … \\* *λ̂**i*1 and the Wilson string begins at the fixed base point as the abelian anyon ${\\bf a}$, which moves and transforms along C. In order for *W*(C) to be closed, the abelian anyon has to close back onto itself and the *S*3-transformations have to satisfy the closeness relation\n \n$$\\left(\\Lambda\\_{i\\_m}\\ldots\\Lambda\\_{i\\_2}\\Lambda\\_{i\\_1}\\right)\\cdot{\\bf a}={\\bf a}\\label{ATcloseness}$$ \nThe alphabets inside the bracket in the middle of has a fixed ordering. Interchanging the order of the alphabets in general will give an entirely different Wilson string that might not even be closed. The product order on the far right of eq. however will only contribute an overall Z*k* phase to *W*(C), and will not affect its intersection with other Wilson operator. Abelian anyon fusion imply the simplification\n \n$$\\left[\\hat{\\lambda}\\_i\\right]\\_{\\bf a}\\left[\\hat{\\lambda}\\_i\\right]\\_{\\bf b}=\\left[\\hat{\\lambda}\\_i\\right]\\_{{\\bf a}+{\\bf b}}$$ \nFor instances, the Z*k*-torsion $k{\\bf a}=0$ implies *λ̂**i**k* × \\scriptsize ord = 1 for ord = 2, 3 is the order of twofold and threefold defects; the tricolor redundancy requires *λ̂**i*3 = 1 if the *i**t**h* defect is a threefold defect, and *λ̂**i**λ̂**j**λ̂**i**λ̂**j**λ̂**i**λ̂**j* = 1 if the *i**t**h* and *j**t**h* defects are twofold defects of different colors. If the system of *N* defects is closed on a compact surface without boundary, we have\n \n*λ̂**N*…*λ̂*2*λ̂*1 = 1 \nIgnoring the overall Z*k*-phase, are of the form\n \n$$W[\\{{\\bf a}\\_i\\}]=\\prod\\_{i=1}^N\\left[\\hat{\\lambda}\\_i\\right]\\_{{\\bf a}\\_i},\\quad\\mbox{for}\\quad\\sum\\_{i=1}^N{\\bf a}\\_i=\\sum\\_{i=1}^N\\Lambda\\_i{\\bf a}\\_i$$ \nThe alphabetic presentation of the Wilson loops considered in the previous subsections are summarized in table [tab:prototypesalphabets].\n[ht]\ncl Wilson loop prototypes & Word presentation \n A*i*, *i* + 1*χ*, • & $\\left[\\hat{\\lambda}\\_{i+1}\\right]\\_{-{\\bf e}\\_\\chi}\\left[\\hat{\\lambda}\\_i\\right]\\_{{\\bf e}\\_\\chi}$ \nA*i*, *i* + 1*χ*, ∘ & $\\left[\\hat{\\lambda}\\_{i+1}\\right]\\_{-{\\bf f}^\\chi}\\left[\\hat{\\lambda}\\_i\\right]\\_{{\\bf f}^\\chi}$ \nΣ*i*• / ∘ & $\\left[\\hat{\\lambda}\\_i\\right]\\_{\\overline{\\boldsymbol\\kappa}\\_{\\bullet/\\circ}}$ \n Z*i*, *i* + 1*χ* & $\\left[\\hat{\\lambda}\\_{i+1}\\hat{\\lambda}\\_i\\right]\\_{{\\bf e}\\_\\chi}$ \nΘ*i**χ* & $\\left[\\hat{\\lambda}\\_i\\hat{\\lambda}\\_i\\right]\\_{{\\bf e}\\_\\chi}$ \n[tab:prototypesalphabets]\nBy a slight deformation, one can assume all intersections between two Wilson operators are at the fixed base point. The algebraic relation of Wilson operators are given by\n \n$$W[\\{{\\bf a}\\_i\\}]W[\\{{\\bf b}\\_i\\}]=e^{i\\frac{2\\pi}{k}\\langle\\{{\\bf a}\\_i\\},\\{{\\bf b}\\_i\\}\\rangle}W[\\{{\\bf b}\\_i\\}]W[\\{{\\bf a}\\_i\\}]\\label{multidefectalgebra}$$ \nwhere the intersection form I = ⟨ \\* , \\* ⟩ is\n \n$$\\langle\\{{\\bf a}\\_i\\},\\{{\\bf b}\\_i\\}\\rangle=\\sum\\_{i=1}^N\\left({\\bf a}\\_i-\\Lambda\\_i{\\bf a}\\_i\\right)\\cdot\\left[\\sum\\_{j=1}^i{\\bf b}\\_j-\\sum\\_{j=1}^{i-1}\\Lambda\\_j{\\bf b}\\_j\\right]\\label{multidefectintersection}$$ \nfor ${\\bf a}\\_i=[{\\bf a}^i\\_\\bullet,{\\bf a}^i\\_\\circ]=[y^i\\_1{\\bf e}\\_Y+r^i\\_1{\\bf e}\\_R,y^i\\_2{\\bf f}^Y+r^i\\_2{\\bf f}^R]$ and ${\\bf b}\\_i=[{\\bf b}^i\\_\\bullet,{\\bf b}^i\\_\\circ]=[y'^i\\_1{\\bf e}\\_Y+r'^i\\_1{\\bf e}\\_R,y'^i\\_2{\\bf f}^Y+r'^i\\_2{\\bf f}^R]$, where the anti-symmetric dot product is defined by\n \n$$\\begin{aligned}\n{\\bf a}\\_i\\cdot{\\bf b}\\_j&=({\\bf a}\\_\\circ^j)^Ti\\sigma\\_y{\\bf b}\\_\\bullet^i-({\\bf b}\\_\\circ^j)^Ti\\sigma\\_y{\\bf a}\\_\\bullet^i\\nonumber\\\\&=y^i\\_2r'^j\\_1-r^i\\_1y'^j\\_2-r^i\\_2y'^j\\_1+y^i\\_1r'^j\\_2\\label{anyondotproduct}\\end{aligned}$$ \nUsing the prototype basis identified in table [tab:prototypesalphabets], the intersection form reproduces the intersection matrices for threefold defects and for twofold defects. As a result of self-intersections such as those in Θ*i**χ*, in general $W[\\{{\\bf a}\\_i\\}]$ need not be a *k**t**h* root of unity.\nWe notice the dot product and subsequently the intersection form and Wilson algebra are invariant under global symmetry transformation\n \n$$\\begin{aligned}\n({\\bf a}\\_i,{\\bf b}\\_i)&\\to({\\bf a}'\\_i,{\\bf b}'\\_i)=\\left(\\Lambda\\cdot{\\bf a}\\_i,\\Lambda\\cdot{\\bf b}\\_i\\right)\\\\\\Lambda\\_i&\\to\\Lambda'\\_i=\\Lambda\\Lambda\\_i\\Lambda^{-1}\\label{defectdualitytransformation}\\end{aligned}$$ \nfor any Λ in *S*3. The color and sublattice transformations of abelian anyon have already been described in and in the previous section. The symmetry transformation for defects are summarized by the following.\n \n$$\\begin{aligned}\n\\Lambda\\_3:&\\left(\\left[\\frac{1}{3}\\right],\\left[\\overline{\\frac{1}{3}}\\right],\\left[\\frac{1}{2}\\right]\\_\\chi\\right)\\to\\left(\\left[\\frac{1}{3}\\right],\\left[\\overline{\\frac{1}{3}}\\right],\\left[\\frac{1}{2}\\right]\\_{\\chi+1}\\right)\\label{defectST}\\\\\\Lambda\\_{B}:&\\left(\\left[\\frac{1}{3}\\right],\\left[\\overline{\\frac{1}{3}}\\right],\\left[\\frac{1}{2}\\right]\\_\\chi\\right)\\to\\left(\\left[\\overline{\\frac{1}{3}}\\right],\\left[\\frac{1}{3}\\right],\\left[\\frac{1}{2}\\right]\\_{\\Lambda\\_{B}\\cdot\\chi}\\right)\\label{defectS}\\end{aligned}$$ \nwhere *χ* = *Y*, *R*, *B* are the color types of twofold defects, and Λ3 and Λ*B* are cyclic color permutation and transposition respectively described in and. For *k* divisible by 3, the threefold defect species labels ${\\bf s}=(s\\_\\bullet,s\\_\\circ)\\in\\mathbb{Z}\\_3\\oplus\\mathbb{Z}\\_3$ transform according to\n \n$$\\Lambda\\_3:{\\bf s}\\to{\\bf s}'={\\bf s},\\quad\\Lambda\\_B:{\\bf s}\\to{\\bf s}'=-\\sigma\\_x{\\bf s}\\label{speciestrans1}$$ \nThe species labels for twofold defects ${\\bf l}=(l\\_Y,l\\_R)$ transform according to\n \n$$\\begin{aligned}\n\\Lambda\\_3:{\\bf l}\\to{\\bf l}'&=\\left(\\Lambda\\_3^{-1}\\right)^T{\\bf l}\\label{speciestrans2}\\\\\\Lambda\\_B:{\\bf l}\\to{\\bf l}'&=\\left\\{\\begin{array}{\\*{20}c}\\left(\\Lambda\\_3^{-1}\\right)^T{\\bf l},&\\mbox{for $[1/2]\\_Y$}\\\\\\Lambda\\_3^T{\\bf l}\\hfill,&\\mbox{for $[1/2]\\_R$}\\\\{\\bf l}\\hfill,&\\mbox{for $[1/2]\\_B$}\\end{array}\\right.\\label{speciestrans3}\\end{aligned}$$ \nwhere $\\Lambda\\_3=\\left(\\begin{array}{\\*{20}c}0&-1\\\\1&-1\\end{array}\\right)$ represents cyclic color permutation. As a result of and, twist defects do not only transform the labels of an orbiting abelian anyon but also the type of an orbiting twist defect.\nNon-Commutative Fusion\n======================\nWe describe the quantum bases and transformations for twist defects. Our system consists of deconfined abelian anyon excitations of the Z*k*-gauge theory as well as semiclassical twist defects that locally violate the *S*3-symmetry.\nAbelian anyons form the basis for all quantum states and measurements. There is a correspondence between anyon operators, non-local Jordan-Wigner strings that leave local excitations at their ends, and excitation states formed when Jordan-Wigner operators act on a ground state. The anyon charge of an excited state can be locally measured by plaquette stabilizers or closed Wilson strings accumulated by dragging a conjugate anyon around it.\nNone of the above holds for defects. Twist defects are not excitations of a Hamiltonian describing a topological gauge theory. There are no ``defect operators\" that correspond to quantum states by their action on the vacuum. Instead, states are generated by non-contractible closed strings of anyon trajectories around defects. If one switches the type of a defect even within the same conjugacy class, for example $[1/3]\\leftrightarrow[\\overline{1/3}]$ or [1/2]*Y* ↔ [1/2]*R*, a closed anyon trajectory may become open by violating eq.. Hence in the semiclassical description, one cannot take superposition between states for different defect configurations — the probability amplitude for a defect to be a particular *S*3-element is either 0 or 1. Moreover, defects cannot be used as a tool for measurement. No quantum states are observed by expectation values of unitary operations involving moving a twist defect in a cycle. Therefore, anyons and twist defects are fundamentally different. The distinction stems from the fact that, unlike the underlying Z*k*-gauge symmetry, the tricolor and bipartite *S*3-symmetry is a classical non-dynamical physical symmetry and is *weakly* broken by anyon labels. We assume there is only a finite (in particular non-dense) population of twist defects so that the system admits an almost global tricoloration and bipartite structure except along finite length branch cuts (a set with measure zero), and hence a *gauging* of *S*3-symmetry is unnecessary.\nAnalogous physical examples include any defect heterostructures between superconductors, (anti)-ferromagnets and (fractional) topological insulators or strong spin-orbit coupled semiconductors , where the pairing phase, magnetic spin order, band inversion mass gap and fermi energy, are all treated as non-dynamical variables. For vortices in chiral *p* + *i**p* superconductors and across the topological insulator to superconductor interface , the pairing phase vortex and branch cuts in the fermion sign, are treated as classical objects. In the study of crystalline dislocations and disclinations in topological insulators and superconductors , the underlying lattice is also regarded as stationary.\nIn practice, although an abelian"},"data_id":{"kind":"string","value":"arxiv_0000362"}}},{"rowIdx":98,"cells":{"content":{"kind":"string","value":" the harvested energy be stored in the battery before it can be used for transmission. An alternative energy storage model, considered in, is to allow the harvested energy *E**t* to be used instantaneously at time *t* and the remaining amount be stored in the battery. The capacities under the two energy storage models are different in general.[↩](#fnref3)\n4. The continuous-time version of this problem has been considered in many works in the literature where it is more commonly referred to as the power control problem and the function *r*(*g*) is called the power-rate or the power-utility function in this case. Note that in our discrete-time setup power corresponds to energy per channel use, and therefore the terms energy (per channel use) and power can be used interchangeably. We prefer to refer to the problem as the energy allocation problem.[↩](#fnref4)\n5. The strategy discussed in is a slight variation of the *Uniform Policy* in that the amount of energy utilized in each channel use is ± *ε* of the average energy arrival rate. *ε* decreases to zero as *B**m**a**x* → ∞ and the decay of *ε* controls the battery discharge probability and the convergence speed to the optimal average rate as *B**m**a**x* → ∞. However, in its essence the strategy is a *Uniform Policy*, so its behavior is captured by the *Uniform Policy* plots.[↩](#fnref5)\n6. Indeed, this fact can be shown analytically; for example taking *B**m**a**x* = *E*, one can show that the gap between the upper bound and the rate achieved by the *Uniform Policy* increases to infinity as *B**m**a**x* = *E* → ∞. However, we do not provide a proof of this fact since the trend is already quite obvious from the graphs.[↩](#fnref6)\n7. This proof was pointed out to the authors by one of the anonymous reviewers.[↩](#fnref7)\n8. Theorem 2 of is stated for multiple access finite state channel. We use a simplified version of it for single-user channels, combined with part 2 of Corollary 1 in the same paper which shows that when the initial state is known to the transmitter, the capacity region is independent of the initial pmf.[↩](#fnref8)\n\nNear Optimal Energy Control and Approximate Capacity of Energy Harvesting Communication\n=======================================================================================\nWe consider an energy-harvesting communication system where a transmitter powered by an exogenous energy arrival process and equipped with a finite battery of size *B**m**a**x* communicates over a discrete-time AWGN channel. We first concentrate on a simple Bernoulli energy arrival process where at each time step, either an energy packet of size *E* is harvested with probability *p*, or no energy is harvested at all, independent of the other time steps. We provide a near optimal energy control policy and a simple approximation to the information-theoretic capacity of this channel. Our approximations for both problems are universal in all the system parameters involved (*p*, *E* and *B**m**a**x*), i.e. we bound the approximation gaps by a constant independent of the parameter values. Our results suggest that a battery size *B**m**a**x* ≥ *E* is (approximately) sufficient to extract the infinite battery capacity of this channel. We then extend our results to general i.i.d. energy arrival processes. Our approximate capacity characterizations provide important insights for the optimal design of energy harvesting communication systems in the regime where both the battery size and the average energy arrival rate are large.\nEnergy Harvesting Channel, Information-Theoretic Capacity, Online Power Control, Constant Gap Approximation, Receiver Side Information.\nIntroduction\n============\nIn many future wireless networks, we may encounter nodes (such as sensor nodes) that harvest the energy they need for communication from the natural resources in their environment. The simplest model that captures this communication scenario is the discrete-time AWGN channel depicted in Fig. [fig: model], where the transmitter is powered by an exogenous stochastic energy arrival process *E**t* stored in a battery of size *B**m**a**x*. The energy of the transmitted symbol at each channel use is limited by the available energy in the battery, which unlike in traditionally powered communication systems is a random quantity due to the randomness in the energy harvesting process *E**t* and moreover depends on the energy expenditure in the previous channel uses. Understanding the capacity of such newly emerging communication systems and the optimal principles to design and operate them has received significant attention over the recent years.\nIn the limiting case *B**m**a**x* = ∞, the capacity of the energy harvesting communication system in Fig. [fig: model] has been characterized by Ozel and Ulukus in. Their result shows that in this asymptotic case the capacity of the energy-harvesting system is equal to the capacity of a classical AWGN channel with an average power constraint equal to the average energy harvesting rate $\\E[E\\_t]$. Perhaps even more importantly, the result of offers a number of important engineering insights for the large battery limit: first it shows that, via simple modifications, the standard communication and coding techniques developed for the classical AWGN channel can be used to achieve the capacity of an energy-harvesting AWGN channel; second, it shows that, in this asymptotic case, the only relevant property of the energy-harvesting process in determining capacity is the average energy harvesting rate. Two energy harvesting mechanisms are equivalent as long as they have the same average energy arrival rate.\n [fig: model]\nCan we obtain analogous engineering insights for the more realistic case of finite battery? For example, how does the capacity of the energy-harvesting AWGN channel in Fig. [fig: model] depend on major system parameters such as *B**m**a**x* and *E**t*? Are there different operating regimes where this dependence is qualitatively different? Given a communication system powered with a certain energy harvesting mechanism *E**t*, how can we optimally choose the battery size *B**m**a**x*? What are the properties of the energy harvesting process *E**t* that are critical to capacity in the finite battery regime? Consequently, what are more desirable and less desirable energy-harvesting profiles? These are foremost engineering questions, the answers of which can guide the design of optimal communication architectures for such systems.\nDespite significant recent effort to characterize the capacity of the energy-harvesting channel in Fig. [fig: model] in the finite battery case, we currently lack an understanding of the above questions. For example, provides a formulation of the capacity in terms of the Verdu-Han general framework and based on a conjecture on the properties of the optimal energy management strategy derives a lower bound to the capacity which is numerically computable for a given setup. However, it is difficult to obtain the above high-level insights from the numerical evaluations. Indeed, even in the case of zero battery, *B**m**a**x* = 0, where provides an exact single letter characterization of the capacity as an optimization problem over the so called Shannon-strategies, the resultant optimization is difficult to solve and requires numerical evaluations, therefore providing limited high-level insights.\nOverview of Our Results\n-----------------------\nIn this paper, we take an alternative approach. Instead of seeking the exact capacity, we seek to provide a simple approximation to the capacity (with bounded guarantee on the approximation gap) which can provide insights on the above engineering questions. As a starting point, we concentrate on an i.i.d. Bernoulli energy arrival process, i.e. *E**t* = *E* with probability *p* and zero otherwise, and is independent across different channel uses. We show that in this case the capacity of the energy harvesting AWGN channel in Fig. [fig: model] is approximately given by\n \n$$\\label{eq:approxcap}\nC\\approx\\left\\{ \\begin{array}{ll}\n\\frac{1}{2}\\log(1+pB\\_{max}) & \\mbox{when $B\\_{max} \\leq E$} \\\\\n\\frac{1}{2}\\log(1+pE) & \\mbox{when $B\\_{max} > E$}.\n\\end{array} \\right.$$ \nThe approximation gap is bounded by 2.58 bits for all values of the system parameters *p*, *E*, and *B**m**a**x*. See Fig. [fig: CapBound].\nThe capacity approximation in provides couple of important insights: First, it identifies the dependence of the capacity to major system parameters. There are two regimes where this dependence is qualitatively different: in the large battery regime (*B**m**a**x* > *E*), the capacity is mainly determined by the average energy arrival rate and is (almost) independent of the exact value of the battery size; on the other hand, in the small battery regime (*B**m**a**x* < *E*), the capacity depends critically on the battery size and is increasing logarithmically with increasing *B**m**a**x*. The formula also suggests that choosing *B**m**a**x* ≈ *E* is sufficient to extract most of the capacity of the system (achieved at *B**m**a**x* = ∞). One can also observe that while in the large battery regime the only property of the energy harvesting profile that impacts capacity is the average rate, in the small battery regime energy profiles that are less peaky over time lead to larger capacity. See Figure [fig: CapBound2] which compares two energy profiles with the same average rate.\nThe main ingredient of the above result is a near optimal online strategy we develop for energy/power control over the Bernoulli energy harvesting channel. The problem of optimal energy/power allocation over an energy harvesting channel has been extensively studied in the literature. While the offline version is well-understood, the online version of the problem is known to be difficult and several works suggest simple online power control policies with and without performance guarantees. For example, shows that a simple strategy that allocates constant energy, equal to the average energy arrival rate, to each channel use as long the battery is not empty becomes asymptotically optimal as *B**m**a**x* → ∞. However as we show in Section [sec: simulations], this strategy can be arbitrarily away from optimality for finite values of *B**m**a**x*. In contrast, we show analytically that the online energy allocation policy we propose remains within 0.973 bits of the optimal value for all values of the system parameters. This result can be of interest in its own right.\nIn the final section of the paper, we extend our approximation results to general i.i.d. energy arrival processes where each *E**t* is i.i.d. according to some arbitrary distribution. We show that a simple modification allows to apply the near optimal energy allocation policy and the information-theoretic coding strategy we develop in the earlier sections for the Bernoulli process to general i.i.d. energy arrival processes. We show that for many distributions of *E**t*, this yields an approximate characterization of the capacity within a constant gap, though with a larger constant. However, we also illustrate that one can engineer specific distributions for which our approach would fail to provide a constant gap approximation.\nThe constant gap approach we propose in this paper is most useful to understand the capacity of energy harvesting communication channels operating at moderate to high SNRs. This, for example, can be the operating regime of a base station in a rural area powered by renewable energy sources (ex. wind or solar). However, the insights obtained from such an analysis can be applicable even at low SNR. Note that the 2.58 bits/s/Hz is an analytical upper bound on the worst case gap over all SNR regimes and the actual gap between the rate achieved by the strategies we propose and the true capacity of the system can be much smaller especially at low SNR. More generally, we believe the approximation philosophy we propose in this paper will be useful in providing a basis for comparing the performances of different strategies, developing insights into the capacity of the system and giving a sense of the remaining gap in characterizing the problem, when obtaining an exact *insightful* capacity formula proves to be difficult.\n [fig: CapBound]\n [fig: CapBound2]\nSystem Model\n============\nWe consider an AWGN communication system powered by an energy harvesting mechanism with limited battery (See Fig. [fig: model]). At each time step *t*, *E**t* amount of energy is harvested from the exogenous energy source and is stored in the battery of size *B**m**a**x*. In the case the harvested energy *E**t* exceeds the available space in the battery at time *t*, the battery is charged to maximum capacity and the remaining energy is discarded. Let *X**t* denote the scalar real input to the channel at time *t*, *Y**t* the output of the channel and *N**t* the additive noise with unit normal distribution N(0, 1). We have\n \n*Y**t* = *X**t* + *N**t*. \nLet *B**t* represent the amount of energy available in the battery for transmission at time step *t*. Then the transmitted signal *X**t* is amplitude constrained by the available energy in the battery *B**t*. The system energy constraints can be summarized as follows:\n \n$$\\begin{aligned}\n|X\\_t|^2 & \\le B\\_t \\label{eq: ampConst}\\\\\nB\\_{t+1} & = \\min(B\\_t + E\\_{t+1} - |X\\_t|^2, B\\_{max}) \\label{eq: batteryUpdate}\\end{aligned}$$ \nEquation represents the amplitude constraint on the input, and represents the update rule for the energy available in the battery. [3](#fn3)\nThe harvested energy at each time step *E**t* is a discrete-time stochastic process dictated by the energy harvesting mechanism. We assume that the energy arrival process is causally known at the transmitter (i.e. at time *t* the transmitter knows {*E**t*, *E**t* − 1, …}), but not at the receiver. In the sequel, we first focus on a simple Bernoulli process (Sections [sec: OnlinePolicy] and [sec: InfoCap]), where *E**t*’s are i.i.d. Bernoulli random variables:\n \n$$\\begin{aligned}\n\\label{eq: Et}\nE\\_t = \\left\\{ \\begin{array}{rl}\nE & \\mbox{w.p. $p$} \\\\\n0 & \\mbox{w.p. $1-p$},\n\\end{array} \\right.\\end{aligned}$$ \nso at each time step, either an energy packet of size *E* is harvested, or no energy is harvested at all, independent of the other time steps. We then extend our results to more general i.i.d. energy harvesting processes in Section [sec: GeneralEP], where we assume that *E**t* are i.i.d. with common cdf *F*(*x*).\nThe information-theoretic capacity of the channel is defined in the usual way as the largest rate at which the transmitter can reliably communicate to the receiver under the system energy constraints in and and the assumption that the realizations of *E**t* are known only to the transmitter in a causal fashion. While the main focus of our paper is the information-theoretic capacity of this channel, in Section [sec: OnlinePolicy] we also study a related problem, optimal (online) energy allocation for maximizing the average rate (or utility) of an energy-harvesting system. This problem is defined more precisely in the corresponding section and forms the basis for the information-theoretic results we prove in Section [sec: InfoCap].\nMain Result\n===========\nThe following theorem is the main result of the paper.\n[Main Result] [thm: MainResult] The capacity *C* of the channel described in Section [sec: SysModel] with Bernoulli energy arrival process *E**t* satisfies\n \n$$\\begin{aligned}\n\\frac{1}{2} \\log \\left(1 + p \\cdot \\min\\{ B\\_{max}, E\\} \\right)& - 2.58 \\le \\; C \\\\\n \\le \\frac{1}{2} \\log & \\left( 1 + p \\cdot \\min\\{B\\_{max}, E \\} \\right).\\end{aligned}$$ \nThe bound in Theorem [thm: MainResult] is illustrated in Figure [fig: CapBound]. The result shows that the capacity of a finite battery energy harvesting system is within 2.58 bits of $\\frac{1}{2} \\log \\left( 1 + p \\cdot \\min\\{B\\_{max}, E \\} \\right)$ for any choice of the system parameters *p*, *E* and *B**m**a**x*. We therefore refer to $\\frac{1}{2} \\log \\left( 1 + p \\cdot \\min\\{B\\_{max}, E \\} \\right)$ as the approximate capacity of the channel. Figure [fig: CapBound2] compares the approximate capacity under two different Bernoulli energy harvesting profiles.\nThe main ingredient of the above theorem is a near optimal online energy allocation policy we develop for the Bernoulli energy harvesting channel in Section [sec: OnlinePolicy]. We show that this strategy is within 0.973 bits of optimality for all values of the system parameters. We state the corresponding results in Theorems [thm: constgaprate] and [thm: constgaprate2], after we precisely define the online energy allocation problem in Section [sec: OnlinePolicy]. Finally, we discuss how our results from Sections [sec: OnlinePolicy] and [sec: InfoCap] for the Bernoulli process can be extended to more general i.i.d. energy harvesting profiles in Section [sec: GeneralEP].\nA Near Optimal Online Policy for Energy Control\n===============================================\nIn this section, we study an online energy allocation problem for the energy harvesting communication system with finite battery described in Section [sec: SysModel]. The near optimal energy control policy we develop in this section turns out to be a critical ingredient of the approximate capacity characterization we develop in the next section.\nWe consider the energy harvesting transmitter described in Section [sec: SysModel] and assume that we are specified an energy-rate (or energy-utility) function *r*(*g*) which specifies the rate or utility *r*(*g*(*t*)) obtained at a given channel use *t* as a function of the energy *g*(*t*) allocated for transmission at time *t*.[4](#fn4) We assume the energy arrival process is a Bernoulli process described by, and again only online, i.e. causal, information of the energy packet arrivals is available at the transmitter. An online policy *g*(*t*) denotes the amount of energy transmitter decides to allocate for transmission at time *t*. We call *g*(*t*) a feasible online policy, if *g*(*t*) satisfies:\n \n$$\\begin{aligned}\n& 0 \\le g(t) \\le B\\_t \\label{eq: allocConst}\\\\\n& B\\_{t+1} = \\min(B\\_t + E\\_{t+1} -g( t), B\\_{max}) \\label{eq: batteryUpdate2}\\end{aligned}$$ \nNotice that constraints and are analogous to the system energy constraints and from Section [sec: SysModel], except ∣*X**t*∣2, the amount of energy used for transmission at time step *t*, from Section [sec: SysModel] is replaced with *g*(*t*) here. Moreover, the energy allocated at time *t* can only depend on the past realizations of the energy harvesting process, i.e. we have the causality constraint\n \n*g*(*t*) = *f*(*t*, {*E**i*}*i* = 0*t*). \nSince the energy arrival process *E**t* is a stochastic process, the quantity *g*(*t*) is random. Let G denote the class of online policies satisfying constraints, and. Then, our goal is to maximize the long term average rate over the class of feasible online policies:\n \n$$\\begin{aligned}\n \\label{eq: obj}\n\\max\\_{g \\in \\mathcal{G}} \\liminf\\_{N \\to \\infty} \\mathbb{E}\\left[ \\frac{1}{N} \\displaystyle\\sum\\limits\\_{t = 1}^N r(g( t)) \\right],\\end{aligned}$$ \nwhere the expectation here is over the ensembles of {*E**t*}*t* = 0*N* and *r*( ⋅ ) is the energy-rate function of interest. Since our main motivation for studying this problem is to use the solution to approximate the capacity of the AWGN channel in the next section, we restrict our attention to the following classical AWGN rate function:\n \n$$\\begin{aligned}\n\\label{eq: AWGN\\_rate}\nr(x) = \\frac{1}{2} \\log\\left(1 + x \\right)\\end{aligned}$$ \nin bits per channel use. Therefore, the problem of interest here is:\n \n$$\\begin{aligned}\n \\label{eq: AWGN\\_cap\\_obj}\n\\max\\_{g \\in \\mathcal{G}} & \\liminf\\_{N \\to \\infty} \\mathbb{E} \\left[ \\frac{1}{N} \\displaystyle\\sum\\limits\\_{t = 1}^N \\frac{1}{2} \\log\\left( 1 + g( t)\\right) \\right]. \\end{aligned}$$ \nThe case *B**m**a**x* ≤ *E*\n---------------------------\nWe first analyze the case where *B**m**a**x* ≤ *E*. In this case, according to our system definition, every time a non-zero energy packet arrives, the battery is charged to full, and the left over energy is wasted. Since, the system is reset to the initial state of full battery each time a non-zero energy packet arrives, each epoch (the period of time instances between two adjacent non-zero packet arrivals) is independent and statistically indistinguishable from every other epoch. Motivated by this observation, we propose to use a strategy where the energy *g*(*t*) allocated to transmission at time *t* depends only on the number of channel uses since the last time the battery was recharged, i.e., *g*(*t*) = *g̃*(*j*) where *j* = *t* − max{*t*ʹ ≤ *t* : *E**t*ʹ = *E*}, for a function *g̃*(*j*) that satisfies\n \n$$\\begin{aligned}\n\\label{eq: constraint}\n\\displaystyle\\sum\\limits\\_{j=0}^\\infty \\tilde{g}(j) \\le B\\_{max} \\;\\;\\;\\; \\text{and} \\;\\;\\;\\ \\tilde{g}(j) \\ge 0 \\;\\;\\; \\forall j. \\end{aligned}$$ \nNote that an energy allocation policy that satisfies the above properties clearly satisfies the feasibility constraints and. Moreover, it uses only information about the past realizations of the process. We choose *g̃*(*j*) = *p*(1 − *p*)*j**B**m**a**x*, which clearly satisfies, and show in the following theorem that it achieves an objective value that’s no more than 0.973 bits away from the optimum value of the optimization problem given in for any value of *p* and *B**m**a**x*.\n[thm: constgaprate] Let *g*′(*t*) = *g̃*(*j*) where *j* = *t* − max{*t*ʹ ≤ *t* : *E**t*ʹ = *E*} and *g̃*(*j*) = *p*(1 − *p*)*j**B**m**a**x* for *j* = 0, 1, 2, .... When *B**m**a**x* ≤ *E*, we have the following guarantee:\n \n$$\\begin{aligned}\n\\liminf\\_{N \\to \\infty}&\\; \\mathbb{E} \\left[ \\frac{1}{N} \\displaystyle\\sum\\limits\\_{t = 1}^N \\frac{1}{2} \\log\\left( 1 + g^\\prime(t)\\right) \\right] \\notag \\\\\n & \\ge \\max\\_{g \\in \\mathcal{G}} \\liminf\\_{N \\to \\infty} \\mathbb{E} \\left[ \\frac{1}{N} \\displaystyle\\sum\\limits\\_{t = 1}^N \\frac{1}{2} \\log\\left( 1 + g( t)\\right) \\right] - 0.973. \\label{eq: const\\_gap\\_rate}\\end{aligned}$$ \nThe detailed proof of the theorem is deferred to the Appendix. The proof follows roughly three major steps.\n* Using Jensen’s inequality, we first show that\n \n$$\\begin{aligned}\n\\max\\_{g \\in \\mathcal{G}} \\liminf\\_{N \\to \\infty} \\mathbb{E} & \\left[ \\frac{1}{N} \\displaystyle\\sum\\limits\\_{t = 1}^N \\frac{1}{2} \\log\\left( 1 + g( t) \\right) \\right] \\notag\\\\\n& \\leq \\frac{1}{2}\\log(1+pB\\_{max})\\label{eq: policy\\_upper}\\end{aligned}$$\n* Using the fact that *g*′(*t*) is same across different epochs, we turn our achievable rate (left-hand-side of ) into the following expression as a function of *g̃*(*j*):\n \n$$\\begin{aligned}\n\\label{eq: ach}\n\\displaystyle\\sum\\limits\\_{j = 0}^{\\infty}p &(1-p)^j \\frac{1}{2} \\log(1 + \\tilde{g}(j)) \\notag \\\\\n& = \\displaystyle\\sum\\limits\\_{j = 0}^{\\infty} p(1-p)^j \\frac{1}{2} \\log(1 + p(1-p)^j B\\_{max}) \\end{aligned}$$\n* Finally, we upper bound the gap between the objective value achieved with *g*ʹ(*t*), i.e. Equation, and $\\frac{1}{2}\\log(1+pB\\_{max})$ by a constant.\n [fig: EnergyAlloc]\nFig. [fig: EnergyAlloc] illustrates the energy allocation policy in Theorem [thm: constgaprate]. In this allocation policy we use *p* fraction of the remaining energy in the battery at each time (so the energy in the battery decays like *B**t* = (1 − *p*)*j**B**m**a**x*). The motivation for this energy allocation policy is the following: for the Bernoulli arrival process *E**t*, the inter-arrival time is a Geometric random variable with parameter *p*. We know that the Geometric random variable is memoryless and has mean 1/*p*. Therefore, at each time step, the expected number of time steps to the next energy arrival is 1/*p*. Furthermore, since log( ⋅ ) is a concave function, results from tell us that in order to achieve higher rate, we want to allocate the energy as uniform as possible between energy arrivals, i.e. if the current energy level in the battery is *B**t* and we knew that the next recharge of the battery would be in exactly *m* channel uses, we would allocate *B**t*/*m* energy to each of the next *m* channel uses. For the online case of interest here, we do not know when the next energy arrival would be. Instead, we use the expected time to the next energy arrival as a basis: since at each time step, the expected time to the next energy arrival is 1/*p*, we use a fraction *p* of the currently available energy.\nSome simple online policies with and without performance guarantees have been earlier proposed in. None of these strategies utilize the idea of exponential energy usage we propose here and can achieve the optimal rate within a constant gap uniformly over all parameter ranges. In Section [sec: simulations] below, we provide simulations which illustrate that these strategies can be arbitrarily away from optimality. However, before that we first address the remaining case of *B**m**a**x* > *E*.\nThe case *B**m**a**x* > *E*\n---------------------------\nNote that when *B**m**a**x* ≤ *E*, because the energy must be stored into the battery before it can be used, the extra energy is wasted, and the system *B**m**a**x* ≤ *E* is equivalent to a system where the energy packet size is exactly equal to *B**m**a**x*. Therefore, the average optimal rate we can achieve with online energy management strategies is independent of *E*, and Theorem [thm: constgaprate] characterizes this rate as approximately $\\frac{1}{2}\\log(1+pB\\_{max})$. In the case *B**m**a**x* > *E*, we show that the average optimal rate is given by $\\frac{1}{2}\\log(1+pE)$ which can be achieved by a simple modification of the energy control policy proposed in the earlier section. We have the following theorem.\n[thm: constgaprate2] Let *g*′(*t*) = *g̃*(*j*) where *j* = *t* − max{*t*ʹ ≤ *t* : *E**t*ʹ = *E*} and *g̃*(*j*) = *p*(1 − *p*)*j**E* for *j* = 0, 1, 2, .... When *B**m**a**x* > *E*, we have the following guarantee:\n \n$$\\begin{aligned}\n\\liminf\\_{N \\to \\infty}&\\; \\mathbb{E} \\left[ \\frac{1}{N} \\displaystyle\\sum\\limits\\_{t = 1}^N \\frac{1}{2} \\log\\left( 1 + g^\\prime(t)\\right) \\right] \\notag \\\\\n & \\ge \\max\\_{g \\in \\mathcal{G}} \\liminf\\_{N \\to \\infty} \\mathbb{E} \\left[ \\frac{1}{N} \\displaystyle\\sum\\limits\\_{t = 1}^N \\frac{1}{2} \\log\\left( 1 + g( t)\\right) \\right] - 0.973. \\label{eq: const\\_gap\\_rate\\_2}\\end{aligned}$$ \nNote that in the case *B**m**a**x* > *E*, each time an energy packet arrives *B**t*, the available energy in the battery, becomes at least as large as *E*. Based on this observation, our strategy utilizes a fraction *p* of *E* when *j* = 0, i.e. if the energy packet has just arrived in the current channel use; a fraction *p* of the remaining (1 − *p*)*E* in the next channel use *j* = 1, i.e. if the energy has arrived in the previous channel use and not the current one, etc. It is easy to verify that since\n \n$$\\begin{aligned}\n\\label{eq:needeqnumber}\n\\displaystyle\\sum\\limits\\_{j=0}^\\infty \\tilde{g}(j) \\le E \\;\\;\\;\\; \\text{and} \\;\\;\\;\\ \\tilde{g}(j) \\ge 0 \\;\\;\\; \\forall j,\\end{aligned}$$ \nthis strategy satisfies the energy feasibility constraints in and when *B**m**a**x* > *E*. Indeed, this energy management policy is quite conservative and can be clearly wasteful of resources. Consider the first epoch which starts with the arrival of the first energy packet: the remaining energy in the battery *j*-channel uses after the first energy packet arrives is given by (1 − *p*)*j**E*, so at the time the second packet arrives, there will be some residual energy left in the battery, at least part of which will add up to the arriving energy packet *E* since *B**m**a**x* > *E*. The strategy we propose ignores this residual energy. An equivalent way of thinking about our strategy is that it operates as if *B**m**a**x* = *E* even though *B**m**a**x* > *E*. However this strategy still turns out to be within constant number of bits of the optimal value. One immediate way to improve this strategy would be to use, at each time step, a fraction *p* of the currently available energy in the battery *B**t*. However, this improved strategy turns out to be difficult to analyze analytically. In the next section, we present simulation results which demonstrate the improvement due to this modification.\nTheorem [thm: constgaprate2] can be proved by using similar lines to the proof of Theorem [thm: constgaprate]. In the appendix, again based on Jensen’s inequality, we show that\n \n$$\\begin{aligned}\n\\max\\_{g \\in \\mathcal{G}} \\liminf\\_{N \\to \\infty} \\mathbb{E} \\left[ \\frac{1}{N} \\displaystyle\\sum\\limits\\_{t = 1}^N \\frac{1}{2} \\log\\left( 1 + g( t)\\right) \\right]\n \\leq \\frac{1}{2}\\log(1+pE)\\end{aligned}$$ \nin this case. The remaining step is to show is that the strategy we propose in Theorem [thm: constgaprate2] is within 0.973 bits of this upper bound. While this can be shown from first principles by following the steps of Theorem [thm: constgaprate], it can also be directly observed from the proof of Theorem [thm: constgaprate]: Fix the energy packet size *E*. When the battery size is as large as *E*, i.e. *B**m**a**x* = *E*, the strategy in Theorem [thm: constgaprate] reduces to the strategy in Theorem [thm: constgaprate2]. Moreover, the proof of Theorem [thm: constgaprate] establishes that this strategy achieves $ \\frac{1}{2}\\log(1+pE)$ within 0.973 bits. Now in a system with same *E* but larger *B**m**a**x*, so that *B**m**a**x* > *E*, the same strategy is still feasible and will clearly achieve the same rate. This argument shows that when *B**m**a**x* > *E*, the strategy proposed in Theorem [thm: constgaprate2] achieves an average rate at least as large as $ \\frac{1}{2}\\log(1+pE)-0.973$ bits/channel use.\nNumerical Evaluations\n---------------------\n[ht] [fig: p15]\n[ht] [fig: p5]\nIn this section, we provide simulation results that compare the performance of the near optimal energy control policy we proposed in the earlier section, which we refer to as the *Constant Fraction Policy* in the current section, to some other simple energy/power control policies that have been proposed in the literature for the same problem. In particular, proposes a *Constant Water Level Policy*, which we refer to as the *Uniform Policy* in this section, which allocates a constant amount of energy, equal to the average energy arrival rate, to each channel use as long as there is sufficient energy in the battery. When the energy in the battery is exhausted, no energy is allocated until the next energy arrival. shows that this strategy becomes asymptotically optimal as *B**m**a**x* increases.[5](#fn5) The same intuition is suggested by in the information-theoretic setting.\nFigure [fig: p15] and [fig: p5] summarize the results. In all the plots, the curve with label ``Upper Bound“ is the upper bound on the average rate achieved by any feasible policy, i.e. 1/2log(1 + *p**B**m**a**x*) when *B**m**a**x* ≤ *E* and 1/2log(1 + *p**E*) when *B**m**a**x* > *E*. When *B**m**a**x* ≤ *E*, the curve with label ``Constant Fraction Policy” is the rate achieved by the strategy we proposed in the earlier section (Theorem [thm: constgaprate]), which uses a fixed *p* fraction of the remaining energy in the battery; the curve with label ``Uniform Policy“ is the rate achieved by a strategy that allocates *p**B**m**a**x* amount of energy, which is the average energy arrival rate in this case, if there is enough energy left in the battery, and 0 energy otherwise. Similarly when *B**m**a**x* > *E*, the *Constant Fraction Policy* uses a fixed fraction *p* of the available energy in the battery *B**t* and the *Uniform Policy* uses energy *p**E* at each channel use if possible. There are two curves for each of these strategies. The curves with ``(Lower Bound)” in the label represent an analytical lower bound we can compute on the rate achieved by these strategies by assuming *B**m**a**x* = *E* (so these lower bounds remain the same as long as *B**m**a**x* > *E*). In particular, for the *Constant Fraction Policy*, the lower bound corresponds to the rate achieved by the strategy proposed in Theorem [thm: constgaprate2]. Recall the discussion in the paragraph after which suggests a strategy using a constant fraction *p* of the available energy would actually achieve a larger rate when *B**m**a**x* > *E*. This actual rate is obtained by running Monte Carlo simulation and is given by the curve with ``(Monte Carlo)\" in the label. Similarly, it’s possible to analytically compute a lower bound on the rate achieved by the *Uniform Policy* by assuming *B**m**a**x* = *E* and the actual performance is obtained by Monte Carlo simulations.\nBased on Fig. [fig: BleE15] and [fig: BleE5], we see that in the *B**m**a**x* ≤ *E* regime, the *Constant Fraction Policy* indeed tracks the upper bound within a constant gap for all values of *B**m**a**x*, whereas the gap of the *Uniform Policy* starts to diverge at around 15 to 20 dB depending on *p*. A similar conclusion holds for Fig. [fig: B2E15] and [fig: B2E5] where *B**m**a**x* > *E* with the ratio of *B**m**a**x* to *E* being fixed at 2. Fig. [fig: B8E15] and [fig: B8E5] showed that when *B**m**a**x* ≫ *E* (in this case *B**m**a**x* = 8*E*), the performances of both policies are very similar to each other across all SNR regimes. Moreover, they both track the upper bound very closely. This is not surprising given that, show that the *Uniform Policy* converges to the upper bound as *B**m**a**x*/*E* → ∞. Fig. [fig: B8E15] and [fig: B8E5] empirically show that our *Constant Fraction Policy* performs just as good when *B**m**a**x* ≫ *E*.\nThe figures also illustrate the difference between a constant gap guarantee on optimality and an asymptotic guarantee on optimality as *B**m**a**x* → ∞. While Fig. [fig: B8E15] and [fig: B8E5] show that the *Uniform Policy* becomes optimal in the regime when *B**m**a**x* ≫ *E*, when *B**m**a**x* and *E* are comparable this strategy can be arbitrarily away from the optimal rate (as illustrated in the figures for *B**m**a**x* ≤ *E* and *B**m**a**x* = 2*E*).[6](#fn6) Our *Constant Fraction Policy* on the other hand is within a bounded gap from optimality for all parameter values as guaranteed by our theoretical results. Finally, observe that there is not much qualitative difference between Fig. [fig: p15] and Fig. [fig: p5] which correspond to different values of energy arrival probability *p*. This is expected as our constant gap guarantees hold independent of *p*.\nThe information theoretic capacity of the finite battery system\n===============================================================\nIn this section, we approach the system in Section [sec: SysModel] with Bernoulli energy arrival process from an information theoretic perspective. In particular, we derive an upper and a lower bound on the information-theoretic capacity of the channel and show that the gap between these upper and lower bounds is no more than a constant for all choices of the system parameters. Our lower bound relies on the near-optimal energy allocation policy we developed in Section [sec: OnlinePolicy], and reveals the connection between the two problems by developing a codebook construction which allows to implement a given energy allocation policy. We examine the regime where *B**m**a**x* ≤ *E* in Section [subsec: BlessE], and the regime *B**m**a**x* > *E* in Section [subsec: BgreaterE].\nThe *B**m**a**x* ≤ *E* regime\n-----------------------------\nIn the case *B**m**a**x* ≤ *E*, each time the non-zero energy packet arrives, the battery will be filled up completely regardless of how much energy was remaining in the battery. In particular, at least *E* − *B**m**a**x* amount of energy is wasted in every non-zero incoming energy packet. Therefore, a system with energy packet size *E*ʹ = *E* − (*E* − *B**m**a**x*) = *B**m**a**x* is equivalent to the original system in terms of the available communication resources and hence the two systems must have the same capacity. In the sequel, we consider the equivalent system with *E*ʹ = *B**m**a**x*.\nNote that as a result of the above observation, in the regime *B**m**a**x* ≤ *E*, the capacity of the system can only explicitly depend on *B**m**a**x* and *p* and not *E*. We next provide an upper bound on the capacity as a function of *B**m**a**x* and *p*. We will then provide an achievable scheme and show that the rate it achieves is within a constant gap from this upper bound for all choices of *B**m**a**x* and *p*.\n[Upper Bound on Capacity: *B**m**a**x* ≤ *E*] [thm: UpperBoundC] When *E* ≥ *B**m**a**x*, the capacity *C* of the channel with Bernoulli energy arrival process defined in Section [sec: SysModel] is upper bounded by\n \n$$\\label{eq: CapUpperBound}\nC \\le \\frac{1}{2} \\log \\left(1 + p B\\_{max}\\right) \\triangleq C\\_{ub}( B\\_{max}, p).$$ \nNote that the capacity of the channel in Section [sec: SysModel] should be an increasing function of the battery size, since we can always choose not to use the extra battery space. Therefore, the capacity of our channel with finite battery is upper bounded by the capacity of the same channel with infinite battery size. The infinite battery capacity has been characterized in as\n \n$$\\begin{aligned}\n C\\_{\\infty} & = \\frac{1}{2}\\log(1 + \\mathbb{E}\\left[ E\\_t \\right] ) \\notag \\\\\n & = \\frac{1}{2}\\log(1 + pE) \\label{eq: ClassicAWGN}\\end{aligned}$$ \nsince E[*E**t*] = *p**E* in our current case. Based on the earlier discussion, when *E* ≥ *B**m**a**x*, the capacity of the system is the same as the capacity of a system with reduced energy packet size *E* = *B**m**a**x*. Plugging *E* = *B**m**a**x* in gives the desired upper bound in.\nNext, we will provide an achievable strategy for the channel in Fig. [fig: model] when the energy arrival process {*E**t*} is also causally known at the receiver. Later, we will use this result to derive an achievable rate for our original system in Section [sec: SysModel] where we assume that the receiver has no information about the energy arrival process.\n[Achievable Scheme with CSIR] [thm: Achievable] Assume that for the system defined in Section [sec: SysModel], the Bernoulli energy arrival process {*E**t*} is causally known not only at the transmitter but also at the receiver. Then we can achieve any rate:\n \n$$R\\_{ach} \\leq \\displaystyle\\sum\\limits\\_{j = 0}^{\\infty} p (1-p)^j \\max\\_{p(x): |X|^2 \\le \\mathcal{E}\\_j} I(X; Y) \\label{eq: Achievable}$$ \nfor any non-negative E0, E1, ... satisfying\n \n$$\\sum\\limits\\_{j = 0}^{\\infty} \\mathcal{E}\\_j \\le B\\_{max}. \\label{eq: EnergyConstr}$$ \nThe proof of Theorem [thm: Achievable] is provided in the Appendix.\nThe idea for the achievable scheme is that if both the transmitter and receiver know when the energy packet *E* arrives, they can agree on an energy allocation strategy ahead of time. As we did in Section [sec: OnlinePolicy], here we concentrate on an energy allocation policy E*j* that is invariant across different epochs (the period of time between two adjacent non-zero packet arrivals). E*j* denotes the amount of energy allocated to transmission, *j* channel uses after the last time the battery was recharged, i.e. if energy arrives at the current channel use, we allocate E0 amount of energy for transmission; if energy arrived in the previous channel use but not the current channel use, then we allocate E1 amount of energy for transmission, etc. The transmitter and receiver agree ahead of time on a sequence of codebooks C(*j*) where each codebook is amplitude-constrained to E*j*, i.e. the symbols of each codeword in C(*j*) are such that ∣*X*∣2 ≤ E*j*. This ensures that the symbol transmitted at the corresponding time will not exceed the energy constraint E*j*. We assume that the transmitter has one codeword *c**j* ∈ C(*j*) from each codebook to communicate to the receiver and the symbols of these codewords are interleaved as dictated by the realization of the energy arrival process. For example, upon the arrival of the first energy packet, the transmitter sends the first symbol of codeword *c*0; if there is no energy packet arrival in the next channel use, it transmits the first symbol of codeword *c*1 in the next channel use, etc. Once the second energy packet arrives, the cycle is reset and the transmitter moves to transmitting the second symbol of the codeword *c*0, then the second symbol in codeword *c*1, etc. (See the Appendix for a detailed description of the strategy and its performance analysis.) gives the rate we can achieve with such a strategy in the large blocklength limit. ensures that the total energy spent does not exceed the available energy in the battery at *j* = 0 which is equal to *B**m**a**x* (since when *E* ≥ *B**m**a**x*, the battery is recharged to full every time an energy packet arrives).\nWe next show that when there is no channel state information at the receiver we can achieve the rate in Theorem [thm: Achievable] with at most *H*(*p*) bits of penalty. The main idea of the result is similar to Theorem 1 in which shows that over an information stable channel the maximum possible capacity improvement due to receiver side information is bounded by the amount of the side information itself. In the current case, the energy constraints in and introduce memory into the system and its not a priori clear if the channel is information stable or not. The following theorem extends Theorem 1 of to general, not necessarily information-stable, channels.\n[Capacity improvement due to RX Side Info][thm:CapCSIR] Consider a general channel, not necessarily stationary memoryless, defined as a sequence {*W**n*( ⋅ ∣ ⋅ ) = *P**Y*(*n*)∣*X*(*n*) : X(*n*) → Y(*n*)}*n* = 1∞, of arbitrary conditional probability distributions together with an input and an output alphabet for each *n* (which need not be Cartesian products of a basic input and an output alphabet). The improvement in channel capacity due to the availability of side information at the receiver is upper-bounded by the *spectral sup-entropy rate* of the side information process *G* = {*G**n*}*n* = 1∞, which is defined as\n \n$$\\bar{H}(G)=\\text{p-}\\limsup\\_{n\\rightarrow\\infty}\\frac{1}{n}\\log \\frac{1}{P\\_{G^n}(g^n)},$$ \nwhere p-limsup denotes *limsup in probability* (see ).\nThe proof of the theorem is given in the Appendix. In order to apply the theorem to our current channel with causal transmitter side information, we can use Shannon’s technique in to first transform the channel to an equivalent channel without states but with an enlarged input alphabet over so called Shannon strategies (this transformation has been developed in ) and then apply Theorem [thm:CapCSIR] to the equivalent channel. Since the side information process *G* in our case is the i.i.d. Bernoulli energy arrival process {*E**t*}, which has entropy rate *H̄*(*G*) = *H*(*p*) we immediately get the following theorem.\n[Achievable Rate without CSIR] [thm: Achievable2] Consider the system defined in Section [sec: SysModel] where the energy arrival process {*E**t*} is causally known only at the transmitter, but not at the receiver. The capacity *C* of this system is lower bounded by\n \n$$\\begin{aligned}\nC \\geq \\displaystyle\\sum\\limits\\_{j = 0}^{\\infty} & \\; p (1-p)^j \\max\\_{p(x): |X|^2 \\le \\mathcal{E}\\_j} I(X; Y)- H(p) \\label{eq: Achievable2} \\\\\n& \\text{subject to: } \\;\\;\\; \\displaystyle\\sum\\limits\\_{j = 0}^{\\infty} \\mathcal{E}\\_j \\le B\\_{max} \\label{eq: EnergyConstr2}\\end{aligned}$$ \nwhere *H*(*p*) is the binary entropy function.\nWhile the proof of Theorem [thm: Achievable2] follows directly from Theorem [thm:CapCSIR], in the Appendix we provide an alternative proof for Theorem [thm: Achievable2] that makes specific use of our channel structure and the achievable scheme we propose in Theorem [thm: Achievable].\nNote that the mutual information maximization problem in is over the class of distributions with bounded support in $\\left[ - \\sqrt{\\mathcal{E}\\_j}, \\sqrt{\\mathcal{E}\\_j} \\right]$. This is a nontrivial optimization problem and as shown in, the optimal input distribution turns out to be discrete rather than continuous. Below we lower bound the optimal value of this optimization problem by considering the uniform distribution over the interval $\\left[ - \\sqrt{\\mathcal{E}\\_j}, \\sqrt{\\mathcal{E}\\_j} \\right]$.\n[LB on Amplitude-constrained AWGN Capacity] [lem: GapUniform]\n \n$$\\label{eq: UniformCap}\n\\max\\_{p(x): |X|^2 \\le \\mathcal{E}\\_j} I(X; Y) \\ge \\frac{1}{2} \\log \\left(1 + \\mathcal{E}\\_j \\right) - 1.04$$ \nTake *p*(*x*) to be the uniform distribution on $\\left[ - \\sqrt{\\mathcal{E}\\_j}, \\sqrt{\\mathcal{E}\\_j} \\right]$, then the average power is $\\mathbb{E}[X^2] = \\frac{\\mathcal{E}\\_j}{3}$. We use the following result which is proved in :\nConsider a discrete-time AWGN channel with average power constraint *P*, i.e. E[∣*X*∣2] ≤ *P*, and noise variance *σ*2. Let the input *X* of the channel be uniformly distributed over $[-\\sqrt{3P},\\sqrt{3P}]$ (so that E[∣*X*∣2] = *P*) and *Y* be the corresponding output random variable. Then \n$$I(X;Y)\\geq \\log\\left(1+\\frac{P}{\\sigma^2}\\right)- \\frac{1}{2}\\log\\left(\\frac{\\pi e}{6}\\right).$$ \nBy using the result in the proposition, we get\n \n$$\\begin{aligned}\n\\max\\_{p(x): |X|^2 \\le \\mathcal{E}\\_j} & I( X; Y) \\ge \\frac{1}{2} \\log \\left(1 + \\frac{\\mathcal{E}\\_j}{3} \\right) - \\frac{1}{2}\\log\\left(\\frac{\\pi e}{6}\\right) \\notag\\\\\n & \\ge \\frac{1}{2} \\log(1 + \\mathcal{E}\\_j) - \\left(\\frac{1}{2}\\log(3) + \\frac{1}{2}\\log\\left(\\frac{\\pi e}{6}\\right) \\right) \\notag \\\\\n & = \\frac{1}{2} \\log(1 + \\mathcal{E}\\_j) - 1.04\\end{aligned}$$ \nCombining Lemma [lem: GapUniform] and Theorem [thm: Achievable2], we can conclude that the following rate is achievable in the system defined in Section [sec: SysModel] where the energy arrival process is causally known only at the transmitter, but not at the receiver:\n \n$$\\begin{aligned}\n\\label{eq: achievable\\_2}\nR\\_{ach} \\ge \\displaystyle\\sum\\limits\\_{j = 0}^{\\infty} p & (1-p)^j \\frac{1}{2} \\log \\left(1 + \\mathcal{E}\\_j \\right) - H(p) - 1.04 \\end{aligned}$$ \nwith E*j* subject to the constraint. Notice that up to a fixed constant, we are back to the online rate optimization problem studied in Section [sec: OnlinePolicy] (see eq.). Employing the near-optimal allocation policy from the previous section, which assigns E*j* = *p*(1 − *p*)*j**B**m**a**x*, gives us the following achievable rate.\n[Lower Bound on Capacity: *B**m**a**x* ≤ *E*] [lem: LowerBoundC] When *E* ≥ *B**m**a**x*, the capacity of a system defined in Section [sec: SysModel] with Bernoulli energy arrival process {*E**t*} only causally known at the transmitter, but not at the receiver, is lower bounded by\n \n$$\\begin{aligned}\nC& (B\\_{max}, p) \\ge C\\_{lb}(B\\_{max}, p) \\notag\\\\\n& \\triangleq \\left( \\displaystyle\\sum\\limits\\_{j = 0}^{\\infty} p(1-p)^j \\frac{1}{2} \\log\\left( 1 + (1-p)^j p B\\_{max} \\right) - K(p)\\right)^+ \\label{eq: CapLowerBound}\\end{aligned}$$ \nwhere *a*+ = max(*a*, 0) and *K*(*p*) = 1.04 + *H*(*p*) where *H*(*p*) is the binary entropy function.\nThis lemma is a direct consequence of Theorem [thm: Achievable2] and Lemma [lem: GapUniform] with particular choice of E*j* = *p*(1 − *p*)*j**B**m**a**x*. We complete the proof by verifying {E*j*} satisfies the constraint since $\\displaystyle\\sum\\limits\\_{j = 0}^{\\infty} p (1-p)^j B\\_{max} = B\\_{max}$, and use the fact that the capacity is non-negative.\nThe next theorem states that for all values of *B**m**a**x* and *p*, the gap between the lower bound in and the upper bound in is bounded by a constant.\n[Constant Gap] [thm: ConstantGapBlessE] For all 0 < *p* < 1, *B**m**a**x* > 0, we have the following inequality:\n \n$$\\begin{aligned}\n\\label{eq: gap}\nC\\_{ub}(B\\_{max}, p) - C\\_{lb}(B\\_{max}, p) \\le 2.58 \\mbox{ bits}\\end{aligned}$$ \nThe detailed proof is given in the Appendix. Notice that in the proof of Theorem [thm: constgaprate], we have already bounded the gap between $\\displaystyle\\sum\\limits\\_{j = 0}^{\\infty} p(1-p)^j \\frac{1}{2} \\log\\left( 1 + (1-p)^j p B\\_{max} \\right)$ and $\\frac{1}{2}\\log\\left(1+pB\\_{max} \\right)$ by a constant, so it’s no surprise we have a constant bound here. The additional term *K*(*p*) in Lemma [lem: LowerBoundC] is why we have a larger constant, i.e. 2.58 bits, here.\nThe *B**m**a**x* > *E* regime\n-----------------------------\nIn Section [subsec: BlessE], we provided an upper bound and a lower bound on the capacity when *B**m**a**x* ≤ *E*. Because the energy must be stored into the battery before it can be used, the extra energy is wasted when *B**m**a**x* ≤ *E*. Therefore, the capacity depends only on *B**m**a**x* and not *E* in that case. When *B**m**a**x* > *E*, a natural question is how the extra battery space impacts the capacity of the system, and whether *B**m**a**x* or *E* (or both) is the determining factor for the capacity. We show below that the capacity depends critically on *E* in this case and not so much on the extra battery space *B**m**a**x*.\nThe bounds in Section [subsec: BlessE] can be summarized as:\n \n$$\\label{eq: two\\_sided}\n\\frac{1}{2} \\log \\left(1 + p B\\_{max} \\right) - 2.58 \\le C \\le \\frac{1}{2} \\log \\left(1 + p B\\_{max} \\right)$$ \nNotice, in the case *B**m**a**x* = *E*, this bound turns into:\n \n$$\\label{eq: B\\_equal\\_E}\n\\frac{1}{2} \\log \\left(1 + p E \\right) - 2.58 \\le C(B\\_{max} = E, p) \\le \\frac{1}{2} \\log \\left(1 + p E \\right)$$ \nNote that the right hand side of is the same as the infinite battery capacity given by. This gives rise to the following theorem.\n[Bounds on Capacity: *B**m**a**x* > *E*] [thm: BgreaterE] When the battery size *B**m**a**x* > *E*, the capacity of a system with Bernoulli energy arrival process defined in Section [sec: SysModel] is bounded by:\n \n$$\\label{eq: B\\_greater\\_E}\n\\frac{1}{2} \\log \\left(1 + p E \\right) - 2.58 \\le C \\le \\frac{1}{2} \\log \\left(1 + p E \\right)$$ \nSince having a larger battery can only help the capacity of the system, the capacity when *B**m**a**x* > *E* is at least as large as the capacity when *B**m**a**x* = *E*. Using the left hand side of, we have: when *B**m**a**x* > *E*\n \n$$\\begin{aligned}\nC & \\ge C(B\\_{max} = E, p) \\\\\n& \\ge \\frac{1}{2} \\log \\left(1 + p E \\right) - 2.58\\end{aligned}$$ \nSimilarly, using, we have the upper bound:\n \n$$\\begin{aligned}\nC & \\le C(B\\_{max} = \\infty, p) \\\\\n& \\leq \\frac{1}{2} \\log \\left(1 + p E \\right) \\end{aligned}$$ \nThis completes the proof of the theorem.\nGeneralization to other energy profiles\n=======================================\nThe near optimal energy allocation policy and the coding strategy we developed in the earlier sections, as well as the resultant constant gap approximation for the information-theoretic capacity of the AWGN energy harvesting communication channel were specific to the i.i.d. Bernoulli energy arrival process and it may seem difficult to extend these results to the more general settings. In this section, we present a simple way to apply these strategies in the more general settings of i.i.d. energy arrival processes.\nThe idea we propose is very simple: given an i.i.d. energy arrival process where *E**t* has an arbitrary distribution (discrete or continuous), fix an energy level *E* and find the probability *p* of having an energy arrival with packet size at least *E*, i.e. *p* = P(*E**t* ≥ *E*); then apply the near optimal energy allocation policy of Section [sec: OnlinePolicy] or the near optimal coding strategy of Section [sec: InfoCap] as if the exogenous energy process were i.i.d. Bernoulli with parameters *E* and *p*. Clearly, this is an energy feasible strategy for the general i.i.d. energy arrival process. However, a priori it may seem highly suboptimal and wasteful of energy since by treating the energy arrival process as Bernoulli with parameters *E* and *p*, we assume that there is no energy arrival when the arriving energy packet size is smaller than *E*, and we assume that the energy packet size is equal to *E* each time we receive an energy packet of size larger than or equal to *E*. Effectively, our strategy may not be utilizing a large fraction of the energy accumulating in the battery under the general i.i.d. energy arrival process. However, as we illustrate next this simple strategy turns out to be sufficient to achieve the optimal value for the energy allocation problem in Section [sec: OnlinePolicy] and the information-theoretic capacity of the channel within a constant gap for a large class of i.i.d. energy arrival processes, though with a larger constant. Note that as already discussed in Section [sec: simulations], the earlier strategies proposed in the literature would fail to provide a constant gap approximation.\nWe have the following theorem which is the extension of our main result in Theorem [thm: MainResult] to the general i.i.d. energy arrival processes. It is straightforward to write down the corresponding extensions of Theorems [thm: constgaprate] and [thm: constgaprate2] to this more general setting.\n[thm: Extension] Assume that the energy arrival process {*E**t*} is an i.i.d. random process with each *E**t* distributed according to an arbitrary cumulative distribution function *F*(*x*) such that *F*(*x*) = 0, ∀*x* < 0. Then for each 0 ≤ *x* ≤ *B**m**a**x*, the information theoretic capacity *C* of the corresponding AWGN energy harvesting channel with battery size *B**m**a**x* is bounded by\n \n$$\\label{eq: mainres\\_general}\n\\begin{split}\n\\frac{1}{2}\\log(1+ & x (1-F(x))) - 2.58 \\leq C\n\\\\ &\\leq \\frac{1}{2}\\log\\left(1+\\int\\_0^{B\\_{max}} \\! (1-F(y) ) \\, \\mathrm{d}y \\right).\n\\end{split}$$ \nFix an energy level *x*. The capacity *C* of our channel can be lower bounded by the capacity of the same channel when the exogenous energy arrival process is i.i.d. Bernoulli with energy packet size *E* = *x* and probability of energy packet arrival *p* = 1 − *F*(*x*) since we can always discard the additional energy. Then, applying Theorem [thm: MainResult] we immediately get the lower bound in.\nFor the upper bound, we notice when the arrival energy *E**t* > *B**m**a**x*, the extra *E**t* − *B**m**a**x* amount of energy is wasted. Therefore, the available resource for communication for this system is equivalent to a system where *E**t*, is distributed according to a cumulative distribution function:\n \n$$\\begin{aligned}\n\\label{eq: equiv\\_cdf}\n\\tilde{F}(x) = \\left\\{ \\begin{array}{rl}\n0 & \\mbox{for $x < 0$} \\\\\nF(x) & \\mbox{for $0 \\le x < B\\_{max}$ } \\\\\n1 & \\mbox{for $B\\_{max} \\le x$}\n\\end{array} \\right.\\end{aligned}$$ \nAgain, using the infinite battery capacity, we can upper bound the capacity of the system by:\n \n$$\\begin{aligned}\nC & \\le C\\_{\\infty} \\\\\n & = \\frac{1}{2}\\log(1+\\mathbb{E}(E\\_t)) \\\\\n & \\stackrel{(a)}{=} \\frac{1}{2} \\log\\left(1+\\int\\_{0}^{\\infty}(1-\\tilde{F}(y)) \\mathrm{d}y \\right) \\\\\n & \\stackrel{(b)}{=} \\frac{1}{2} \\log\\left(1+\\int\\_{0}^{B\\_{max}}(1-F(y)) \\mathrm{d}y \\right)\\end{aligned}$$ \nwhere (a) comes from a well-known identity relating the cdf of a non-negative random variable to its mean, (b) uses.\nNote that the above theorem holds for any value of *x* between 0 and *B**m**a**x*. For a given energy arrival process, we would want to optimize *x* to maximize the lower bound on capacity. In other words, the lower bound in the theorem can be equivalently rewritten as\n \n$$\\label{eq: gen\\_lower\\_bound}\n{\\frac{1}{2}\\log\\left( 1+ \\sup\\_{0\\leq x \\leq B\\_{max}}{x (1-F(x))}\\right) - 2.58} \\leq C.$$ \nUsing this lower bound, the approximation gap in the theorem (the difference between the lower and upper bounds) is given by\n \n$$\\label{eq: gen\\_gap}\n\\frac{1}{2}\\log\\left( \\frac{1+\\int\\_{0}^{B\\_{max}}(1-F(y)) \\mathrm{d}y}{1+ \\displaystyle\\sup\\limits\\_{0\\leq x \\leq B\\_{max}}{x (1-F(x))}} \\right) +2.58,$$ \nwhich can be upper bounded by\n \n$$\\label{eq: ratio}\n\\frac{1}{2}\\log\\left( \\frac{\\int\\_{0}^{B\\_{max}}(1-F(y)) \\mathrm{d}y}{\\displaystyle\\sup\\limits\\_{0\\leq x \\leq B\\_{max}}{x (1-F(x))}} \\right) +2.58.$$ \nsince the numerator is always greater than or equal to the denominator inside the log in Equation.\nFigure [fig: one-CDF] illustrates the ratio inside the logarithm for a given energy arrival process, characterized by the graph 1 − *F*(*x*) between 0 and *B**m**a**x*. The numerator in this fraction is the area under this graph, while the denominator is the largest area of a rectangle lying below this graph (as illustrated by the shaded rectangle). Therefore, as long as the cumulative distribution function has the property that this ratio is not too large, Theorem [thm: Extension] will yield a constant gap approximation of the capacity. As we illustrate next, two examples of such distributions are the uniform distribution and an energy arrival process with two discrete energy levels (as opposed to one energy level in the Bernoulli case). For these distributions we get a total approximation gap of 3.08 bits (as opposed to 2.58 bits in the Bernoulli case). However, not all cumulative distribution functions yield a constant gap approximation. We also provide a counter example (a sequence of energy profiles) for which the approximation gap provided by the theorem becomes arbitrarily large.\n [fig: one-CDF]\n [fig: Uniform-CDF]\nUniform Distribution\n--------------------\nAssume the energy arrival process is i.i.d. with *E**t* uniformly distributed over the interval [*A*1, *A*2] for some arbitrary 0 ≤ *A*1 < *A*2. We first assume that *B**m**a**x* ≥ *A*2. We have\n \n$$\\begin{split}\n\\int\\_{0}^{B\\_{max}}(1-F(y)) \\mathrm{d}y &= \\frac{A\\_1 +A\\_2}{2}\n\\\\ & = 2\\times\\frac{1}{2}\\times\\frac{A\\_1 +A\\_2}{2}\n\\\\ &\\leq 2\\times {\\sup\\_{0\\leq x \\leq B\\_{max}}{x (1-F(x))}}.\n\\end{split}$$ \nwe have the inequality in the last step because by choosing *x* = (*A*1 + *A*2)/2, we can achieve *x*(1 − *F*(*x*)) = (*A*1 + *A*2)/4 (See Figure [fig: Uniform-CDF]). With this choice of energy level *E*, i.e. the midpoint of the interval [*A*1, *A*2], the ratio inside the log in Equation is guaranteed to be a constant of 2 for any i.i.d. uniformly distributed energy arrival process independent of the values of *A*1 and *A*2. In particular, Theorem [thm: Extension] will approximate the capacity as\n \n$$\\label{eq: upper\\_uniform}\n\\frac{1}{2}\\log\\left(1+ \\frac{A\\_1+A\\_2}{2}\\right)\\quad\\text{bits}$$ \nwhen *B**m**a**x* ≥ *A*2 within a gap of 3.08 bits.\nWhen (*A*1 + *A*2)/2 ≤ *B**m**a**x* < *A*2, we can again choose *x* = (*A*1 + *A*2)/2, and achieve within 3.08 bits of, which clearly is also an upper bound for this case. When *A*1 ≤ *B**m**a**x* < (*A*1 + *A*2)/2, we can no longer choose *x* = (*A*1 + *A*2)/2, since *x* must be no more than *B**m**a**x*. However, in this case, if we simply choose *x* = *B**m**a**x*, the ratio of interest is still no more than 2, therefore, Theorem [thm: Extension] will approximate the capacity as 1/2log(1 + *B**m**a**x*) within 3.08 bits. Finally, when *B**m**a**x* ≤ *A*1, we are back to a degenerate Bernoulli case with arrival probability *p* = 1, and packet size *B**m**a**x*, so we have upper bound 1/2log(1 + *B**m**a**x*), and lower bound of 1/2log(1 + *B**m**a**x*) − 2.58.\nWe summarize the approximate capacity and the approximation gap for various regimes in the following table:\n[H]\n| Regime | Approximate Capacity | Gap |\n| --- | --- | --- |\n| $\\frac{A\\_1 + A\\_2}{2} \\le B\\_{max}$ | $\\frac{1}{2} \\log \\left(1 + \\frac{A\\_1+A\\_2}{2} \\right)$ | ≤ 3.08 |\n| $A\\_1 \\le B\\_{max} < \\frac{A\\_1 + A\\_2}{2}$ | $\\frac{1}{2}\\log\\left(1 + B\\_{max} \\right)$ | ≤ 3.08 |\n| *B**m**a**x* < *A*1 | $\\frac{1}{2} \\log \\left(1 + B\\_{max}\\right)$ | ≤ 2.58 |\n[table: Uniform]\nSimilar to the Bernoulli arrival process, based on Table [table: Uniform], we infer there are two qualitatively different regimes for the capacity of a system with uniform arrival process. In particular, when *B**m**a**x* < (*A*1 + *A*2)/2, the capacity is increasing roughly logarithmic in *B**m**a**x*, while when *B**m**a**x* ≥ (*A*1 + *A*2)/2 the capacity approximately saturates to $\\frac{1}{2} \\log \\left(1 + \\frac{A\\_1+ A\\_2}{2} \\right)$. Perhaps, what’s interesting here is that the threshold where this regime shift happens is neither *A*1 nor *A*2 by itself, but instead happens at the midpoint between *A*1 and *A*2.\nk-Level Distribution\n--------------------\nAssume now that the energy arrival process is again i.i.d. but *E**t* is a discrete random variable that takes the value *A**i* with probability *p**i* for *i* = 1, 2, …, *k*. Further, assume that 0 < *A*1 < *A*2 < ⋯ < *A**k* ≤ *B**m**a**x*. Note that the probability of having no energy packets is 1 − ∑*i* = 1*k**p**i*. Then we have\n \n$$\\begin{aligned}\n\\int\\_{0}^{B\\_{max}}(1-F(y)) \\mathrm{d}y &= \\sum\\_{i=1}^{k}{p\\_i A\\_i}\n\\\\ &\\le \\sum\\_{i=1}^{k}{A\\_i \\sum\\_{j=i}^{k}{p\\_j}}\n\\\\ &\\le k\\max\\_{1\\leq i\\leq k}{A\\_i \\sum\\_{j=i}^{k}{p\\_j}}\n\\\\ &= k {\\sup\\_{0\\leq x \\leq B\\_{max}}{x (1-F(x))}}.\\end{aligned}$$ \nThe last equality holds because the supremum is achieved when *x* in Theorem [thm: Extension] approaches *A**i* from left for which *A**i*∑*j* = *i**k**p**j* is maximized. This shows we can approximate the capacity as $\\frac{1}{2}\\log(1+\\mathbb{E}[E\\_t])$, when 0 < *A**i* ≤ *B**m**a**x* for all *i*, within a gap of at most\n \n$$\\label{eq: k\\_level\\_gap}\n\\frac{1}{2}\\log(k)+2.58\\quad\\text{bits}.$$ \nFor example, for *k* = 1, i.e. the Bernoulli case, we recover the gap of 2.58 bits. For the case *k* = 2, we obtain an approximation gap of 3.08 bits. Note that Equation is just an upper bound on the gap, and not necessarily tight. In particular, not all classes of distributions have gap increasing with k. For example, if all the *p**i*’s are the same and *A**i*’s are equally spaced within an interval [*A**a*, *A**b*]. Then as *k* → ∞, the distribution of *E**t* starts to approach a uniform distribution we examined in the previous subsection, which we know has a bounded gap of 3.08 bits. However, there does exist sequence of profiles, where as *k* increases, our approximation gap increases unboundedly. Indeed, in the last subsection, we will find an example of discrete profiles for which the gap can be arbitrarily large.\nIf *A**i* − 1 ≤ *B**m**a**x* < *A**i*, we can consider the equivalent distribution where we replace all *A**j*, *j* ≥ *i* with *B**m**a**x* and assign a probability mass ∑*j* = *i**k**p**j* to *B**m**a**x*. This case can be handled similarly as we did above but note that in this case the expression for the approximate capacity will depend on *A**j* and *p**j*’s for *j* < *i* as well as *B**m**a**x*.\nCounterexample\n--------------\nWhile the strategy has worked well for the above cases, here we provide a sequence of profiles for which the gap between the upper and lower bounds in Theorem [thm: Extension] can be made arbitrarily large. Define a sequence of profiles with cdf {*F**n*(*x*)}*n* = 1∞ as\n \n$$\\begin{aligned}\n1-F\\_n(x) = \\left\\{ \\begin{array}{rl}\n1 & \\mbox{$x < 1$} \\\\\n\\frac{1}{x} & \\mbox{$1\\leq x < n$} \\\\\n0 & \\mbox{$n \\le x$}\n\\end{array} \\right.\\end{aligned}$$ \nNow, suppose we have a channel with energy arrival process described by the cdf *F**n*(*x*). Without loss of generality, let *B**m**a**x* = *n*, i.e. the process takes on a support in [0, *B**m**a**x*]. Then,\n \n$$\\begin{aligned}\n\\frac{1+\\int\\_{0}^{B\\_{max}}(1-F\\_n(y)) \\mathrm{d}y}{1+{\\displaystyle\\sup\\limits\\_{0\\leq x \\leq B\\_{max}}{x (1-F\\_n(x))}}} & \\stackrel{(a)}{=} \\frac{1+\\int\\_{0}^{1}1 \\mathrm{d}y +\\int\\_{1}^{n}\\frac{1}{y} \\mathrm{d}y}{1+1} \\notag \\\\ \n&= 1+\\frac{\\ln(n)}{2} \\label{eq: unbounded\\_ratio}\\end{aligned}$$ \nwhere (a) is true due to the way we constructed *F**n*(*x*), in particular:\n \n$$\\begin{aligned}\nx(1-F\\_n(x)) = \\left\\{ \\begin{array}{rl}\nx & \\mbox{for $x < 1$} \\\\\n1 & \\mbox{for $1 \\le x < n$ } \\\\\n0 & \\mbox{for $n \\le x $}\n\\end{array} \\right.\\end{aligned}$$ \nSubstitute the ratio from into Equation, we obtain a gap between the upper and lower bounds in Theorem [thm: Extension] of:\n \n$$\\label{eq: unbounded\\_gap}\n\\frac{1}{2}\\log\\left(1 + \\frac{\\ln(n)}{2} \\right) + 2.58$$ \nwhich can be made arbitrarily large as *n* grows unboundedly. This suggests that the approximation gap using our strategy can not be bounded by a constant as we did in the previous examples. Alternatively, the unboundedness of the ratio can be seen from Figure [fig: Counterexample] where the area of every shaded bounded rectangle is no more than 1, but the area under the graph becomes arbitrarily large as *n* goes to infinity.\n [fig: Counterexample]\nAs mentioned earlier, it is possible to come up with a discrete counterexample by discretizing this sequence of profiles at integer levels. In particular, let the energy arrival process be a *k*-level distribution with *A**i* = *i* for 1 ≤ *i* ≤ *k*, $p\\_i=\\frac{1}{i}-\\frac{1}{i+1}$ for 1 ≤ *i* ≤ *k* − 1 and $p\\_k = \\frac{1}{k}$. Again, let *B**m**a**x* = *k* to avoid complication due to truncation of the cdf. Then the ratio inside the log in can be written as:\n \n$$\\begin{aligned}\n\\frac{1+\\int\\_{0}^{B\\_{max}}1-F\\_k(y) \\mathrm{d}y}{1+{\\displaystyle\\max\\limits\\_{1\\leq i \\leq k}{A\\_i (1-F\\_k(A\\_i^-))}}} &= \\frac{1+\\sum\\_{i=1}^{k}{\\frac{1}{i}}}{1+\\displaystyle\\max\\limits\\_{1\\leq i \\leq k}{i \\times \\frac{1}{i}}}\n\\\\ &= \\frac{1+\\sum\\_{i=1}^{k}{\\frac{1}{i}}}{2}\n\\\\ &\\geq \\frac{1+\\ln(k+1)}{2}\\end{aligned}$$ \nwhere *F**k*(*A**i*−) = lim*x* ↑ *A**i**F**k*(*x*). This shows the ratio can again grow unboundedly as *k* → ∞. The counterexamples in this subsection show that there are both discrete and continuous profiles, for which our strategy can not achieve the upper bounds within a constant gap, although such profiles may not be common in practice. A natural future direction is to obtain approximate characterizations of the energy harvesting communication channel under general energy harvesting profiles.\nAcknowledgment\n==============\nWe would like to thank S. Ulukus for stimulating discussions on the topic and the anonymous reviewers for their detailed comments and suggestions which significantly improved the paper.\nAppendix\n========\n[Proof of Theorem [thm: constgaprate]] Here, we provide the detailed proof to the three steps mentioned in the main context.\n*Step One:* Since we are in the case *B**m**a**x* ≤ *E*, each time a non-zero energy packet arrives, the battery will be filled up completely regardless of how much energy was remaining in the battery. In particular, at least *E* − *B**m**a**x* amount of energy is wasted in every non-zero incoming energy packet. Therefore, a system with Bernoulli arrival process *Ẽ**t* where the energy packet size is *Ẽ* = *E* − (*E* − *B**m**a**x*) = *B**m**a**x* is completely equivalent to our original system in terms of the amount of the available energy for communication.\nNow, let *g*(*t*) be any policy in G. We have:\n \n$$\\begin{aligned}\n\\liminf\\_{N \\to \\infty} \\mathbb{E} & \\left[ \\frac{1}{N} \\displaystyle\\sum\\limits\\_{t = 1}^N \\frac{1}{2} \\log\\left( 1 + g( t)\\right) \\right]\n\\\\\n& \\stackrel{(a)}{\\leq} \\liminf\\_{N \\to \\infty} \\mathbb{E} \\left[ \\frac{1}{2} \\log\\left( 1 + \\frac{1}{N} \\displaystyle\\sum\\limits\\_{t = 1}^N g( t)\\right) \\right] \\\\\n& \\stackrel{(b)}{\\leq} \\liminf\\_{N \\to \\infty} \\mathbb{E} \\left[ \\frac{1}{2} \\log\\left( 1 + \\frac{1}{N} \\displaystyle\\sum\\limits\\_{t = 1}^N \\tilde{E}\\_{t} \\right) \\right] \\\\\n& \\stackrel{(c)}{=} \\mathbb{E} \\left[ \\liminf\\_{N \\to \\infty} \\frac{1}{2} \\log\\left( 1 + \\frac{1}{N} \\displaystyle\\sum\\limits\\_{t = 1}^N \\tilde{E}\\_{t}\\right) \\right] \\\\\n& = \\; \\frac{1}{2} \\log\\left( 1 + pB\\_{max}\\right)\\end{aligned}$$ \nwhere (a) follows from the concavity of the log which allows to apply Jensen’s Inequality; (b) follows from the argument at the beginning that we can equivalently consider a Bernoulli arrival process *Ẽ**t* where the energy packet is *Ẽ* = *B**m**a**x* and the fact that *g*(*t*) is feasible, so we can not spend more energy up to time *N* than the amount of exogenous energy we receive by time *N*, i.e. \n$$\\sum\\limits\\_{t = 1}^N g( t)\\leq \\sum\\limits\\_{t = 1}^N \\tilde{E}\\_{t};$$ \n (c) follows from the Dominated Convergence Theorem. The Dominated Convergence Theorem holds because first,\n \n$$\\begin{aligned}\n \\lim\\_{N \\to \\infty} \\frac{1}{2} &\\log\\left( 1 + \\frac{1}{N} \\displaystyle\\sum\\limits\\_{t = 1}^N \\tilde{E}\\_{t}\\right)\\\\ &\\stackrel{(d)}{=} \\frac{1}{2} \\log\\left( 1 + \\liminf\\_{N \\to \\infty} \\frac{1}{N} \\displaystyle\\sum\\limits\\_{t = 1}^N \\tilde{E}\\_{t} \\right) \\\\\n& \\stackrel{(e)}{=} \\frac{1}{2} \\log\\left( 1 + \\mathbb{E}[\\tilde{E}\\_{t}] \\right) \\\\\n& = \\; \\frac{1}{2} \\log\\left( 1 + pB\\_{max}\\right)\\end{aligned}$$ \nalmost surely, where (d) follows from the fact $\\frac{1}{2}\\log(1+x)$ is smooth and monotonically increasing so we can move the liminf inside; and (e) follows from the Law of Large Numbers; and second \n$$\\frac{1}{2} \\log \\left( 1 + \\frac{1}{N} \\displaystyle\\sum\\limits\\_{t=1}^{N} \\tilde{E}\\_{t} \\right) \\le \\frac{1}{2} \\log \\left(1 + B\\_{max} \\right)$$ \n for all *N* which is finite. Since the above upper bound applies to all feasible policies *g* ∈ G, we have the desired upper bound.\n*Step Two:* Let {*T**i*}*i* = 1*L* be the inter-arrival times between the *i*’th and *i* + 1’th non-zero energy packets, where *L* is the total number of non-zero packets received by time instance N, i.e. ∑*i* = 1*L**T**i* ≤ *N* < ∑*i* = 1*L* + 1*T**i*. Notice that for fixed *N*, *T**i*’s and *L* defined in this way are random variables which are functions of *N* and the random energy arrival process {*E**t*}*t* = 0*N*. We can lower bound the rate achieved by *g*ʹ(*t*) in terms of these new variables as\n \n$$\\begin{aligned}\n &\\liminf\\_{N \\to \\infty} \\; \\mathbb{E} \\left[ \\frac{1}{N} \\displaystyle\\sum\\limits\\_{t = 1}^N \\frac{1}{2} \\log\\left( 1 + g^\\prime(t)\\right) \\right] \\nonumber\\\\\n& \\;\\;\\geq \\; \\liminf\\_{N \\to \\infty} \\mathbb{E} \\left[ \\displaystyle\\sum\\limits\\_{i = 1}^L \\displaystyle\\sum\\limits\\_{j = 0}^{T\\_i-1} \\frac{1}{2} \\log\\left( 1 + \\tilde{g}(j)\\right) \\bigg / \\displaystyle\\sum\\limits\\_{i = 1}^{L+1} T\\_i \\right]\\label{eq:lln1}\\end{aligned}$$ \nwhich follows from the fact that the strategy *g*′(*t*) we consider is of the form *g*′(*t*) = *g̃*(*j*) where *j* = *t* − max{*t*ʹ ≤ *t* : *E**t*ʹ = *E*}, i.e., the strategy is invariant across different epochs and the allocated energy depends only on the number of time steps since the last energy arrival.\nNotice that as *N* → ∞, *L* → ∞ with probability 1. Divide both the numerator and the denominator of the last equation by *L* and apply the Law of Large Numbers to both the numerator and the denominator, we obtain\n \n$$\\begin{aligned}\n & \\liminf\\_{N \\to \\infty} \\;\\displaystyle\\sum\\limits\\_{i = 1}^L \\displaystyle\\sum\\limits\\_{j = 0}^{T\\_i-1} \\frac{1}{2} \\log\\left( 1 + \\tilde{g}(j)\\right) \\bigg / \\displaystyle\\sum\\limits\\_{i = 1}^{L+1} T\\_i \\nonumber \\\\\n&\\;\\; = \\;\\; \\mathbb{E}\\left[ \\displaystyle\\sum\\limits\\_{j = 0}^{T\\_1-1} \\frac{1}{2} \\log\\left( 1 + \\tilde{g}(j)\\right)\\right] \\bigg / \\mathbb{E}[T\\_1].\\label{eq:lln0}\\end{aligned}$$ \nNote that {*T**i*}’s are i.i.d. Geometric(*p*) so the Law of Large Numbers is directly applicable to the denominator and it also applies to the numerator since the random variables $\\sum\\limits\\_{j = 0}^{T\\_i-1} \\frac{1}{2} \\log\\left( 1 + \\tilde{g}(j)\\right)$ are i.i.d. with finite mean. Now, because the sequence of random variables in converges almost surely, and for every *N* the sequence is upper bounded, we can apply the Dominated Convergence Theorem to exchange the limit and the expectation in to obtain\n \n$$\\begin{aligned}\n \\liminf\\_{N \\to \\infty}\\; &\\mathbb{E} \\left[ \\frac{1}{N} \\displaystyle\\sum\\limits\\_{t = 1}^N \\frac{1}{2} \\log\\left( 1 + g^\\prime(t)\\right) \\right] \\\\\n& \\geq \\mathbb{E}\\left[ \\displaystyle\\sum\\limits\\_{j = 0}^{T\\_1-1} \\frac{1}{2} \\log\\left( 1 + \\tilde{g}(j)\\right)\\right] \\bigg / \\mathbb{E}[T\\_1].\\end{aligned}$$ \nWe have\n \n$$\\begin{aligned}\n\\mathbb{E}&\\left[ \\displaystyle\\sum\\limits\\_{j = 0}^{T\\_1-1} \\frac{1}{2} \\log\\left( 1 + \\tilde{g}(j)\\right)\\right] \\bigg / \\mathbb{E}[T\\_1]\\;\\;\\;\\\\\n = \\;\\; & p \\displaystyle\\sum\\limits\\_{i = 1}^{\\infty} \\mathbb{P}(T\\_1 = i) \\displaystyle\\sum\\limits\\_{j = 0}^{i-1} \\frac{1}{2} \\log(1+\\tilde{g}(j)) \\\\\n\\stackrel{(a)}{=} \\;\\; & p \\displaystyle\\sum\\limits\\_{i = 1}^{\\infty} (1-p)^{i-1} p \\displaystyle\\sum\\limits\\_{j = 0}^{i-1} \\frac{1}{2} \\log(1+\\tilde{g}(j)) \\\\\n \\stackrel{(b)}{=} \\;\\; & p \\displaystyle\\sum\\limits\\_{j = 0}^{\\infty} \\left( \\displaystyle\\sum\\limits\\_{i = j+1}^{\\infty} (1-p)^{i-1} p \\right) \\frac{1}{2} \\log(1+\\tilde{g}(j)) \\\\\n \\stackrel{(c)}{=} \\;\\; & \\displaystyle\\sum\\limits\\_{j = 0}^{\\infty} p(1-p)^j \\frac{1}{2} \\log(1 + \\tilde{g}(j)) \\\\\n = \\;\\; & \\displaystyle\\sum\\limits\\_{j = 0}^{\\infty} p(1-p)^j \\frac{1}{2} \\log(1 + p(1-p)^j B\\_{max}) \\end{aligned}$$ \nwhere (a) follows from the fact that {*T*1} is Geometric(*p*), (b) follows from switching the order of summations, and (c) uses the formula for the sum of geometric series.\n*Step Three:* Finally, to complete the proof, we need to show:\n \n$$\\begin{aligned}\n\\label{eq: constant\\_gap\\_upper}\n\\frac{1}{2} \\log(1 + p B\\_{max}) - \\displaystyle\\sum\\limits\\_{j = 0}^{\\infty} p (1-p)^j \\frac{1}{2} \\log(1 + \\tilde{g}(j)) \\le 0.973\\end{aligned}$$ \nIn order to show, we will first restrict the range of (*B**m**a**x*, *p*) to be considered by noticing:\n \n$$\\begin{aligned}\n\\frac{1}{2} \\log(1 + p B\\_{max}) - & \\displaystyle\\sum\\limits\\_{j = 0}^{\\infty} p (1-p)^j \\frac{1}{2} \\log(1 + \\tilde{g}(j)) \\\\\n& \\le \\frac{1}{2} \\log(1 + p B\\_{max}) \\\\\n& \\le 0.973\\end{aligned}$$ \nfor all (*B**m**a**x*, *p*) such that *p**B**m**a**x* ≤ 2.853.\nSo let’s restrict ourself to the set of (*B**m**a**x*, *p*) such that *p**B**m**a**x* > 2.853. We have\n \n$$\\begin{aligned}\n& \\frac{1}{2} \\log(1 + p B\\_{max}) - \\displaystyle\\sum\\limits\\_{j = 0}^{\\infty} p (1-p)^j \\frac{1}{2} \\log(1 + \\tilde{g}(j)) \\\\\n= \\; & \\frac{1}{2} \\log(1 + p B\\_{max}) \\\\\n& - \\displaystyle\\sum\\limits\\_{j = 0}^{\\infty} p (1-p)^j \\frac{1}{2} \\log(1 + p(1-p)^j B\\_{max}) \\\\\n\\stackrel{(a)}{\\le} \\; & \\frac{1}{2} \\log(1 + p B\\_{max}) - \\displaystyle\\sum\\limits\\_{j = 0}^{\\infty} p (1-p)^j \\frac{1}{2} \\log(p(1-p)^j B\\_{max})) \\\\\n = \\; & \\frac{1}{2}\\log(p) + \\frac{1}{2}\\log(B\\_{max}) + \\frac{1}{2}\\log\\left(\\frac{1}{pB\\_{max}} + 1\\right) \\\\\n& - \\displaystyle\\sum\\limits\\_{j = 0}^{\\infty} p(1-p)^j \\frac{1}{2} \\left[\\log(p) + j\\log\\left(1-p\\right) +\\log( B\\_{max})\\right] \\\\\n\\stackrel{(b)}{=} \\; & \\frac{1}{2}\\log(p) + \\frac{1}{2}\\log(B\\_{max}) + \\frac{1}{2}\\log\\left(\\frac{1}{pB\\_{max}} + 1\\right) \\\\\n& - \\frac{1}{2}\\log(p) - \\frac{1-p}{p} \\frac{1}{2} \\log\\left(1-p\\right) -\\frac{1}{2} \\log( B\\_{max}) \\\\\n= \\; & \\frac{1}{2}\\log\\left(\\frac{1}{pB\\_{max}} + 1\\right) - \\frac{1-p}{p} \\frac{1}{2} \\log\\left(1-p\\right) \\\\\n\\stackrel{(c)}{\\le} \\; & \\frac{1}{2\\ln(2)}\\frac{1}{pB\\_{max}} - \\frac{1-p}{p} \\frac{1}{2} \\log\\left(1-p\\right) \\\\\n\\stackrel{(d)}{\\le} \\; & \\frac{1}{2\\ln(2)}\\frac{1}{2.853} + \\frac{1-p}{2p} \\log\\left(\\frac{1}{1-p}\\right) \\\\\n\\stackrel{(e)}{\\le} \\; & 0.253 + 0.72 \\\\\n= \\; & 0.973\\end{aligned}$$ \nwhere (a) follows from the fact that removing the 1 inside the second log results in an upper bound; (b) uses the identity $\\displaystyle\\sum\\limits\\_{j=0}^{\\infty} j \\cdot p(1-p)^j = (1-p) \\mathbb{E}\\left[X\\right] = \\frac{1-p}{p}$, where *X* ∼ Geometric(*p*), and $\\displaystyle\\sum\\limits\\_{j = 0}^{\\infty} p(1-p)^j = 1$; (c) uses the inequality ln(1 + *x*) ≤ *x* for all *x*; (d) follows from the fact we restrict ourself to the case *p**B**m**a**x* > 2.853; and finally (e) follows from the fact that\n \n$$\\begin{aligned}\ng(p) & \\triangleq \\frac{1-p}{2p} \\log\\left(\\frac{1}{1-p}\\right)\\end{aligned}$$ \nis a continuous bounded function on *p* ∈ (0, 1). Furthermore, it’s monotonically decreasing and is upper bounded by $\\displaystyle\\lim\\_{p \\to 0} g(p) = \\frac{1}{2\\ln(2)} = 0.72$.\n[Proof of Theorem [thm: constgaprate2]] The highlight of the proof has been outlined in the comment after the statement of Theorem [thm: constgaprate2] in Section [subsec: rateBgreaterE]. The proof is essentially the same as Theorem [thm: constgaprate] with *B**m**a**x* replaced by E. Therefore, we will simply reiterate the major steps of the proof without boring the reader with all the details.\n*Step One:* Using Jensen’s Inequality, Dominated Convergence Theorem and Law of Large Numbers, we have the following upper bound:\n \n$$\\begin{aligned}\n\\max\\_{g \\in \\mathcal{G}} \\liminf\\_{N \\to \\infty} \\mathbb{E} \\left[ \\frac{1}{N} \\displaystyle\\sum\\limits\\_{t = 1}^N \\frac{1}{2} \\log\\left( 1 + g(t)\\right) \\right] \\leq \\frac{1}{2}\\log(1+pE)\\end{aligned}$$ \n*Step Two:* Replace *B**m**a**x* by *E* in Step Two in the proof of Theorem [thm: constgaprate] and follow the exact same sequence of arguments, we can lower bound the rate achieved by *g*′(*t*) as:\n \n$$\\begin{aligned}\n\\liminf\\_{N \\to \\infty} & \\; \\mathbb{E} \\left[ \\frac{1}{N} \\displaystyle\\sum\\limits\\_{t = 1}^N \\frac{1}{2} \\log\\left( 1 + g^\\prime(t)\\right) \\right] \\\\\n& \\ge \\displaystyle\\sum\\limits\\_{j = 0}^{\\infty} p(1-p)^j \\frac{1}{2} \\log(1 + p(1-p)^j E) \\end{aligned}$$ \n*Step Three:* Finally, replace *B**m**a**x* everywhere by *E* in Step Three in the proof of Theorem [thm: constgaprate], we have the following bound:\n \n$$\\begin{aligned}\n\\frac{1}{2} \\log(1 + p E) - \\displaystyle\\sum\\limits\\_{j = 0}^{\\infty} p (1-p)^j \\frac{1}{2} \\log(1 + p(1-p)^j & E ) \\\\\n & \\le 0.973\\end{aligned}$$ \nwhich completes the proof.\n[Proof of Theorem [thm: Achievable]] We want to show that for any {E*j*} satisfying and any *ε* > 0, we can construct a communication strategy that achieves a rate\n \n$$\\begin{aligned}\n R \\ge \\displaystyle\\sum\\limits\\_{j = 0}^{\\infty} p & (1-p)^j \\max\\_{p(x): |X|^2 \\le \\mathcal{E}\\_j} I(X; Y) - \\epsilon\\end{aligned}$$ \nand has arbitrarily small probability of error.\nWe start by fixing a positive integer *M*(*ε*), and positive real numbers *δ*2(*ε*, *M*) and *δ*1(*ε*, *M*, *δ*2) such that\n \n$$\\begin{aligned}\n& (1 - p)^{M+1} K \\le \\epsilon/3 \\label{eq: M\\_Constraint} \\\\\n\\frac{\\delta\\_2}{1+\\delta\\_2} & \\left( 1 - (1-p)^{M+1} \\right) K \\le \\epsilon/3 \\label{eq: delta2\\_Constraint} \\\\\n& \\delta\\_1 \\cdot \\frac{(M+1)p}{1 + \\delta\\_2} K \\le \\epsilon/3 \\label{eq: delta1\\_Constraint}\\end{aligned}$$ \nwhere $K \\triangleq \\displaystyle\\max\\_{j} \\left\\{ \\displaystyle\\max\\_{p(x): |X|^2 \\le \\mathcal{E}\\_j} I(X; Y) \\right\\}$ is a finite constant. Furthermore, let *L* be a large positive integer and let\n \n$$\\begin{aligned}\nn^{(j)} = L\\left( (1-p)^j - \\delta\\_1 \\right)\\end{aligned}$$ \nFor a given energy allocation policy {E*j*} for *j* = 0, 1, 2, ... that satisfies, the transmitter and the receiver agree on a sequence of *M* + 1 codebooks:\n \n$$\\begin{aligned}\n\\mathcal{C}^{(0)}, \\mathcal{C}^{(1)}, \\mathcal{C}^{(2)},..., \\mathcal{C}^{(M)}\\end{aligned}$$ \nwhere each codebook C(*j*) consists of 2*n*(*j*)*R*(*j*) codewords generated randomly from a distribution *p*(*x*) whose support is limited to $[-\\sqrt{\\mathcal{E}\\_j},\\sqrt{\\mathcal{E}\\_j}]$ so that the symbols of each codeword are amplitude constrained according to\n \n$$\\begin{aligned}\n|X\\_i^{(j)}(k)|^2 \\le \\mathcal{E}\\_j\\end{aligned}$$ \nfor all *i* = 1, 2, ..., *n*(*j*), *j* = 0, 1, 2..., *M* and *k* = 1, 2, ..., 2*n*(*j*)*R*(*j*).\nDuring the course of communication the transmitter will aim to send one codeword from each of the *M* + 1 codebooks. Communication proceeds as follows: At each time step *t*, the transmitter sees the realization of the energy process *E**t*, let *j* = *t* − max{*t*ʹ ≤ *t* : *E**t*ʹ = *E*}, i.e. the number of time steps since the last time battery was recharged. Then the transmitter selects the next untransmitted symbol from the codeword it wants to transmit from the codebook C(*j*) (if all *n*(*j*) symbols have been transmitted, it simply transmits the zero symbol). Similarly if *j* > *M*, it transmits the zero symbol. Communication ends when the transmitter observes the arrival of the *L* + 1’th energy packet. (We assume that communication starts with the arrival of the first energy packet). The receiver can track the codebook used by the transmitter and decode each codeword separately by collecting together the corresponding channel observations since we assume that {*E**t*} are also causally known at the receiver.\nWe will say that communication is in error if one of the following events occur:\n* *E*(*j*), *j* = 0, …, *M* : by the end of the transmission, the codeword from C(*j*) has not been completely transmitted.=-10\n* *E*0 : the total duration of the communication exceeds $\\frac{L}{p}(1 + \\delta\\_2)$.=-10\n* *F*(*j*), *j* = 0, …, *M* : the codeword from codeword C(*j*) is decoded erroneously at the receiver.=-10\nBelow we will argue that the probability of each one of these events can be made arbitrarily small by taking *L* large enough.\nTo bound the probability of *E*(*j*), recall the inter arrival time of energy packets, denoted by *T*1, *T*2, ..., *T**L*, are i.i.d. Geometric(*p*) r.v’s. Therefore,\n \n$$\\begin{aligned}\n\\mathbb{P}(E^{(j)}) & = \\mathbb{P}\\left( \\displaystyle\\sum\\limits\\_{i = 1}^{L} \\mathbbm{1}\\{ T\\_i >j \\} \\le n^{(j)}\\right) \\\\\n& = \\mathbb{P}\\left( \\frac{1}{L} \\displaystyle\\sum\\limits\\_{i = 1}^{L} \\mathbbm{1}\\{ T\\_i >j \\} \\le (1-p)^j - \\delta\\_1 \\right) \\\\\n & \\le \\epsilon\\_1 ( L) \\rightarrow 0\\end{aligned}$$ \nas *L* → ∞ by the weak law of large number, since $\\mathbb{E}[ \\mathbbm{1}\\{ T\\_i >j \\} ] = \\mathbb{P}( T\\_i >j) = (1-p)^j$.\nFurthermore, notice the total duration of communication *N* = *T*1 + ⋯ + *T**L* is a random quantity. Then, we have\n \n$$\\begin{aligned}\n\\mathbb{P}(E\\_0) & = \\mathbb{P}\\left( \\displaystyle\\sum\\limits\\_{i = 1}^{L} T\\_i > \\frac{L}{p}(1 + \\delta\\_2) \\right) \\\\\n& = \\mathbb{P}\\left( \\frac{1}{L} \\displaystyle\\sum\\limits\\_{i = 1}^{L} T\\_i > \\frac{1}{p}(1 + \\delta\\_2) \\right) \\\\\n & \\le \\epsilon\\_2 ( L) \\rightarrow 0\\end{aligned}$$ \nas *L* → ∞, again by the weak law of large numbers, since E[*T**i*] = 1/*p*.\nOn the receiver side, the decoder can decode each individual codeword via joint typical decoding once the transmission is finished. The classic channel coding theorem shows that as long as \n*R*(*j*) ≤ max*p*(*x*) : ∣*X*∣2 ≤ E*j**I*(*X*; *Y*) \n the probability of each one of the events *F*(0), …, *F*(*M*) can be made arbitrarily small by making *L*, hence *n*(*j*), large enough, using codewords from each codebook C(*j*).\nFinally, the union bound allows us to conclude that the probability of the union of all these error events can be made arbitrarily small as *L* → ∞ (note that *δ*1, *δ*2 > 0, and *M* are fixed at the beginning).\nWe finally compute the average rate achieved by this communication strategy\n \n$$\\begin{aligned}\n& R \\ge \\frac{p}{L(1+ \\delta\\_2)} \\displaystyle\\sum\\limits\\_{j = 0}^{M} n^{(j)} R^{(j)} \\\\\n = & \\frac{p}{L(1+ \\delta\\_2)} \\displaystyle\\sum\\limits\\_{j = 0}^{M} L\\left( (1-p)^j - \\delta\\_1 \\right)\\max\\_{p(x): |X|^2 \\le \\mathcal{E}\\_j} I(X; Y) \\\\\n = & \\displaystyle\\sum\\limits\\_{j = 0}^{M} \\frac{p}{1+\\delta\\_2} \\left( (1-p)^j - \\delta\\_1 \\right) \\max\\_{p(x): |X|^2 \\le \\mathcal{E}\\_j} I(X; Y) \\\\\n= & \\displaystyle\\sum\\limits\\_{j = 0}^{\\infty} p(1-p)^j \\max\\_{p(x): |X|^2 \\le \\mathcal{E}\\_j} I(X; Y) \\\\ \n& - \\displaystyle\\sum\\limits\\_{j = M+1}^{\\infty} p(1-p)^j \\max\\_{p(x): |X|^2 \\le \\mathcal{E}\\_j} I(X; Y) \\\\ \n& - \\displaystyle\\sum\\limits\\_{j = 0}^{M} \\frac{\\delta\\_2 p}{1+\\delta\\_2} (1-p)^j \\max\\_{p(x): |X|^2 \\le \\mathcal{E}\\_j} I(X; Y) \\\\\n & - \\displaystyle\\sum\\limits\\_{j = 0}^{M} \\frac{p}{1+\\delta\\_2} \\delta\\_1 \\max\\_{p(x): |X|^2 \\le \\mathcal{E}\\_j} I(X; Y) \\\\\n\\ge & \\displaystyle\\sum\\limits\\_{j = 0}^{\\infty} p(1-p)^j \\max\\_{p(x): |X|^2 \\le \\mathcal{E}\\_j} I(X; Y) - (\\epsilon/3 +\\epsilon/3+\\epsilon/3) \\end{aligned}$$ \nwhich is what we set up to prove. The last step follows from the fact *M*, *δ*2 and *δ*1 are chosen to satisfy the bounds, and.\n[Proof of Theorem [thm:CapCSIR]][7](#fn7) The capacity of a general channel is given by as the maximum *spectral inf-mutual information rate* between the input and the output, \n$$C= \\sup\\_X\\text{p-} \\liminf\\_{n\\rightarrow\\infty} \\frac{1}{n}\\log \\frac{P\\_{Y^n\\vert X^n}({y^n\\vert x^n})}{P\\_{Y^n}(y^n)},$$ \n where the supremum is over all input processes *X* = {*X**n*}*n* = 1∞. When the receiver has access to a side information process *G* = {*G**n*}*n* = 1∞, this process can be viewed as part of the output of the channel and the corresponding capacity becomes \n$$C\\_G =\\sup\\_X\\; \\text{p-}\\liminf\\_{n\\rightarrow\\infty}\\: \\frac{1}{n}\\log \\frac{P\\_{Y^n,G^n\\vert X^n}(y^n,g^n \\vert x^n)}{P\\_{Y^n,G^n}(y^n,g^n)}.$$ \n We have\n \n$$\\begin{aligned}\nC\\_G &= \\sup\\_X\\; \\text{p-}\\liminf\\_{n\\rightarrow\\infty}\\: \\frac{1}{n}\\log \\frac{P\\_{Y^n,G^n\\vert X^n}(y^n,g^n \\vert x^n)}{P\\_{Y^n,G^n}(y^n,g^n)} \\nonumber\\quad\\\\\n&= \\sup\\_X\\; \\text{p-}\\liminf\\_{n\\rightarrow\\infty}\\: \\Big[ \\frac{1}{n}\\log \\frac{P\\_{Y^n\\vert X^n}(y^n \\vert x^n)}{P\\_{Y^n}(y^n)}\\nonumber \\\\\n&\\qquad\\qquad + \\frac{1}{n}\\log \\frac{P\\_{G^n\\vert Y^n,X^n}(g^n \\vert y^n, x^n)}{P\\_{G^n\\vert Y^n}(g^n\\vert y^n)} \\Big] \\nonumber\n\\\\ \n& \\le \\sup\\_X \\text{p-}\\liminf\\_{n\\rightarrow\\infty}\\: \\frac{1}{n}\\log \\frac{P\\_{Y^n\\vert X^n}(y^n \\vert x^n)}{P\\_{Y^n}(y^n)} \\nonumber\\\\\n&\\quad + \\sup\\_X \\text{p-}\\limsup\\_{n\\rightarrow\\infty} \\frac{1}{n}\\log \\frac{P\\_{G^n\\vert Y^n,X^n}(g^n \\vert y^n, x^n)}{P\\_{G^n\\vert Y^n}(g^n\\vert y^n)} \\label{sum2sup} \\\\\n& = \\: C \\: + \\sup\\_X \\text{p-}\\limsup\\_{n\\rightarrow\\infty} \\frac{1}{n}\\log \\frac{P\\_{G^n\\vert Y^n,X^n}(g^n \\vert y^n, x^n)}{P\\_{G^n\\vert Y^n}(g^n\\vert y^n)} \\nonumber\\\\\n& \\le \\: C \\: + \\sup\\_X \\text{p-}\\limsup\\_{n\\rightarrow\\infty} \\frac{1}{n}\\log \\frac{1}{P\\_{G^n\\vert Y^n}(g^n\\vert y^n)} \\label{Probless1}\\\\\n& = C \\: + \\sup\\_{X} \\bar{H}(G\\vert Y) \\label{VH:CondEntDef}\\\\\n& \\le C \\: + \\sup\\_{X} \\bar{H}(G). \\label{VH:I}\\end{aligned}$$ \nHere, ([sum2sup]) follows from the fact that for any two sequences of random variables (*V**n*)*n* = 1∞ and (*Z**n*)*n* = 1∞ we have \n \np-liminf*n* → ∞(*V**n* + *Z**n*) ≤ p-liminf*n* → ∞*V**n* + p-limsup*n* → ∞*Z**n*. \nNote that *P**G**n*|*Y**n*, *X**n*(*g**n*|*y**n*, *x**n*) ≤ 1, thus ([Probless1]) holds. ([VH:CondEntDef]) follows by the definition of the *conditional spectral sup-entropy rate*, given as\n \n$$\\bar{H}(G\\vert Y)=\\text{p-}\\limsup\\_{n\\rightarrow\\infty}\\frac{1}{n}\\log \\frac{1}{P\\_{G^n \\vert Y^n}(g^n\\vert y^n)}.$$ \nFinally, non-negativity of the inf-mutual information rate \n \n$$0 \\le \\underline{I}(G;Y) \\le \\bar{H}(G) - \\bar{H}(G\\vert Y),$$ \nyields ([VH:I]). In ([VH:I]), we assume that the side information process *G* is independent of the input process, which yields the conclusion of the theorem. Note that when the side information process is a function of the input, the capacity gain due to receiver side information can be bounded by considering the sup of the spectral sup-entropy rate of *G* over the input process.\n[Proof of Theorem [thm: Achievable2]] Here, we provide an alternative proof for Theorem [thm: Achievable2] that does not use Theorem [thm:CapCSIR]. We first define a specific finite state channel and establish a relation between its capacities with and without channel state information at the receiver, denoted by *C*1 and *C*2 respectively. In particular, we show that *C*2 ≥ *C*1 − *H*(*p*). We then relate the capacity of this finite state channel to our original communication setup defined in Section [sec: SysModel]. We show that the capacity *C* of our original energy harvesting channel without channel state information at the receiver satisfies *C* ≥ *C*2, while *C*1 is larger than the right-hand side of in Theorem [thm: Achievable]. This completes the proof of the theorem.\nGiven any E0, E1, E2, … that satisfy define a finite state channel *p*(*y**t*∣*x**t*, *s**t*) where the output *Y**t* ∈ R depends on the input *X**t* ∈ R to the channel and the channel state *S**t* ∈ {0, 1, ..., *M* + 1} in the following way:\n \n$$\\begin{aligned}\n\\label{eq:fsc}\nY\\_t = \\left\\{\\begin{array}{rl}\nX\\_t + Z\\_t & \\mbox{if } \\;\\;\\; |X\\_t|^2 \\le \\mathcal{E}^\\prime\\_{S\\_t}\\\\\nZ\\_t & \\mbox{otherwise}\n\\end{array} \\right.\\end{aligned}$$ \nwhere *Z**t* is i.i.d. N(0, 1) Gaussian noise. The channel state process {*S**t*} is a Markov Chain and its (*M* + 2) × (*M* + 2) transition matrix is given by\n \n$$\\label{eq: Tm}\nT=\\left( \\begin{array}{ccccc}\np & 1-p & 0 & \\cdots & 0 \\\\\np & 0 & 1- p & \\ddots & \\vdots \\\\\n\\vdots & \\vdots & \\ddots & \\ddots & 0\\\\\np & 0 & & 0 & 1-p\\\\\np & 0 & \\cdots & 0 & 1-p\n \\end{array} \\right)$$ \nwhere *T**i**j* = *p*(*S**t* + 1 = *j*∣*S**t* = *i*). The sequence E0′, …, E*M* + 1′ satisfies E*j*′ = E*j* for 0 ≤ *j* ≤ *M* and E*M* + 1′ = 0. There are no constraints on the transmitted signal *X**t* ∈ R over this channel. We assume that the initial state of the channel *S*0 = 0 and the realization of the state sequence {*S**t*} is known causally at the transmitter. We will discuss the capacity of this channel under two different assumptions:\n1. The realization of the state sequence {*S**t*} is also known at the receiver. In this case we will denote the capacity by *C*1.\n2. The realization of the state sequence {*S**t*} is not known at the receiver. In this case we will denote the capacity by *C*2.\nObviously, *C*2 ≤ *C*1. Below, we show that *C*2 can not be much smaller than *C*1.\nFirst note that the Markov Chain {*S**t*} is time-invariant, aperiodic and indecomposable. For such channels Theorem 2 of [8](#fn8) establishes the following expression for the capacity,\n \n$$\\begin{aligned}\n\\label{eq: cap\\_rec\\_info}\nC\\_D = \\sup\\_{\\underline{F}} \\liminf\\_{n \\to \\infty} \\frac{1}{n} I (F^n ; Y^n |D^n)\\end{aligned}$$ \nwhere *D**n* = (*D*1, ..., *D**n*) is the channel state information available at the receiver which is of the form *D**t* = *g*(*S**t*) for a given mapping *g*( ⋅ ). Here, we will be interested in the two extremal cases when *D**n* = *S**n*, in which case the capacity is denoted by *C*1, and *D**n* = ∅ in which case the capacity is denoted by *C*2. Here, *F**n* denotes (*F*1, …, *F**n*) where *F**t* stands for a random variable that takes values in the space of all mappings {*f**t* : *E**t* → X}. Also, $\\underline{F}$ denotes a sequence {*F**n*}*n* = 1∞. Now, observe that\n \n$$\\begin{aligned}\nC\\_1 & = \\sup\\_{\\underline{F}} \\liminf\\_{n \\to \\infty} \\frac{1}{n} I (F^n ; Y^n |S^n)\\nonumber \\\\\n& \\le \\sup\\_{\\underline{F}} \\liminf\\_{n \\to \\infty} \\frac{1}{n} I (F^n ; Y^n, S^n)\\nonumber \\\\\n& = \\sup\\_{\\underline{F}} \\liminf\\_{n \\to \\infty} \\frac{1}{n} \\left( I (F^n ; Y^n) + I(F^n;S^n|Y^n) \\right) \\nonumber\\\\\n& \\le \\sup\\_{\\underline{F}} \\liminf\\_{n \\to \\infty} \\frac{1}{n} I (F^n ; Y^n) + \\frac{1}{n}H(S^n) \\label{eq:eq1}\\\\\n& = H(p) + \\sup\\_{\\underline{F}} \\liminf\\_{n \\to \\infty} \\frac{1}{n} I (F^n ; Y^n) \\label{eq:eq2}\\\\\n& = H(p)+ C\\_{2} \\label{eq:CSIRCapDiff}\\end{aligned}$$ \nwhere follows from the fact that\n \n$$\\begin{aligned}\n\\lim\\_{n\\to \\infty}\\frac{1}{n}H(S^n) & = \\lim\\_{n \\to \\infty}{H(S\\_{n+1} | S\\_{n},\\ldots S\\_{1})}\\\\\n& = \\lim\\_{n \\to \\infty}{H(S\\_{n+1} | S\\_{n})}\\\\\n & = H(p)\\end{aligned}$$ \nsince for the transition matrix *T* in, we have *H*(*S**n* + 1∣*S**n*) = *H*(*p*) independent of the state *S**n*. *H*(*p*) refers to the binary entropy function evaluated at *p*. Note that when $\\lim\\_{n\\to \\infty}\\frac{1}{n}H(S^n)$ exists, we can decompose the limit inferior of the sum of the two sequences in to the sum of the limit inferior of those sequences.\nThis shows that the capacity of this channel without side information at the receiver is at most *H*(*p*) bits smaller than its capacity with receiver side information. We next relate the capacity of this finite state channel to the capacity of our original channel.\nIn the proof of Theorem [thm: Achievable], we have shown that when the realization of the external energy arrival process {*E**t*} is causally known to both the transmitter and the receiver, for any given energy allocation policy {E*j*} that satisfies, we can achieve a rate\n \n$$\\begin{aligned}\n R(M) \\ge \\displaystyle\\sum\\limits\\_{j = 0}^{\\infty} p & (1-p)^j \\max\\_{p(x): |X|^2 \\le \\mathcal{E}\\_j} I(X; Y) - \\epsilon(M),\\end{aligned}$$ \nwhere *M* determines the maximal number of channel uses employed after each energy packet arrival and as *M* → ∞, *ε*(*M*) → 0. Observe that the same strategy can be employed over the finite state channel defined in and would achieve exactly the same rate *R*(*M*). (Here, the state *S**t* will dictate which of the *M* + 1 codebooks C(0), C(1), C(2), ..., C(*M*) to transmit from at given *t*.) This is because the code constructed in the achievability scheme for our original channel (with receiver side information) will satisfy ∣*X**t*∣2 ≤ E*S**t* at every *t* when applied over the finite state channel. This ensures that the input-output relations for the two channels are the same. Moreover, the transition matrix *T* in of the finite state channel is designed so that the process {E*t*′} induced by {*S**t*} is probabilistically equivalent to the energy allocation process {E*t*} induced by the Bernoulli energy arrival process {*E**t*} over the original channel. This allow us to conclude that \n*C*1 ≥ *R*(*M*), \n or using the result of \n*C*2 ≥ *R*(*M*) − *H*(*p*). \n Finally, we want to argue that the capacity of our original system receiver side information is lower bounded by the capacity of this finite state channel receiver side information, i.e. *C* ≥ *C*2. Once this is established, we can conclude that *C* ≥ *R*(*M*) − *H*(*p*). Taking *M* → ∞, hence *ε*(*M*) → 0, completes the proof of the theorem.\nTo show that *C* ≥ *C*2, we argue that any code designed for the finite state channel (without receiver state information) can be applied to our original energy harvesting communication channel (again without receiver side information).\nWhen the state information {*S**t*} (or equivalently the realization of the process {E*t*′}) is known causally at the transmitter, a code for communicating over the finite state channel is given by a set of mappings \n*W* → *f**W**n* = (*f**W*, 1, *f**W*, 2, ..., *f**W*, *n*) \n for each message *W* (a so-called Shannon strategy), where for given *W*, *f**W*, *t* is a mapping from all the transmitter side information received up to time step *t*, i.e. *S**t* = (*S*0, ..., *S**t*), to the space of channel input symbols, X. This code can not be immediately applied to our original energy harvesting communication channel since the input alphabet for the finite state channel X = R and it can potentially imply using symbols with magnitude ∣*X**t*∣2 > E*S**t*′. However, this code can be modified, without impacting its probability of error and rate, to a form that can be immediately applied over our energy harvesting channel.\nLet us modify the code by zeroing all the symbols that violate ∣*X**t*∣2 > E*S**t*′. In particular, we define a new sequence of functions $(\\overline{f}\\_{W,1}, \\overline{f}\\_{W,2},..., \\overline{f}\\_{W,n})$ associated with the message *W* as follow: \n$$\\overline{f}\\_{W,t}(S^t) = \\mathbbm{1}\\_{\\left\\{ |f\\_{W,t}(S^t)|^2 \\le \\mathcal{E}^\\prime\\_{S\\_t} \\right\\}} f\\_{W,t}(S^t).$$ \nBased on the way our finite state channel is constructed, the output distributions are exactly the same whether we use the code with *f**n* or $\\overline{f}^n$, i.e.\n \n$$\\begin{aligned}\np(y^n|f\\_W^n, s^n) = p(y^n|\\overline{f}\\_W^n, s^n) \\end{aligned}$$ \nTherefore, if we keep our decision regions at the decoder exactly the same, we can achieve the exact same rate and probability of error with both codes over the finite state channel. Now that all the potential channel input symbol generated from this new code do satisfy the constraint ∣*X**t*∣2 ≤ E*S**t*′, we can apply this new code over our energy harvesting communication channel: Given the Bernoulli energy harvesting process {*E**t*} define the process {*S**t*′} such that \n$$S\\_t^\\prime=\\left\\{\\begin{array}{rl} j &\\textrm{if}\\quad 0\\leq j\\leq M,\\\\\nM+1 &\\textrm{if}\\quad j>M.\n\\end{array} \\right.$$ \n where *j* = *t* − max{*t*ʹ ≤ *t* : *E**t*ʹ = *E*}. Applying the codebook {*f**W**n*}*W* with side information process {*S**t*′} over the original energy harvesting communication channel achieves the exact same rate and probability of error as in the finite state channel. This is because the input-output relations for the two channels are the same and {*S**t*′} and {*S**t*} are probabilistically equivalent. Also note that because of our initial assumption that E0, E1, E2, … satisfy, this is an energy feasible strategy for our harvesting system. This completes the proof of the theorem.\n[Proof of Theorem [thm: ConstantGapBlessE]] The proof follows almost the same line of argument as in Theorem [thm: constgaprate], except we have a different constant here due to the additional *K*(*p*) = 1.04 + *H*(*p*) in the *C**l**b*(*B**m**a**x*, *p*).\nAgain, we will first restrict the range of (*B**m**a**x*, *p*) to be considered by noticing:\n \n$$\\begin{aligned}\nC\\_{ub}(B\\_{max}, p) - & C\\_{lb}(B\\_{max}, p ) \\le C\\_{ub}(B\\_{max}, p) \\\\\n & = \\frac{1}{2} \\log(1 + pB\\_{max}) \\\\\n & \\le 2.58\\end{aligned}$$ \nfor all (*B**m**a**x*, *p*) such that *p**B**m**a**x* ≤ 34.75.\nNext, let’s restrict ourself to the set of (*B**m**a**x*, *p*) such that *p**B**m**a**x* > 34.75:\n \n$$\\begin{aligned}\n& C\\_{ub}(B\\_{max}, p) - C\\_{lb}(B\\_{max}, p ) \\\\\n= & \\frac{1}{2} \\log \\left(1 + p B\\_{max}\\right) \\\\\n&- \\left( \\displaystyle\\sum\\limits\\_{j = 0}^{\\infty} p(1-p)^j \\frac{1}{2} \\log\\left( 1 + (1-p)^j p B\\_{max} \\right) - K(p)\\right)^+ \\\\\n\\stackrel{(a)} \\le & \\frac{1}{2} \\log \\left(1 + p B\\_{max}\\right) \\\\ \n& - \\displaystyle\\sum\\limits\\_{j = 0}^{\\infty} p(1-p)^j \\frac{1}{2} \\log\\left( 1 + (1-p)^j p B\\_{max} \\right) + H(p) + 1.04 \\\\\n \\stackrel{(b)} \\le & \\frac{1}{2\\ln(2)}\\frac{1}{pB\\_{max}} - \\frac{1-p}{p} \\frac{1}{2} \\log\\left(1-p\\right) + H(p) + 1.04 \\\\\n \\stackrel{(c)} \\le & \\frac{1}{2\\ln(2)}\\frac{1}{34.75} - \\frac{1-p}{p} \\frac{1}{2} \\log\\left(1-p\\right) + H(p) + 1.04 \\\\\n = & 1.06 + \\frac{1-p}{2p} \\log\\left(\\frac{1}{1-p}\\right) + H(p) \\\\\n \\stackrel{(d)} \\le & 2.58\\end{aligned}$$ \nwhere (a) comes from removing the ( ⋅ )+ resulting in an upper bound and using the definition of *K*(*p*). (b) uses exactly the same sequence of inequalities in the proof of Theorem [thm: constgaprate], so we won’t repeat here. (c) comes from the fact we restrict ourself to *p**B**m**a**x* > 34.75. Finally, (d) comes from the fact:\n \n$$\\begin{aligned}\ng(p) & \\triangleq \\frac{1-p}{2p} \\log\\left(\\frac{1}{1-p}\\right) + H(p) \\\\\n&= \\frac{1-p}{2p} \\log\\left(\\frac{1}{1-p}\\right) + p \\log\\left( \\frac{1}{p} \\right) + (1-p) \\log\\left( \\frac{1}{1-p} \\right)\\end{aligned}$$ \nis a continuous bounded function on *p* ∈ (0, 1). Furthermore, it’s concave and attains a maximum value of 1.52 at *p* = 0.413.\n---\n1. Manuscript received April 1, 2014; revised September 1, 2014. This work was supported by the Center for Science of Information (CSoI), an NSF Science and Technology Center, under grant agreement CCF-0939370. An initial version of this paper was presented at the IEEE International Symposium on Information Theory (ISIT), Honolulu, July 2014.[↩](#fnref1)\n2. The authors are with Stanford University, Packard Electrical Engineering, 350 Serra Mall, Stanford, California 94305-9510, USA (e-mails: {ydong2, farnia, aozgur}@ stanford.edu).[↩](#fnref2)\n3. Here, we require the harvested energy be stored in the battery before it can be used for transmission. An alternative energy storage model, considered in, is to allow the harvested energy *E**t* to be used instantaneously at time *t* and the remaining amount be stored in the battery. The capacities under the two energy storage models are different in general.[↩](#fnref3)\n4. The continuous-time version of this problem has been considered in many works in the literature where it is more commonly referred to as the power control problem and the function *r*(*g*) is called the power-rate or the power-utility function in this case. Note that in our discrete-time setup power corresponds to energy per channel use, and therefore the terms energy (per channel use) and power can be used interchangeably. We prefer to refer to the problem as the energy allocation problem.[↩](#fnref4)\n5. The strategy discussed in is a slight variation of the *Uniform Policy* in that the amount of energy utilized in each channel use is ± *ε* of the average energy arrival rate. *ε* decreases to zero as *B**m**a**x* → ∞ and the decay of *ε* controls the battery discharge probability and the convergence speed to the optimal average rate as *B**m**a**x* → ∞. However, in its essence the strategy is a *Uniform Policy*, so its behavior is captured by the *Uniform Policy* plots.[↩](#fnref5)\n6. Indeed, this fact can be shown analytically; for example taking *B**m**a**x* = *E*, one can show that the gap between the upper bound and the rate achieved by the *Uniform Policy* increases to infinity as *B**m**a**x* = *E* → ∞. However, we do not provide a proof of this fact since the trend is already quite obvious from the graphs.[↩](#fnref6)\n7. This proof was pointed out to the authors by one of the anonymous reviewers.[↩](#fnref7)\n8. Theorem 2 of is stated for multiple access finite state channel. We use a simplified version of it for single-user channels, combined with part 2 of Corollary 1 in the same paper which shows that when the initial state is known to the transmitter, the capacity region is independent of the initial pmf.[↩](#fnref8)\n"},"data_id":{"kind":"string","value":"arxiv_0000373"}}},{"rowIdx":99,"cells":{"content":{"kind":"string","value":"[\n0.3in ]\nIn this paper, we study distributed algorithms for large-scale AUC maximization with a deep neural network as a predictive model. Although distributed learning techniques have been investigated extensively in deep learning, they are not directly applicable to stochastic AUC maximization with deep neural networks due to its striking differences from standard loss minimization problems (e.g., cross-entropy).Towards addressing this challenge, we propose and analyze a communication-efficient distributed optimization algorithm based on a *non-convex concave* reformulation of the AUC maximization, in which the communication of both the primal variable and the dual variable between each worker and the parameter server only occurs after multiple steps of gradient-based updates in each worker. Compared with the naive parallel version of an existing algorithm that computes stochastic gradients at individual machines and averages them for updating the model parameters, our algorithm requires a much less number of communication rounds and still achieves a linear speedup in theory. To the best of our knowledge, this is the **first** work that solves the *non-convex concave min-max* problem for AUC maximization with deep neural networks in a communication-efficient distributed manner while still maintaining the linear speedup property in theory. Our experiments on several benchmark datasets show the effectiveness of our algorithm and also confirm our theory.\nIntroduction\n============\nLarge-scale distributed deep learning has achieved tremendous successes in various domains, including computer vision , natural language processing , generative modeling , reinforcement learning , etc. From the perspective of learning theory and optimization, most of them are trying to minimize a surrogate loss of a specific error measure using parallel minibatch stochastic gradient descent (SGD). For example, on the image classification task, the surrogate loss is usually the cross entropy between the estimated probability distribution according to the output of a certain neural network and the vector encoding ground-truth labels , which is a surrogate loss of the misclassification rate. Based on the surrogate loss, parallel minibatch SGD is employed to update the model parameters.\nHowever, when the data for classification is imbalanced, AUC (short for Area Under the ROC Curve) is a more suitable measure . AUC is defined as the probability that a positive sample has a higher score than a negative sample . Despite the tremendous applications of distributed deep learning in different fields, the study about optimizing AUC with distributed deep learning technologies is rare. The commonly used parallel mini-batch SGD for minimizing a surrogate loss of AUC will suffer from high communication costs in a distributed setting due to the non-decomposability nature of AUC measure. The reason is that positive and negative data pairs that define a surrogate loss for AUC may sit on different machines. To the best of our knowledge, is the only work trying to optimize a surrogate loss of *AUC with a deep neural network* that explicitly tackles the non-decomposability of AUC measure. Nevertheless, their algorithms are designed only for the single-machine setting and hence are far from sufficient when encountering a huge amount of data. Although a naive parallel version of the stochastic algorithms proposed in can be used for distributed AUC maximization with a deep neural network, it would still suffer from high communication overhead due to a large number of communication rounds.\n[t] [tab:1] [tab:2]\nl|l|l|l Algorithm & Setting&Iteration Complexity &Communication Complexity \nPPD-SG & Single & *O*(1/(*μ*2*ε*))& - \nNP-PPD-SG& Distributed & *O*(1/(*K**μ*2*ε*))& *O*(1/(*K**μ*2*ε*)) \nCoDA& Distributed & *O*(1/(*K**μ*2*ε*))& *O*(1/(*μ*3/2*ε*1/2)) \nIn this paper, we bridge the gap between stochastic AUC maximization and distributed deep learning by proposing a communication-efficient distributed algorithm for stochastic AUC maximization with a deep neural network. **The focus is to make the total number of communication rounds much less than the total number of iterative updates.** We build our algorithm upon the nonconvex-concave min-max reformulation of the original problem. The key ingredient is to design a communication-efficient distributed algorithm for solving the regularized min-max subproblems using multiple machines. Specifically, we follow the proximal primal-dual algorithmic framework proposed by , i.e., by solving a sequence of quadratically regularized min-max saddle-point problems with periodically updated regularizers successively. **The key difference** is that the inner min-max problem solver is built on a distributed periodic model averaging technique, which consists of a fixed number of stochastic primal-dual updates over individual machines and a small number of averagings of model parameters from multiple machines. This mechanism can greatly reduce the communication cost, which is similar to . However, their analysis cannot be applied to our case since their analysis only works for convex or non-convex *minimization* problems. In contrast, our algorithm is designed for a particular *non-convex concave min-max* problem induced by the original AUC maximization problem. Our contributions are summarized as follows:\n* We propose a communication-efficient distributed stochastic algorithm named CoDA for solving a nonconvex-concave min-max reformulation of AUC maximization with deep neural networks by local primal-dual updating and periodically global variable averaging. To our knowledge, this is the first communication-efficient distributed stochastic algorithm for learning a deep neural network by AUC maximization.\n* We analyze the iteration complexity and communication complexity of the proposed algorithm under the commonly used Polyak- Łojasiewicz (PL) condition as in . Comparing with, our theoretical result shows that the iteration complexity can be reduced by a factor of *K* (the number of machines) in a certain region, while the communication complexity (the rounds of communication) is much lower than that of a naive distributed version of the stochastic algorithm proposed in . The summary of iteration and communication complexities is given in Table [tab:2].\n* We verify our theoretical claims by conducting experiments on several large-scale benchmark datasets. The experimental results show that our algorithm indeed exhibits good acceleration performance in practice.\nRelated Work\n============\n#### Stochastic AUC Maximization.\nIt is challenging to directly solve the stochastic AUC maximization in the online learning setting since the objective function of AUC maximization depends on a sum of pairwise losses between samples from positive and negative classes. addresses this problem by maintaining a buffer to store representative data samples, employing the reservoir sampling technique to update the buffer, calculating gradient information based on the data in the buffer, and then performing a gradient-based update rule to update the classifier. does not maintain a buffer, while they instead maintained first-order and second-order statistics of the received data to update the classifier by gradient-based update. Both of them are infeasible in big data scenarios since suffers from a large amount of training data and is not suitable for high dimensional data. addresses these issues by introducing a min-max reformulation of the original problem and solving it by a primal-dual stochastic gradient method , in which no buffer is needed and the per-iteration complexity is the same magnitude as the dimension of the feature vector. improves the convergence rate by adding a strongly convex regularizer upon the original formulation. Based on the same saddle point formulation as in , gets an improved convergence rate by developing a multi-stage algorithm without adding the strongly convex regularizer. However, all of these studies focus on learning a linear model. Recently, considers stochastic AUC maximization for learning a deep non-linear model, in which they designed a proximal primal-dual gradient-based algorithm under the PL condition and established non-asymptotic convergence results.\n#### Communication Efficient Algorithms.\nThere are multiple approaches for reducing the communication cost in distributed optimization, including skipping communication and compression techniques. Due to limit of space, we mainly review the literature on skipping communication. For compression techniques, we refer the readers to and references therein. Skipping communication is realized by doing multiple local gradient-based updates in each worker before aggregating the local model parameters. One special case is so-called one-shot averaging , where each machine solves a local problem and averages these solutions only at the last iterate. Some works consider one-shot averaging with one-pass of the entire data and establish statistical convergence, which is usually not able to guarantee the convergence of the training error. The scheme of local SGD update at each worker with skipping communication is analyzed for convex and nonconvex problems . There are also several empirical studies showing that this scheme exhibits good empirical performance for distributed deep learning. However, all of these works only consider minimization problems and do not apply to the nonconvex-concave min-max formulation as considered in this paper.\n#### Nonconvex Min-max Optimization.\nStochastic nonconvex min-max optimization has garnered increasing attention recently . considered the case where the objective function is weakly-convex and concave and proposed an algorithm based on the spirit of proximal point method , in which a proximal subproblem with periodically updated reference points is approximately solved by an appropriate stochastic algorithm. They established the convergence to a nearly stationary point for the equivalent minimization problem. Under the same setting, designed a block-based algorithm and showed that it can converge to a solution with a small stationary gap, and considered solving the problem using vanilla stochastic gradient descent ascent and established its convergence to a stationary point under the smoothness assumption. There are also several papers trying to solve non-convex non-concave min-max problems. proposed an inexact proximal point method for solving a class of weakly-convex weakly-concave problems, which was proven to converge to a nearly stationary point. exploited the PL condition for the inner maximization problem and designed a multi-step alternating optimization algorithm which was able to converge to a stationary point. considered solving a class of nonconvex-nonconcave min-max problems by designing an adaptive gradient method and established an adaptive complexity for finding a stationary point. However, none of them is particularly designed for the distributed stochastic AUC maximization problem with a deep neural network.\nPreliminaries and Notations\n===========================\nThe area under the ROC curve (AUC) on a population level for a scoring function *h* : X → R is defined as\n \n$$AUC(h) = \\text{Pr}(h(\\x) \\geq h(\\x') | y=1, y'=-1),$$ \nwhere $\\z = (\\x, y)$ and $\\z'=(\\x', y')$ are drawn independently from P. By employing the squared loss as the surrogate for the indicator function which is commonly used by previous studies , the deep AUC maximization problem can be formulated as\n \n$$\\min\\limits\\_{\\w\\in\\mathbb{R}^d} \\E\\_{\\z, \\z'}\\left[(1-h(\\w;\\x)+h(\\w;\\x'))^2| y=1, y'=-1\\right],\n\\label{original\\_deep\\_auc}$$ \nwhere $h(\\w; \\x)$ denotes the prediction score for a data sample $\\x$ made by a deep neural network parameterized by $\\w$. It was shown in that the above problem is equivalent to the following min-max problem:\n \n$$\\label{min-max}\n\\min\\limits\\_{\\w\\in \\mathbb{R}^d \\atop (a, b)\\in \\mathbb{R}^2} \\max\\limits\\_{\\alpha\\in \\mathbb{R}} f(\\w, a, b, \\alpha)=\\E\\_\\z[F(\\mathbf{w}, a, b, \\alpha, \\z)],$$ \nwhere\n \n$$\\begin{split}\n& F(\\textbf{w}, a, b, \\alpha; \\z) = (1-p) (h(\\w; \\x)- a)^2 \\mathbb{I}\\_{[y=1]} \\\\\n&+ p(h(\\w; \\x) - b)^2\\mathbb{I}\\_{[y=-1]} \n+ 2(1+\\alpha)(p h(\\w; \\x)\\mathbb{I}\\_{[y=-1]} \\\\\n&- (1-p) h(\\w, \\x)\\mathbb{I}\\_{[y=1]}) - p(1-p)\\alpha^2,\n\\end{split}$$ \nwhere *p* = Pr(*y* = 1) denotes the prior probability that an example belongs to the positive class, and $\\mathbb I$ denotes an indicator function. The above min-max reformulation allows us to decompose the expectation over all data into the expectation over data on individual machines.\nIn this paper, we consider the following distributed AUC maximization problem:\n \n$$\\label{opt:problem}\n\\min\\limits\\_{\\w\\in \\mathbb{R}^d \\atop (a, b)\\in \\mathbb{R}^2} \\max\\limits\\_{\\alpha\\in \\mathbb{R}} f(\\w, a, b, \\alpha)=\\frac{1}{K}\\sum\\limits\\_{k=1}^{K} f\\_k(\\mathbf{w}, a, b, \\alpha),$$ \nwhere *K* is the total number of machines, $f\\_k(\\w, a, b, \\alpha)= \\E\\_{\\z^k}[F\\_k(\\w, a,b,\\alpha; \\z^k)]$, $\\z^k = (\\x^k, y^k)\\sim \\mathbb{P}\\_k$, P*k* is the data distribution on machine *k*, and $F\\_k(\\w, a,b,\\alpha; \\z^k) = F(\\w, a,b,\\alpha; \\z^k)$. Our goal is to utilize *K* machines to jointly solve the optimization problem ([opt:problem]). We emphasize that the *k*-th machine can only access data $\\z^k\\sim \\mathbb P\\_k$ of its own. It is notable that our formulation includes both the batch-learning setting and the online learning setting. For the batch-learning setting, $\\mathbb P\\_k$ represents the empirical distribution of data on the *k*-th machine and *p* denotes the empirical positive ratio for all data. For the online learning setting, $\\mathbb P\\_k= \\mathbb P,\\forall k$ represents the same population distribution of data and *p* denotes the positive ratio in the population level.\n#### Notations.\nWe define the following notations:\n \n$$\\begin{aligned}\n&\\v = (\\w^T, a, b)^T, \\quad \\phi(\\v) = \\max\\_{\\alpha} f(\\v, \\alpha),\\\\\n& \\phi\\_s (\\v) = \\phi(\\v) + \\frac{1}{2\\gamma}\\|\\v-\\v\\_{s-1}\\|^2,\\\\\n&\\v^\\*\\_{\\phi} = \\arg\\min\\limits\\_{\\v}\\phi(\\v),\\quad \\v^\\*\\_{\\phi\\_s} = \\arg\\min\\limits\\_{\\v} \\phi\\_s(\\v).\\end{aligned}$$ \nWe make the following assumption throughout this paper.\n \n[ass:1] (i) There exist $\\v\\_0, \\Delta\\_0>0$ such that $\\phi(\\v\\_0) - \\phi(\\v^\\*\\_\\phi)\\leq \\Delta\\_0$. (ii) For any $\\x$, $\\|\\nabla h(\\w; \\x)\\| \\leq G\\_h$. \n(iii) $\\phi(\\v)$ satisfies the *μ*-PL condition, i.e., $\\mu(\\phi(\\v)-\\phi(\\v\\_\\*))\\leq \\frac{1}{2}\\|\\nabla\\phi(\\v)\\|^2$; $\\phi(\\v)$ is *L*1-smooth, i.e., $\\|\\phi(\\v\\_1) - \\phi(\\v\\_2)\\| \\leq L\\_1 \\|\\v\\_1 - \\v\\_2\\|$. \n(iv) For any $\\x$, $h(\\w; \\x)$ is *L**h*-smooth, and $h(\\w; \\x) \\in [0, 1]$. [assumption1]\n**Remark:** Assumptions (i), (ii), (iii) and $h(\\w; \\x)\\in [0, 1]$ of (iv) are also assumed in , which have been justified as well. *L*-smoothness of function *h* is a standard assumption in the optimization literature. Finally, it should be noted that *μ* is usually much smaller than 1 . This is important for us to understand our theoretical result later.\nMain Result and Theoretical Analysis\n====================================\nIn this section, we first describe our algorithm, and then present its convergence result followed by its analysis. For simplicity, we assume that the ratio *p* of data with the positive label is known. For the batch learning setting, *p* is indeed the empirical ratio of positive examples. For the online learning setting with an unknown distribution, we can follow the online estimation technique in to do the parameter update.\nAlgorithm [outerloop] describes the proposed algorithm CoDA for optimizing AUC in a communication-efficient distributed manner. CoDA shares the same algorithmic framework as proposed in . In particular, we employ a proximal-point algorithmic scheme that successively solves the following convex-concave problem approximately:\n \n$$\\min\\_{\\v}\\max\\_{\\alpha}f(\\v,\\alpha)+\\frac{1}{2\\gamma}\\|\\v-\\v\\_0\\|^2,$$ \nwhere *γ* is an appropriate regularization parameter to make sure that the regularized function is strongly-convex and strongly-concave. The reference point $\\v\\_0$ is periodically updated after a number of iterations. At the *s*-th stage our algorithm invokes a communication-efficient algorithm for solving the above strongly-convex and strongly-concave subproblems. After obtaining a primal solution $\\v\\_s$ at the *s*-th stage, we sample some data from individual machines to obtain an estimate of the corresponding dual variable *α**s*.\n[t]\n[1]\n[outerloop]\nOur new contribution is the communication-efficient distributed algorithm for solving the above strongly-convex and strongly-concave subproblems. The algorithm referred to as DSG is presented in Algorithm [innerloop]. Each machine makes a stochastic proximal-gradient update on the primal variable and a stochastic gradient update on the dual variable at each iteration. After every *I* iterations, all the *K* machines communicate to compute an average of local primal solutions $\\v^k\\_t$ and local dual solutions *α**t**k*. It is not difficult to show that when *I* = 1, our algorithm reduces to the naive parallel version of the PPD-SG algorithm proposed in , i.e., by averaging individual primal and dual gradients and then updating the primal-dual variables according to the averaged gradient [1](#fn1). Our novel analysis allows us to use *I* > 1 to skip communications, leading to a much less number of communications. The intuition behind this is that, as long as the step size *η**s* is sufficiently small we can control the distance between individual solutions $(\\v^k\\_t, \\alpha^k\\_t)$ to their global averages, which allows us to control the error term that is caused by the discrepancy between individual machines. We will provide more explanations as we present the analysis.\nBelow, we present the main theoretical result of CoDA. Note that in the following presentation, $L\\_\\v, H, B, \\sigma\\_\\v, \\sigma\\_\\alpha$ are appropriate constants, whose values are given in the proofs of Lemma [properties] and Lemma [lemmaonestage] in the supplement.\n[thm:main] Set $\\gamma = \\frac{1}{2L\\_{\\v}}$, $c =\\frac{\\mu/L\\_{\\v}}{5+\\mu/L\\_{\\v}}$, *η**s* = *η*0*K*exp( − (*s* − 1)*c*) ≤ *O*(1), $T\\_s = \\frac{\\max(8, 8G\\_h^2)}{L\\_{\\v}\\eta\\_0 K} \\exp( (s-1)c)$, $I\\_s = \\max(1, 1/\\sqrt{K\\eta\\_s})$ and $m\\_{s}=\\max\\left(\\frac{1+C}{\\eta\\_{s+1}^2 T\\_{s+1} p^2(1-p)^2}, \\frac{\\log(K)}{\\log(1/\\Tilde{p})}\\right)$, where $C = \\frac{3\\tilde{p}^{\\frac{1}{\\ln(1/\\Tilde{p})}}}{2\\ln(1/\\Tilde{p})}$ and $\\Tilde{p} = \\max(p, 1-p)$. To return $\\v\\_S$ such that $\\E[\\phi(\\v\\_S) - \\phi(\\v^\\*\\_{\\phi})] \\leq \\epsilon$, it suffices to choose $S\\geq \\frac{5L\\_{\\v}+\\mu}{\\mu} \\max\\bigg\\{\\log \\left(\\frac{2\\Delta\\_0}{\\epsilon}\\right), \\log S + \\log\\bigg[ \\frac{2\\eta\\_0}{\\epsilon} \\frac{6HB^2 + 12(\\sigma\\_{\\v}^2+\\sigma\\_{\\alpha}^2)}{5}\\bigg]\\bigg\\}$. As a result, the number of iterations is at most $T=\\widetilde{O}\\bigg(\\max\\left(\\frac{\\Delta\\_0}{\\mu \\epsilon \\eta\\_0 K}, \\frac{L\\_{\\v}}{\\mu^2 K\\epsilon}\\right)\\bigg)$ and the number of communications is at most $\\widetilde{O}\\bigg(\\max\\left(\\frac{K}{\\mu} + \\frac{\\Delta\\_0^{1/2}}{\\mu(\\eta\\_0 \\epsilon)^{1/2}}, \\frac{K}{\\mu} + \\frac{L\\_{\\v}^{1/2}}{\\mu^{3/2}\\epsilon^{1/2}}\\right)\\bigg)$, where $\\widetilde{O}$ suppresses logarithmic factors, and $H, B, \\sigma\\_\\v, \\sigma\\_\\alpha$ are appropriate constants. [theoremallstage]\n[t]\n[innerloop]\nWe have the following remarks about Theorem [thm:main].\n* First, we can see that the step size *η**s* is reduced geometrically in a stagewise manner. This is due to the PL condition. We note that a stagewise geometrically decreasing step size is usually used in practice in deep learning . Second, by setting *η*0 = *O*(1/*K*) we have $I\\_s =\\Theta(\\frac{1}{\\sqrt{K}}\\exp((s-1)c/2)$. It means two things: (i) the larger the number of machines the smaller value of *I**s*, i.e., more frequently the machines need to communicate. This is reasonable since more machines will create a larger discrepancy between data among different machines; (ii) the value *I**s* can be increased geometrically across stages. This is because that the step size *η**s* is reduced geometrically, which causes one step of primal and dual updates on individual machines to diverge less from their averaged solutions. As a result, more communications can be skipped.\n* Second, we can see that when *K* ≤ Θ(1/*μ*), we have the total iteration complexity given by $\\widetilde O(\\frac{1}{\\mu^2K \\epsilon})$. Compared with the iteration complexity of the PPD-SG algorithm proposed in that is $\\widetilde O(\\frac{1}{\\mu^2 \\epsilon})$, the proposed algorithm CoDA enjoys an iteration complexity that is reduced by a factor of *K*. This means that up to a certain large threshold Θ(1/*μ*) for the number *K* of machines, CoDA enjoys a linear speedup.\n* Finally, let us compare CoDA with the naive parallel version of PPD-SG, which is CoDA by setting *I* = 1. In fact, our analysis of the iteration complexity for this case is still applicable, and it is not difficult to show that the iteration complexity of the naive parallel version of PPD-SG is given by $\\widetilde O(\\frac{1}{\\mu^2K \\epsilon})$ when *K* ≤ 1/*μ*. As a result, its communication complexity is also $\\widetilde O(\\frac{1}{\\mu^2K \\epsilon})$. In contrast, CoDA’s communication complexity is $\\widetilde O(\\frac{1}{\\mu^{3/2}\\epsilon^{1/2}})$ when $K\\leq \\frac{1}{\\mu}\\leq \\frac{1}{\\mu^{1/2}\\epsilon^{1/2}}\\leq \\frac{1}{\\epsilon}$ according to Theorem [thm:main] [2](#fn2). Hence, our algorithm is more communication efficient, i.e., $\\widetilde O(\\frac{1}{\\mu^{3/2}\\epsilon^{1/2}})\\leq \\widetilde O(\\frac{1}{\\mu^2K \\epsilon})$ when $K\\leq \\frac{1}{\\mu}$ [3](#fn3). This means that up to a certain large threshold Θ(1/*μ*) for the number *K* of machines, CoDA has a smaller communication complexity than the naive parallel version of PPD-SG.\nAnalysis\n--------\nBelow, we present a sketch of the proof of Theorem [thm:main] by providing some key lemmas. We first derive some useful properties regarding the random function $F\\_k(\\v, \\alpha, \\z)$.\n[properties] Suppose that Assumption [assumption1] holds and $\\eta \\leq \\min(\\frac{1}{2p(1-p)}, \\frac{1}{2(1-p)}, \\frac{1}{2p})$. Then there exist some constants $L\\_2, B\\_\\alpha, B\\_\\v, \\sigma\\_{\\v}, \\sigma\\_{\\alpha}$ such that\n \n$$\\begin{aligned}\n&\\|\\nabla\\_{\\v} F\\_k(\\v\\_1, \\alpha; \\z) - \\nabla\\_{\\v} F\\_k(\\v\\_2, \\alpha; \\z)\\| \\leq L\\_2 \\|\\v\\_1-\\v\\_2\\|,\\\\\n&\\|\\nabla\\_{\\v} F\\_k(\\v, \\alpha; \\z)\\|^2 \\leq B\\_{\\v}^2, |\\nabla\\_{\\alpha} F\\_k(\\v, \\alpha; \\z)|^2 \\leq B\\_{\\alpha}^2,\\\\ \n&\\E[\\|\\nabla\\_{\\v} f\\_k(\\v, \\alpha) - \\nabla\\_{\\v} F\\_k(\\v, \\alpha; \\z)\\|^2] \\leq \\sigma\\_{\\v}^2,\\\\\n&\\E[|\\nabla\\_{\\alpha}f\\_k(\\v, \\alpha) - \\nabla\\_{\\alpha} F\\_k(\\v, \\alpha; \\z)|] \\leq \\sigma^2\\_{\\alpha}.\\end{aligned}$$ \n**Remark:** We include the proofs of these properties in the Appendix. In the following, we will denote $B^2 = \\max(B\\_{\\v}^2, B\\_{\\alpha}^2)$ and $L\\_{\\v} = \\max(L\\_1, L\\_2)$.\nNext, we introduce a key lemma, which is of vital importance to establish the upper bound of the objective gap of the regularized subproblem.\n(One call of Algorithm [innerloop]) Let $\\psi(\\v) = \\max\\limits\\_{\\alpha} f(\\v, \\alpha) + \\frac{1}{2\\gamma}\\|\\v - \\v\\_0\\|^2$, $\\tilde{\\v}$ be the output of Algorithm [innerloop], and $\\v^\\*\\_{\\psi} = \\arg\\min \\psi(\\v)$, $\\alpha^\\*(\\tilde{\\v}) = \\arg\\max\\limits\\_{\\alpha}f(\\tilde{\\v}, \\alpha)+\\frac{1}{2\\gamma}\\|\\tilde{\\v}-\\v\\_0\\|^2$. By running Algorithm [innerloop] with given input $\\v\\_0, \\alpha\\_0$ for *T* iterations, $\\gamma = \\frac{1}{2L\\_{\\v}}$, and $\\eta \\leq \\min\\{\\frac{1}{L\\_{\\v}+3G^2\\_{\\alpha}/\\mu\\_{\\alpha}}, \\frac{1}{L\\_{\\alpha}+3G\\_{\\v}^2/L\\_{\\v}}, \\frac{3}{2\\mu\\_{\\alpha}}, \\frac{1}{2p(1-p)}, \\frac{1}{2(1-p)}, \\frac{1}{2p}\\}$, we have \n \n$$\\begin{aligned}\n\\E[\\psi(\\tilde{\\v}) - \\min\\_{\\v}\\psi(\\v)] \\leq& \\!\\frac{2\\|\\v\\_0 - \\v\\_{\\psi}^\\*\\|^2\\! +\\! \\E[(\\alpha\\_{0} - \\alpha^\\*(\\tilde{\\v}))^2]}{\\eta T} \\\\\n&\\!+\\! H\\!\\eta^2\\! I^2\\! B^2 \\I\\_{I>1}\\! +\\! \\frac{\\eta (2\\sigma\\_{\\v}^2\\! +\\! 3\\sigma\\_{\\alpha}^2)}{2K}, \\end{aligned}$$ \nwhere *μ**α* = 2*p*(1 − *p*), *L**α* = 2*p*(1 − *p*), *G**α* = 2max{*p*, 1 − *p*}, $G\\_{\\v} = 2\\max\\{p, 1-p\\} G\\_h$, and $H = \\left(\\frac{6G\\_{\\v}^2}{\\mu\\_{\\alpha}} + 6L\\_{\\v} + \\frac{6G\\_{\\alpha}^2}{L\\_{\\v}} + \\frac{6L\\_{\\alpha}^2}{\\mu\\_{\\alpha}}\\right)$. [lemmaonestage]\n**Remark:** The above result is similar to Lemma 2 in . The key difference lies in the second and third terms in the upper bound. The second term arises because of the discrepancy of updates between individual machines. The third term is due to the variance reduction by using multiple machines, which is the key to establish the linear speed-up. It is easy to see that by setting $I = \\frac{1}{\\sqrt{\\eta K}}$, the second term and the third term have the same order. With the above lemma, the proof of Theorem [thm:main] follows similar analysis to in .\n**Sketch of the Proof of Lemma [lemmaonestage].** Below, we present a roadmap for the proof of the key Lemma [lemmaonestage]. The main idea is to first bound the objective gap of the subproblem in Lemma [lem:objgap]. Then we further bound every term in the RHS in Lemma [lem:objgap] appropriately, which is realized by Lemma [lem:var1], Lemma [lem:var2] and Lemma [stichlemma]. All the detailed proofs of Lemmas can be found in Appendix.\n[lem:objgap] Define $\\bar\\v\\_t = \\frac{1}{K}\\sum\\_{k=1}^K\\v^k\\_t, \\bar{\\alpha}\\_t = \\frac{1}{K}\\sum\\_{k=1}^K\\alpha^k\\_t$. Suppose Assumption [assumption1] holds and by running Algorithm [innerloop], we have\n \n$$\\begin{aligned}\n&\\psi(\\tilde{\\v}) - \\min\\limits\\_{\\v} \\psi(\\v)\\\\\n&\\!\\leq\\! \\frac{1}{T}\\sum\\limits\\_{t=1}^{T}\\bigg[\\! \\underbrace{\\!\\langle \\nabla\\_{\\v}f(\\bar{\\v}\\_{t-1}, \\bar{\\alpha}\\_{t-1}),\\! \\bar{\\v}\\_t\\! -\\! \\v\\_{\\psi}^\\*\\rangle \\!+\\!2L\\_{\\v}\\! \\langle \\bar{\\v}\\_t\\!-\n\\!\\v\\_0,\\! \\bar{\\v}\\_t\\!-\\!\\v\\_{\\psi}^\\*\\rangle}\\_{A\\_1} \\\\\n& + \\underbrace{ \\langle \\nabla\\_{\\alpha} f(\\bar{\\v}\\_{t-1}, \\bar{\\alpha}\\_{t-1}), \\alpha^\\* - \\bar{\\alpha}\\_t \\rangle }\\_{A\\_2}\\end{aligned}$$ \n \n$$\\begin{aligned}\n&+\\!\\underbrace{\\frac{L\\_{\\v} + 3G\\_{\\alpha}^2/\\mu\\_{\\alpha}}{2}\\|\\bar{\\v}\\_t - \\bar{\\v}\\_{t-1}\\|^2 \\!+\\! \\frac{L\\_{\\alpha} + 3G\\_{\\v}^2/L\\_{\\v}}{2} (\\bar{\\alpha}\\_t - \\bar{\\alpha}\\_{t-1})^2}\\_{A\\_3} \\\\\n&+ \\frac{2L\\_{\\v}}{3} \\| \\bar{\\v}\\_{t-1} - \\v\\_{\\psi}^\\*\\|^2 - L\\_{\\v} \\| \\bar\\v\\_t - \\v\\_{\\psi}^\\* \\|^2 - \\frac{\\mu\\_{\\alpha}}{3} (\\bar{\\alpha}\\_{t-1} - \\alpha^\\*)^2\\bigg].\n\\label{sum\\_v\\_alpha}\\end{aligned}$$ \nNext, we will bound *A*1, *A*2 in Lemma [lem:var1] and Lemma [lem:var2]. *A*3 can be cancelled with similar terms in the following two lemmas. The remaining terms will be left to form a telescoping sum with other similar terms in the following two lemmas.\n[lem:var1] Define $\\hat{\\v}\\_t\\! =\\! \\arg\\min\\limits\\_{\\v}\\! \\left(\\! \\frac{1}{K}\\!\\sum\\limits\\_{k=1}^{K}\\! \\nabla\\_{\\v}\\! f\\!(\\!\\v^k\\_{t-1}\\!,\\! \\alpha^k\\_{t-1}\\!)\\!\\right)^T\\! \\v\\!$ $+ \\frac{1}{2\\eta} \\|\\v - \\bar{\\v}\\_{t-1}\\|^2 + \\frac{1}{2\\gamma}\\|\\v - \\v\\_{0}\\|^2$. We have\n \n$$\\begin{split}\n&A\\_1\n\\!\\leq \\! \\frac{3G\\_{\\alpha}^2}{2L\\_{\\v}}\\frac{1}{K} \\sum\\limits\\_{k=1}^{K} (\\bar{\\alpha}\\_{t-1}-\\alpha\\_{t-1}^{k})^2 \\!+\\! \\frac{3L\\_{\\v}}{2}\\frac{1}{K}\\sum\\limits\\_{k=1}^{K}\\|\\bar{\\v}\\_{t-1}\\!-\\!\\v\\_{t-1}^k\\|^2 \\\\\n&+ \\eta \\left\\|\\frac{1}{K}\\sum\\limits\\_{k=1}^{K}[\\nabla\\_{\\v}f\\_k(\\v\\_{t-1}^k, \\alpha\\_{t-1}^k) \\!-\\! \\nabla\\_{\\v}F\\_k(\\v\\_{t-1}^{k}, \\alpha\\_{t-1}^k; \\z\\_{t-1}^k)]\\right\\|^2 \\\\\n&\\!+\\! \\frac{1}{K}\\!\\sum\\limits\\_{k=1}^{K}\\!\\langle \\!\\nabla\\_{\\v}f\\_k(\\v\\_{t-1}^k,\\! \\alpha\\_{t-1}^k)\\!-\\!\\nabla\\_{\\v}F\\_k(\\v\\_{t-1}^{k},\\! \\alpha\\_{t-1}^k;\\z\\_{t-1}^k),\\! \\hat{\\v}\\_t - \\v\\_{\\psi}^\\*\\rangle\\\\\n& \\!+\\!\\frac{1}{2\\eta} \\!(\\!\\|\\bar{\\v}\\_{t-1}\\!-\\!\\v\\_{\\psi}^\\*\\|^2 \\!-\\! \\|\\bar{\\v}\\_{t-1} \\!-\\! \\bar{\\v}\\_t\\|^2 \\!-\\! \\|\\bar{\\v}\\_t \\!-\\! \\v\\_{\\psi}^\\*\\|^2) \n\\!+\\! \\frac{L\\_{\\v}}{3}\\|\\bar{\\v}\\_t\\! -\\! \\v\\_{\\psi}^\\*\\|^2.\n\\end{split}$$ \n[lem:var2] Define $\\hat{\\alpha}\\_t\\! = \\!\\bar{\\alpha}\\_{t-1} + \\frac{\\eta}{K}\\sum\\limits\\_{k=1}^{K} \\nabla\\_{\\alpha} f\\_k(\\v\\_{t-1}^{k}, \\alpha\\_{t-1}^k)$, and\n \n$$\\tilde{\\alpha}\\_{t}\\! =\\! \\tilde{\\alpha}\\_{t-1}\\! +\\! \\frac{\\eta}{K}\\sum\\limits\\_{k=1}^{K}( \n \\nabla\\_{\\alpha} F\\_k(\\v\\_{t-1}^{k},\\alpha\\_{t-1}^k; \\z\\_{t-1}^k) \n\\!-\\!\\nabla\\_{\\alpha} f\\_k(\\v\\_{t-1}^{k},\\!\\alpha\\_{t-1}^k) ).$$ \nWe have,\n \n$$\\begin{split}\n&A\\_2 \\!\\leq\\! \\frac{3G\\_{\\v}^2}{2\\mu\\_{\\alpha}} \\frac{1}{K}\\sum\\limits\\_{k=1}^{K}\\|\\bar{\\v}\\_{t-1}-\\v\\_{t-1}^k\\|^2 \\!+\\! \\frac{3L\\_{\\alpha}^2}{2\\mu\\_{\\alpha}}\\frac{1}{K}\\sum\\limits\\_{k=1}^{K} (\\bar{\\alpha}\\_{t-1} - \\alpha\\_{t-1}^k)^2\\\\\n&+\\frac{3\\eta}{2} ( \\frac{1}{K} \\sum\\limits\\_{k=1}^{K}[ \\nabla\\_{\\alpha} f\\_k(\\v\\_{t-1}^k, \\alpha\\_{t-1}^k) - \\nabla\\_{\\alpha} F\\_k (\\v\\_{t-1}^k, \\alpha\\_{t-1}^k; \\z\\_{t-1})])^2\\\\ \n&+\\!\\frac{1}{K}\\sum\\limits\\_{k=1}^{K}\\!\\langle\\! \\nabla\\_{\\alpha}\\! f\\_k(\\v\\_{t-1}^k,\\!\\alpha\\_{t-1}^k)\\!- \\! \\nabla\\_{\\alpha}\\!F\\_k(\\v\\_{t-1}^k,\\!\\alpha\\_{t-1}^k;\\!\\z\\_{t-1}^k),\\!\\tilde{\\alpha}\\_{t-1}\\!-\\!\\hat{\\alpha}\\_t\\!\\rangle\\\\ \n&\\!+\\!\\frac{1}{2\\eta} ((\\bar{\\alpha}\\_{t-1} \\!-\\! \\alpha^\\*(\\Tilde{\\v}))^2 \\!- \\!(\\bar{\\alpha}\\_{t-1} \\!-\\! \\bar{\\alpha}\\_t)^2 \\!-\\! (\\bar{\\alpha}\\_t \\!-\\! \\alpha^\\*(\\Tilde{\\v}))^2)\n\\!\\\\ \n& +\\! \\frac{\\mu\\_{\\alpha}}{3} (\\bar{\\alpha}\\_{t} \\!-\\! \\alpha^\\*(\\Tilde{\\v}))^2 \n + \\frac{1}{2\\eta} ((\\alpha^\\*(\\Tilde{\\v}) - \\tilde{\\alpha}\\_t)^2 - (\\alpha^\\*(\\Tilde{\\v})-\\Tilde{\\alpha}\\_{t+1})^2).\n\\end{split}$$ \nThe first two terms in the upper bounds of *A*1, *A*2 are the differences between individual solutions and their averages, the third term is the variance of stochastic gradient, and the expectation of the fourth term will diminish. The lemma below will bound the difference between the averaged solution and the individual solutions.\n[stichlemma] If *K* machines communicate every *I* iterations, and update with step size *η*, then \n \n$$\\begin{aligned}\n&\\frac{1}{K}\\sum\\limits\\_{k=1}^{K} \\E[\\|\\bar{\\v}\\_t - \\v\\_t^k\\|^2] \\leq 4\\eta^2I^2 B\\_{\\v}^2\\mathbb I\\_{I>1}\\\\\n&\\frac{1}{K}\\sum\\limits\\_{k=1}^{K} \\E[\\|\\bar{\\alpha}\\_t - \\alpha\\_t^k\\|^2] \\leq 4\\eta^2 I^2 B\\_{\\alpha}^2\\mathbb I\\_{I>1}.\\end{aligned}$$ \nCombining the results in Lemma [lem:objgap], Lemma [lem:var1], Lemma [lem:var2] and Lemma [stichlemma], we can prove the key Lemma [lemmaonestage].\nExperiments\n===========\nIn this section, we conduct some experiments to verify our theory. In our experiments, one “machine” corresponds to one GPU. We use a cluster of 4 computing nodes with each computer node having 4 GPUs, which gives a total of 16 “machines”. We would like to emphasize that even though 4 GPUs sit on one computing node, they only access to different parts of the data. For the experiment with *K* = 1 GPU, We run one computing node by using one GPU. For experiments with *K* = 4 GPUs, we run one computing node by using all four GPUs, and for those experiments with *K* = 16 GPUs, we use four computing nodes by using all GPUs. We notice that the communication costs among GPUs on one computing node may be less than that among GPUs on different computing nodes. Hence, it should be kept in mind that when comparing with *K* = 4 GPUs on different computing nodes, the margin of using *K* = 16 GPUs over using *K* = 4 GPUs should be larger than what we will see in our experimental results. All algorithms are implemented by PyTorch.\n**Data**. We do experiments on 3 datasets: Cifar10, Cifar100 and ImageNet. For Cifar10, we split the original training data into two classes, i.e., the positive class contains 5 original classes and the negative class is composed of the other 5 classes. The Cifar100 dataset is split in a similar way, i.e., the positive class contains 50 original classes and the negative class is composed of the other 50 classes. Testing data for Cifar10 and Cifar100 is the same as the original dataset. For the ImageNet dataset, we sample 1% of the original training data as testing data and use the remaining data as the training data. The training data is split in a similar way to Cifar10 and Cifar100, i.e., the positive class contains 500 original classes and the negative class is composed of the other 500 classes. For each dataset, we create two versions of training data with different positive ratios. By keeping all data in the positive and negative classes, we have *p* = 50% for all three datasetes. To create imbalanced data, we drop some proportion of the negative data for each dataset and keep all the positive examples. In particular, by keeping all the positive data and 40% of the negative data we construct three datasets with positive ratio *p* = 71%. Training data is shuffled and evenly divided to each GPU, i.e., each GPU has access to 1/*K* of the training data, where *K* is the number of GPUs. For all data, we use ResNet50 as our neural network and initialize the model as the pretrained model from PyTorch. Due to the limit of space, we only report the results on datasets with *p* = 71% positive ratio, and other results are included in the supplement.\n**Baselines and Parameter Setting.** For baselines, we compare with the single-machine algorithm PPD-SG as proposed in , which is represented by *K* = 1 in our results, and the naive parallel version of PPD-SG, which is denoted by *K* = X, *I* = 1 in our results. For all algorithms, we set *T**s* = *T*03*k*, *η**s* = *η*0/3*k*. *T*0 and *η*0 are tuned for PPD-SG and set to the same for all other algorithms for fair comparison. *T*0 is tuned in [2000, 5000, 10000], and *η*0 is tuned in [0.1, 0.01, 0.001]. We fix the batch size for each GPU as 32. For simplicity, in our experiments we use a fixed value of *I* in order to see its tradeoff with the number of machines *K*.\n[ht]\n[fig:imagenet71]\n[fig:cifar10071]\n[fig:cifar1071]\n \n[fig:ablationc100K4P71]\n \n[fig:ablationc10K4P71]\n**Results.** We plot the curve of testing AUC versus the number of iterations and versus running time. We notice that evaluating the training objective function value on all examples is very expensive, so we use the testing AUC as our evaluation metric. It may cause some gap between our results and the theory; however, the trend should be enough for our purpose to verify that our distributed algorithms can enjoy faster convergence in both the number of iterations and running time. We have the following observations.\n* **Varying *K*.** By varying *K* and fixing the value of *I*, we aim to verify the parallel speedup. The results are shown in Figure [fig:imagenet71](a), Figure [fig:cifar10071](a) and Figure [fig:cifar1071](a). They show that when *K* becomes larger, then our algorithm requires a less number of iterations to converge to the target AUC, which is consistent with the parallel speedup result as indicated by Theorem [theoremallstage]. In addition, CoDA with *K* = 16 machines is also the most time-efficient algorithm among all settings.\n* **Varying *I*.** By varying *I* and fixing the value of *K*, we aim to verify that skipping communications up to a certain number of iterations of CoDA does not hurt the iteration complexity but can dramatically reduce the total communication costs. In particular, we fix *K* = 16 and vary *I* in the range {1, 8, 64, 512, 1024}. The results are shown in Figures [fig:imagenet71](b), Figures [fig:cifar10071](b) and Figures [fig:cifar1071](b). They exhibit that even when *I* becomes moderately large, our algorithm is still able to deliver comparable performance in terms of the number of iterations compared with the case when *I* = 1. The largest value of *I* that does not cause a dramatic performance drop compared with *I* = 1 is *I* = 1024, *I* = 64, *I* = 64 on ImageNet, CIFAR100 and CIFAR10, respectively. However, up to these thresholds the running time of CoDA can be dramatically reduced than the naive parallel version with *I* = 1.\n* **Trade-off between *I* and *K*.** Finally, we verify the trade-off between *I* and *K* as indicated in Theorem [thm:main]. To this end, we conduct experiments by fixing *K* = 4 GPUs and varying the value *I*, and comparing the limits of *I* for *K* = 4 and *K* = 16. The results of using *K* = 4 on CIFAR100 and CIFAR10 are reported in Figure [fig:ablationc100K4P71] and Figure [fig:ablationc10K4P71]. We can observe that when *K* = 4 the upper limit of *I* that does not cause a dramatic performance drop compared with *I* = 1 is *I* = 512 for the two datasets, which is larger than the upper limit of *I* = 64 for *K* = 16. This is consistent with our Theorem [theoremallstage].\nConclusion\n==========\nIn this paper, we have designed a communication-efficient distributed stochastic deep AUC maximization algorithm, in which each machine is able to do multiple iterations of local updates before communicating with the central node. We have proven the linear speedup property and shown that the communication complexity can be dramatically reduced for multiple machines up to a large threshold number. Our empirical studies verify the theory and also demonstrate the effectiveness of the proposed distributed algorithm on benchmark datasets.\nAcknowledgements\n================\nThis work is partially supported by National Science Foundation CAREER Award 1844403 and National Science Foundation Award 1933212.\nProof of Theorem [theoremallstage]\n==================================\n*Proof.* Define $\\phi\\_s(\\v) = \\phi(\\v) + \\frac{1}{2\\gamma}\\|\\v-\\v\\_{s-1}\\|^2$. We can see that $\\phi\\_s(\\v)$ is convex and smooth since $\\gamma \\leq 1/L\\_{\\v}$. The smooth coefficient of *ϕ**s* is $\\hat{L}\\_{\\v} = L\\_{\\v} + 1/\\gamma$. According to Theorem 2.1.5 of, we have\n \n$$\\begin{split}\n\\|\\nabla \\phi\\_s(\\v\\_s)\\|^2 \\leq 2\\hat{L}\\_{\\v} (\\phi\\_{s}(\\v\\_s) - \\phi\\_{s}(\\v\\_{\\phi\\_s}^\\*)).\n\\end{split}\n\\label{by\\_nestrov}$$ \nApplying Lemma [lemmaonestage], we have\n \n$$\\begin{split}\nE\\_{s-1}[\\phi\\_s (\\v\\_s) - \\phi\\_s (\\v\\_{\\phi\\_s}^\\*)]\n\\leq \\frac{2}{\\eta\\_s T\\_s} \\|\\v\\_{s-1} - \\v\\_{\\phi\\_s}^\\*\\|^2 + \\frac{1}{\\eta\\_s T\\_s} (\\alpha\\_{s-1} -\\alpha^\\*(\\v\\_s))^2 + H \\eta\\_s^2 I\\_s^2 B^2 \\I\\_{I\\_s>1} + \\frac{\\eta\\_s (2\\sigma\\_{\\v}^2 + 3\\sigma\\_{\\alpha}^2)}{2K}. \n\\end{split}$$ \nDenote $\\x\\_{1:m\\_s}^k = (\\x\\_1^k,..., \\x\\_{m\\_s}^k)$, *y*1 : *m**s**k* = (*y*1*k*, ..., *y**m**s**k*), and $\\Tilde{f}\\_k (\\x\\_{1:m\\_s}^k, y^k\\_{1:m\\_s}) = \\frac{\\sum\\limits\\_{i=1}^{m\\_s}h(\\w\\_s;\\x\\_i^k) \\I\\_{y\\_i^k=y} }{\\sum\\limits\\_{i=1}^{m\\_s}\\I\\_{y^k\\_i=y}} - E\\_{\\x^k}[h(\\w\\_{s};\\x^k)|y]$. If $\\sum\\_{i=1}^{m\\_s} \\I\\_{y\\_i^k=y}>0$, then $\\frac{\\sum\\limits\\_{i=1}^{m\\_s}h(\\w\\_s;\\x\\_i^k) \\I\\_{y\\_i^k=y} }{\\sum\\limits\\_{i=1}^{m\\_s}\\I\\_{y^k\\_i=y}}$ is an unbiased estimation of $E\\_{\\x^k}[h(\\w\\_{s};\\x^k)|y]$. Noting $0\\leq h(\\w; \\x) \\leq 1$, we have $Var(h(\\w; \\x^k)|y) \\leq \\tilde{\\sigma}^2 \\leq 1$. Then we know that\n \n$$\\begin{split} \nE\\_{\\x\\_{1:m\\_s}^k}[(\\Tilde{f}(\\x\\_{1:m\\_s}^k, y\\_{1:m\\_s}^k))^2 | y\\_{1:m\\_s}^k]\n&\\leq \\frac{\\tilde{\\sigma}^2}{\\sum\\_{i=1}^{m\\_s} \\I\\_{y\\_i^k=y}} \\I\\_{(\\sum\\_{i=1}^{m\\_s} \\I\\_{y\\_i^k=y}>0)} + 1\\cdot \\I\\_{(\\sum\\_{i=1}^{m\\_s} \\I\\_{y\\_i^k=y}=0)} \\\\\n&\\leq \\frac{\\I\\_{(\\sum\\_{i=1}^{m\\_s} \\I\\_{y\\_i^k=y}>0)}}{\\sum\\_{i=1}^{m\\_s} \\I\\_{y\\_i^k=y}} + \\I\\_{(\\sum\\_{i=1}^{m\\_s} \\I\\_{y\\_i^k=y}=0)}.\n\\end{split}$$ \nHence,\n \n$$\\begin{split} \n& E\\_{s-1}[ \\Tilde{f}\\_k (\\x\\_{1:m\\_s}^k, y\\_{1:m\\_s}^k) ]\n= E\\_{y\\_{1:m\\_s}^k} \\left[ E\\_{\\x\\_{1:m\\_s}^k} [ (\\Tilde{f}\\_k (\\x\\_{1:m\\_s}^k, y\\_{1:m\\_s}^k))^2 | y\\_{1:m\\_s}^k] \\right] \\\\\n&\\leq E\\_{y\\_{1:m\\_s}^k} \\left[\\frac{\\I\\_{(\\sum\\_{i=1}^{m\\_s} \\I\\_{y\\_i^k=y} > 0)} }{\\sum\\_{i=1}^{m\\_s} \\I\\_{y\\_i^k=y}} + \\I\\_{\\sum\\_{i=1}^{m\\_s} \\I\\_{y\\_i^k=y} = 0} \\right] \\leq \\frac{1}{m\\_s \\text{Pr}(y\\_i^k = y)} + (1-\\text{Pr}(y\\_i^k=y))^{m\\_s}.\n\\end{split}$$ \nDenote\n \n$$\\begin{split}\n\\alpha^\\*(\\v\\_s) &= \\arg\\max\\limits\\_{\\alpha} f(\\v\\_s,\\alpha) \n= \\frac{1}{K}\\sum\\limits\\_{k=1}^{K} E\\left[ \\frac{h(\\w\\_s; \\x^k)\\I\\_{y^k=-1}}{1-p} - \\frac{h(\\w\\_s; \\x^k)\\I\\_{y^k=1}}{p} \\right]\\\\ \n& = \\frac{1}{K} \\sum\\limits\\_{k=1}^{K} \n\\left[E\\left[h(\\w\\_s; \\x^k)|y^k=-1\\right] - E\\left[h(\\w\\_s; \\x^k)|y^k=1\\right] \\right]. \n\\end{split}$$ \nTherefore,\n \n$$\\begin{split}\nE\\_{s-1} [(\\alpha\\_{s-1}-\\alpha^\\*(\\v\\_{s-1}))^2] &= \nE\\_{s-1} \\left[\\frac{1}{K} \\sum\\limits\\_{k=1}^{K}\\frac{\\sum\\limits\\_{i=1}^{m\\_{s-1}} h(\\w\\_{s-1}; \\x\\_i^k) \\I\\_{y\\_i^{k}=-1}}{\\sum\\limits\\_{i=1}^{m\\_{s-1}} \\I\\_{y\\_i^{k}=-1}} - E\\left[\\frac{1}{K} \\sum\\limits\\_{k=1}^{K} h(\\w\\_{s-1}; \\x\\_i^k)|y=-1\\right] \\right.\\\\ \n&~~~~~~~~~~~~~~~~~ \n\\left. + E\\left[\\frac{1}{K}\\sum\\limits\\_{k=1}^{K} h(\\w\\_{s-1}; \\x\\_i^k)|y=1\\right] \n- \\frac{1}{K}\\sum\\limits\\_{k=1}^K \\frac{ \\sum\\limits\\_{i=1}^{m\\_{s-1}}h(\\w\\_{s-1}; \\x\\_i^k)\\I\\_{y\\_i^k=1}}{\\sum\\limits\\_{i=1}^{m\\_{s-1}} \\I\\_{y\\_i^k=1}} \\right]^2 \\\\\n& \\leq \\frac{2}{K m\\_{s-1} \\text{Pr}(y\\_i^k=-1)} \n+ \\frac{2(1-\\text{Pr}(y\\_i^k=-1))^{m\\_{s-1}}}{K} + (1-\\text{Pr}(y\\_i^k=-1))^{2m\\_{s-1}} \\\\\n&~~~ + \\frac{2}{K m\\_{s-1} \\text{Pr}(y\\_i^k=1)} \n+ \\frac{2(1-\\text{Pr}(y\\_i=1))^{m\\_{s-1}} }{K} \n+ (1- \\text{Pr}(y\\_i^k=1))^{2m\\_{s-1}}\\\\ \n& \\leq \\frac{2}{K m\\_{s-1} p(1-p)} + \\frac{3 p^{m\\_{s-1}}}{K} + \\frac{3(1-p)^{m\\_{s-1}}}{K} \n\\leq 2\\left(\\frac{1}{K m\\_{s-1} p(1-p)} + \\frac{3\\Tilde{p}^{m\\_{s-1}}}{K} \\right) \\\\ \n&\\leq 2 \\left(\\frac{1}{K m\\_{s-1} p (1-p)} + \\frac{C}{K m\\_{s-1}}\\right) \\leq \\frac{2(1 + C)}{K m\\_{s-1} p(1-p)}, \n\\end{split}$$ \nwhere $C = \\frac{3\\tilde{p}^{\\frac{1}{\\ln(1/\\Tilde{p})}}}{2\\ln(1/\\Tilde{p})}$ and $\\Tilde{p} = \\max(p, 1-p)$.\nSince $h(\\w; \\x)$ is *G**h*-Lipschitz, $E[h(\\w, \\x)|y=-1] - E[h(\\w, \\x)|y=1]$ is 2*G**h*-Lipschitz. It follows that\n \n$$\\begin{split}\n&E\\_{s-1}[(\\alpha\\_{s-1} - \\alpha^\\*(\\v\\_s))^2] = E\\_{s-1}[(\\alpha\\_{s-1}-\\alpha^\\*(\\v\\_{s-1}) + \\alpha^\\*(\\v\\_{s-1}) - \\alpha^\\*(\\v\\_{s}))^2] \\\\\n&\\leq E\\_{s-1}[2(\\alpha\\_{s-1} - \\alpha^\\*(\\v\\_{s-1}))^2 + 2(\\alpha^\\*(\\v\\_{s-1}) - \\alpha^\\*(\\v\\_{s}))^2] \\\\\n& = E\\_{s-1}[2(\\alpha\\_{s-1} - \\alpha^\\*(\\v\\_s))^2] \\\\\n&+ 2\\left\\|\\frac{1}{K}\\sum\\limits\\_{k=1}^{K} \\bigg[E\\_{s-1}[h(\\w\\_{s-1}; \\x^k)|y^k=-1] - E\\_{s-1}[h(\\w\\_{s-1}; \\x^k)|y^k=1]] - [E\\_{s-1}[h(\\w\\_s; \\x)|y^k=-1] - E\\_{s-1}[h(\\w\\_s; \\x^k)|y^k=1]\\bigg] \\right\\|^2 \\\\ \n&\\leq \\frac{2(1+C)}{m\\_{s-1} K 4p^2(1-p)^2} + 8G\\_{h}^2 E\\_{s-1}[\\|\\v\\_{s-1} - \\v\\_s \\|^2]. \n\\end{split}$$ \nSince $m\\_{s-1} \\geq \\frac{1+C}{\\eta\\_s^2 T\\_s\\sigma\\_{\\alpha}^2 p^2(1-p)^2}$, then we have\n \n$$\\begin{split}\nE[\\phi\\_s (\\v\\_s) - \\phi\\_s (\\v\\_{\\phi\\_s}^\\*)] &\\leq \\frac{2\\|\\v\\_{s-1} - \\v\\_{\\phi\\_s}^\\*\\|^2 + 8 G\\_{h}^2E[\\|\\v\\_{s-1} -\\v\\_{s} \\|]}{\\eta\\_s T\\_s} + \\frac{\\eta\\_s\\sigma\\_{\\alpha}^2}{2 K} + H \\eta\\_s^2 I\\_s^2 B^2 \\I\\_{I\\_s>1} + \\frac{\\eta\\_s (2\\sigma\\_{\\v}^2 + 3\\sigma\\_{\\alpha}^2)}{2 K} \\\\\n&\\leq \\frac{2\\|\\v\\_{s-1} - \\v\\_{\\phi\\_s}^\\*\\|^2 + 8 G\\_{h}^2 E[\\|\\v\\_{s-1} - \\v\\_s\\|^2]}{\\eta\\_s T\\_s} + H \\eta\\_s^2 I\\_s^2 B^2 \\I\\_{I\\_s>1} + \\frac{2\\eta\\_s (\\sigma\\_{\\v}^2 + \\sigma\\_{\\alpha}^2)}{K}. \\\\\n\\end{split} \n\\label{local\\_lemma\\_ved}$$ \nWe define $I'\\_s = 1/\\sqrt{K\\eta\\_s} = \\frac{1}{K\\sqrt{\\eta\\_0}}\\exp(\\frac{c(s-1)}{2})$. Applying this and ([locallemmaved]) to ([bynestrov]), we get\n \n$$\\begin{split}\nE[\\|\\nabla \\phi\\_s (\\v\\_{s})\\|^2] &\\leq 2\\hat{L}\\_{\\v} \\bigg[ \\frac{2\\|\\v\\_{s-1} - \\v\\_{\\phi\\_s}^\\*\\|^2 + 8 G\\_{h}^2E[\\|\\v\\_{s-1} - \\v\\_s\\|^2]}{\\eta\\_s T\\_s} + H\\eta\\_s^2 {I'}\\_s^2 B^2 + \\frac{2\\eta\\_s(\\sigma\\_{\\v}^2 + \\sigma\\_{\\alpha}^2)}{K}\\bigg] \\\\\n&\\leq 2\\hat{L}\\_{\\v} \\bigg[ \\frac{2\\|\\v\\_{s-1} - \\v\\_{\\phi\\_s}^\\*\\|^2 + 8 G\\_{h}^2E[\\|\\v\\_{s-1} - \\v\\_s\\|^2]}{\\eta\\_s T\\_s} + H\\eta\\_s^2 {I'}^2\\_s B^2 + \\frac{2\\eta\\_s(\\sigma\\_{\\v}^2 + \\sigma\\_{\\alpha}^2)}{K} \\bigg].\n\\end{split}\n\\label{random9}$$ \nTaking $\\gamma = \\frac{1}{2L\\_{\\v}}$, then $\\hat{L}\\_{\\v} = 3L\\_{\\v}$. Note that $\\phi\\_{s}(\\v)$ is $(\\gamma^{-1} - L\\_{\\v})$-strongly convex, we have\n \n$$\\begin{split}\n\\phi\\_{s}(\\v\\_{s-1}) \\geq \\phi\\_s(\\v\\_{\\phi\\_s}^\\*) + \\frac{L\\_{\\v}}{2} \\|\\v\\_{s-1} - \\v\\_{\\phi\\_s}^\\*\\|^2.\n\\end{split}\n\\label{random:strong\\_v}$$ \nPlugging ([random:strongv]) into ([locallemmaved]), we get\n \n$$\\begin{split}\n&E\\_{s-1}[\\phi(\\v\\_s) + L\\_{\\v}\\|\\v\\_s - \\v\\_{s-1}\\|^2]\\\\\n&\\leq \\phi\\_s(\\v\\_{\\phi\\_s}^\\*) + \\frac{2\\|\\v\\_{s-1} - \\v\\_{\\phi\\_s}^\\*\\|^2 + 8 G\\_{h}^2E\\_{s-1}[\\|\\v\\_{s-1} - \\v\\_s\\|^2]}{\\eta\\_s T\\_s} + H \\eta\\_s^2 {I'}\\_s^2 B^2 + \\frac{2\\eta\\_s(\\sigma\\_{\\v}^2 + \\sigma\\_{\\alpha}^2)}{K}\\\\\n&\\leq \\phi\\_s(\\v\\_{s-1}) - \\frac{L\\_{\\v}}{2}\\|\\v\\_{s-1}-\\v\\_{\\phi\\_s}^\\*\\|^2 +\\\\\n&~~~~~~~~~~~ \\frac{2\\|\\v\\_{s-1} - \\v\\_{\\phi\\_s}^\\*\\|^2 + 8 G\\_{h}^2E\\_{s-1}[\\|\\v\\_{s-1} - \\v\\_s\\|^2]}{\\eta\\_s T\\_s} + H\\eta\\_s^2 {I'}\\_s^2 B^2 + \\frac{2\\eta\\_s(\\sigma\\_{\\v}^2 + \\sigma\\_{\\alpha}^2)}{K}.\\\\\n\\end{split}$$ \nNoting $\\eta\\_s T\\_s L\\_{\\v} = \\max(8, 8 G\\_{h}^2)$ and $\\phi\\_{s}(\\v\\_{s-1}) = \\phi(\\v\\_{s-1})$, we rearrange terms and get\n \n$$\\begin{split}\n\\frac{2\\|\\v\\_{s-1} - \\v\\_{\\phi\\_s}^\\*\\|^2 + 8 G\\_{h}^2E\\_{s-1}[\\|\\v\\_{s-1} - \\v\\_s\\|^2]}{\\eta\\_s T\\_s} \\leq \\phi(\\v\\_{s-1}) - E\\_{s-1}[\\phi(\\v\\_{s})] + H\\eta\\_s^2 {I'}\\_s^2 B^2 + \\frac{2\\eta\\_s(\\sigma\\_{\\v}^2 + \\sigma\\_{\\alpha}^2)}{K}.\n\\end{split}\n\\label{random\\_11}$$ \nCombining ([random9]) and ([random11]), we get\n \n$$\\begin{split}\nE\\_{s-1}\\|\\nabla \\phi\\_s(\\v\\_s)\\|^2 \\leq 2\\hat{L}\\_{\\v}\\bigg[ \\phi(\\v\\_{s-1}) - E\\_{s-1}[\\phi(\\v\\_s)] + 2H\\eta\\_s^2 {I'}\\_s^2 B^2 + \\frac{4\\eta\\_s(\\sigma\\_{\\v}^2 + \\sigma\\_{\\alpha}^2)}{K} \\bigg]\\\\\n=6L\\_{\\v}\\bigg[ \\phi(\\v\\_{s-1}) - E\\_{s-1}[\\phi(\\v\\_s)] + 2H\\eta\\_s^2 {I'}\\_s^2 B^2 + \\frac{4\\eta\\_s(\\sigma\\_{\\v}^2 + \\sigma\\_{\\alpha}^2)}{K} \\bigg].\\\\\n\\end{split}$$ \nTaking expectation on both sides over all randomness until $\\v\\_{s-1}$ is generated and by tower property, we have\n \n$$\\begin{split}\nE\\|\\nabla \\phi\\_s(\\v\\_s)\\|^2 \\leq 6L\\_{\\v}\\bigg( E[\\phi(\\v\\_{s-1}) - \\phi(\\v\\_{\\phi}^\\*)] - E[\\phi(\\v\\_{s}) - \\phi(\\v\\_{\\phi}^\\*)] + 2H\\eta\\_s^2 {I'}\\_s^2 B^2 + \\frac{4\\eta\\_s(\\sigma\\_{\\v}^2 + \\sigma\\_{\\alpha}^2)}{K} \\bigg)\n\\end{split}\n\\label{random13}$$ \nSince $\\phi(\\v)$ is $L\\_{\\v}$-smooth and hence is $L\\_{\\v}$-weakly convex, we have\n \n$$\\begin{split}\n&\\phi(\\v\\_{s-1}) \\geq \\phi(\\v\\_{s}) + \\langle \\nabla\\phi(\\v\\_s), \\v\\_{s-1}-\\v\\_s \\rangle - \\frac{L\\_{\\v}}{2}\\|\\v\\_{s-1}-\\v\\_{s}\\|^2\\\\\n&=\\phi(\\v\\_s) + \\langle \\nabla \\phi(\\v\\_s) + 2L\\_{\\v}(\\v\\_s - \\v\\_{s-1}), \\v\\_{s-1} - \\v\\_s\\rangle + \\frac{3}{2}L\\_{\\v}\\|\\v\\_{s-1} - \\v\\_{s}\\|^2\\\\\n& = \\phi(\\v\\_s) + \\langle \\nabla \\phi\\_s(\\v\\_s), \\v\\_{s-1}-\\v\\_{s}\\rangle +\\frac{3}{2}L\\_{\\v}\\|\\v\\_{s-1}-\\v\\_s\\|^2\\\\\n& = \\phi(\\v\\_s) - \\frac{1}{2L\\_{\\v}}\\langle \\nabla\\phi\\_s(\\v\\_s),\\nabla\\phi\\_s(\\v\\_s) -\\nabla\\phi(\\v\\_s)\\rangle + \\frac{3}{8L\\_{\\v}}\\|\\nabla \\phi\\_s(\\v\\_s) - \\nabla \\phi(\\v\\_s)\\|^2 \\\\\n& = \\phi(\\v\\_s) - \\frac{1}{8L\\_{\\v}}\\|\\nabla \\phi\\_s(\\v\\_s\\|^2 - \\frac{1}{4L\\_{\\v}}\\langle \\nabla\\phi\\_s(\\v\\_s), \\nabla\\phi(\\v\\_s)\\rangle + \\frac{3}{8L\\_{\\v}}\\|\\nabla \\phi (\\v\\_s)\\|^2\n\\end{split}$$ \nRearranging terms, it yields\n \n$$\\begin{split}\n&\\phi(\\v\\_s) - \\phi(\\v\\_{s-1}) \\leq \\frac{1}{8L\\_{\\v}}\\|\\nabla \\phi\\_s(\\v\\_s)\\|^2 + \\frac{1}{4L\\_{\\v}}\\langle \\nabla\\phi\\_s(\\v\\_s), \\nabla\\phi(\\v\\_s)\\rangle - \\frac{3}{8L\\_{\\v}}\\|\\nabla \\phi (\\V\\_s)\\|^2\\\\\n&\\leq \\frac{1}{8L\\_{\\v}}\\|\\nabla \\phi\\_s(\\v\\_s)\\|^2 + \\frac{1}{8L\\_{\\v}} (\\|\\nabla \\phi\\_s(\\v\\_s)\\|^2 + \\|\\nabla\\phi(\\v\\_s)\\|^2) - \\frac{3}{8L\\_{\\v}}\\|\\nabla \\phi (\\V\\_s)\\|^2\\\\\n&=\\frac{1}{4L\\_{\\v}}\\|\\nabla \\phi\\_s(\\v\\_s)\\|^2 - \\frac{1}{4L\\_{\\v}}\\|\\nabla \\phi(\\v\\_s)\\|^2\\\\\n&\\leq \\frac{1}{4L\\_{\\v}}\\|\\nabla \\phi\\_s(\\v\\_s)\\|^2 - \\frac{\\mu}{2L\\_{\\v}}(\\phi(\\v\\_s) - \\phi(\\v\\_{\\phi}^\\*))\n\\end{split}\n\\label{random15}$$ \nDefine $\\Delta\\_s = \\phi(\\v\\_s) - \\phi(\\v\\_{\\phi}^\\*)$. Combining ([random13]) and ([random15]), we get\n \n$$\\begin{split}\nE[\\Delta\\_s - \\Delta\\_{s-1}] \\leq \\frac{3}{2}E(\\Delta\\_{s-1} - \\Delta\\_s) + 3H\\eta\\_s^2 {I'}\\_s^2 B^2 + \\frac{6\\eta\\_s(\\sigma\\_{\\v}^2+\\sigma\\_{\\alpha}^2)}{K} - \\frac{\\mu}{2L\\_{\\v}}E[\\Delta\\_s]\n\\end{split}$$ \nTherefore,\n \n$$\\begin{split}\n\\bigg( \\frac{5}{2} + \\frac{\\mu}{2L\\_{\\v}}\\bigg) E[\\Delta\\_{s}] \\leq \\frac{5}{2}E[\\Delta\\_{s-1}] + 3 H \\eta\\_s^2 {I'}\\_s^2 B^2 + \\frac{6\\eta\\_s (\\sigma\\_{\\v}^2+\\sigma\\_{\\alpha}^2)}{K}\n\\end{split}$$ \nUsing $c = \\frac{\\mu/L\\_{\\v}}{5+\\mu/L\\_{\\v}}$ as defined in the theorem,\n \n$$\\begin{split}\n&E[\\Delta\\_S] \\leq \\frac{5L\\_{\\v}}{5L\\_{\\v}+\\mu} E[\\Delta\\_{S-1}] + \n\\frac{2L\\_{\\v}}{5L\\_{\\v}+\\mu}\\bigg[ 3H\\eta\\_S^2 {I'}\\_S^2 B^2 + \\frac{6\\eta\\_S(\\sigma\\_{\\v}^2+\\sigma\\_{\\alpha}^2)}{K}\\bigg]\\\\\n&=(1-c)\\bigg[E[\\Delta\\_{S-1}] + \\frac{2}{5}\\bigg(3H\\eta\\_S^2 {I'}\\_S^2 B^2 + \\frac{6\\eta\\_S(\\sigma\\_{\\v}^2+\\sigma\\_{\\alpha}^2)}{K}\\bigg)\\bigg]\\\\\n&\\leq (1-c)^S E[\\Delta\\_0] + \\frac{6H B^2}{5} \\sum\\limits\\_{j=1}^{S} \\eta\\_j^2 {I'}\\_j^2 (1-c)^{S+1-j} + \\frac{12(\\sigma\\_{\\v}^2 + \\sigma\\_{\\alpha}^2)}{5K} \\sum\\limits\\_{j=1}^{S} \\eta\\_j (1-c)^{S+1-j} \\\\\n& = (1-c)^S E[\\Delta\\_0] + \\frac{6H B^2}{5} \\sum\\limits\\_{j=1}^{S} \\eta\\_j^2 {I'}\\_j^2 (1-c)^{S+1-j} + \\frac{12(\\sigma\\_{\\v}^2 + \\sigma\\_{\\alpha}^2)}{5K} \\sum\\limits\\_{j=1}^{S} \\eta\\_j (1-c)^{S+1-j}\n\\end{split}$$ \nWe then have\n \n$$\\begin{split}\n&E[\\Delta\\_S] \\leq (1 - c )^S E[\\Delta\\_0] + \\left(\\frac{6H B^2}{5K} + \\frac{12(\\sigma\\_{\\v}^2 + \\sigma\\_{\\alpha}^2)}{5K}\\right)\\sum\\limits\\_{j=1}^{S} \\eta\\_j (1-c)^{S+1-j} \\\\\n&\\leq \\exp(-c S)\\Delta\\_0 + \\left(\\frac{6H B^2}{5K} + \\frac{12(\\sigma\\_{\\v}^2 + \\sigma\\_{\\alpha}^2)}{5K}\\right) \\sum\\limits\\_{j=1}^{S} \\eta\\_j \\exp(-c(S+1-j))\\\\\n&=\\exp(-c S)\\Delta\\_0 + \\left(\\frac{6H B^2}{5} + \\frac{12(\\sigma\\_{\\v}^2 + \\sigma\\_{\\alpha}^2)}{5}\\right) \\eta\\_0 S\\exp (-c S).\\\\\n\\end{split}$$ \nTo achieve *E*[Δ*S*] ≤ *ε*, it suffices to make\n \n$$\\begin{split}\n\\exp(-c S) \\Delta\\_0 \\leq \\epsilon/2\n\\end{split}$$ \nand\n \n$$\\begin{split}\n\\left(\\frac{6H B^2}{5} + \\frac{12(\\sigma\\_{\\v}^2 + \\sigma\\_{\\alpha}^2)}{5}\\right) \\eta\\_0 S\\exp (-c S) \\leq \\epsilon/2.\n\\end{split}$$ \nSo, it suffices to make\n \n$$\\begin{split}\nS\\geq c^{-1} \\max\\bigg\\{\\log \\left(\\frac{2\\Delta\\_0}{\\epsilon}\\right), \\log S + \\log\\bigg[ \\frac{2\\eta\\_0}{\\epsilon} \\frac{6HB^2 + 12(\\sigma\\_{\\v}^2+\\sigma\\_{\\alpha}^2)}{5}\\bigg]\\bigg\\}.\n\\end{split}$$ \nTaking summation of iteration over *s* = 1, ..., *S*, we have the total iteration complexity as\n \n$$\\begin{split}\n&T = \n\\sum\\limits\\_{s=1}^{S} T\\_s \\leq \\frac{\\max\\{8, 8G\\_{h}^2\\}}{L\\_{\\v}\\eta\\_0 K} \\frac{\\exp(cS) - 1}{\\exp(c) - 1} \\leq \\frac{\\max\\{8, 8G\\_{h}^2\\}}{L\\_{\\v}\\eta\\_0 K}\\frac{5L\\_{\\v} + \\mu}{\\mu} \\exp(cS) \\\\\n&=\\tilde{O}\\left(\\max\\bigg(\\frac{\\Delta\\_0}{\\mu \\epsilon \\eta\\_0 K }, \\frac{S (6H B^2 + 12(\\sigma\\_{\\v}^2 + \\sigma\\_{\\alpha}^2))}{\\mu \\epsilon K} \\bigg)\\right)\n=\\tilde{O}\\bigg(\\max\\left(\\frac{\\Delta\\_0}{\\mu \\epsilon \\eta\\_0 K}, \\frac{L\\_{\\v\n}}{\\mu^2 K\\epsilon}\\right)\\bigg).\n\\end{split}$$ \nTo analyze the total communication complexity, we will analyze two cases: (1) $\\frac{1}{K\\sqrt{\\eta\\_0}} > 1$ ; (2) $\\frac{1}{K\\sqrt{\\eta\\_0}} \\leq 1$.\n(1) If $\\frac{1}{K\\sqrt{\\eta\\_0}} > 1$, then $I\\_s = \\max(1, \\frac{1}{K\\sqrt{\\eta\\_0}} \\exp(\\frac{c(s-1)}{2})) = \\frac{1}{K\\sqrt{\\eta\\_0}} \\exp(\\frac{c(s-1)}{2}) $ for any *s* ≥ 1.\nThe total number of communications:\n \n$$\\begin{split}\n&\\sum\\limits\\_{s=1}^{S} \\frac{T\\_s}{I\\_s} = \\sum\\limits\\_{s=1}^{S} \\frac{\\max(8, 8 G\\_h^2)}{L\\_{\\v} {\\eta^{1/2}\\_0}} \\exp\\bigg(\\frac{c(s-1)}{2}\\bigg) = \\frac{\\max(8, 8 G\\_h^2)}{L\\_{\\v}{\\eta^{1/2}\\_0}} \\frac{\\exp(c S/2) - 1}{\\exp(c/2) - 1}\\\\\n&=\\tilde{O}\\bigg(\\max\\bigg(\\frac{(2\\Delta\\_0/\\epsilon)^{1/2}}{\\mu {\\eta^{1/2}\\_0}},\\frac{(S(6H B^2+12(\\sigma\\_{\\v}^2+\\sigma\\_{\\alpha}^2))^{1/2}}{ \\mu \\epsilon^{1/2}}\\bigg)\\bigg)\n=\\tilde{O} \\left(\\frac{\\Delta\\_0^{1/2}}{\\mu (\\eta\\_0 \\epsilon)^{1/2}}, \\frac{L\\_{\\v}^{1/2}}{\\mu^{3/2}\\epsilon^{1/2}}\\right).\n\\end{split}$$ \n(2) If $\\frac{1}{K\\sqrt{\\eta\\_0}} \\leq 1$, then *I**s* = 1 for $s \\leq \\left\\lceil {2c^{-1}} \\log(K\\sqrt{\\eta\\_0}) + 1 \\right\\rceil := S\\_1$ and $I\\_s = \\frac{1}{K\\sqrt{\\eta\\_0}}\\exp\\left(\\frac{s-1}{2}\\right)$ for $s > \\frac{2(5+\\mu/L\\_{\\v})}{\\mu/L\\_{\\v}} \\log(K\\sqrt{\\eta\\_0}) + 1$.\nObviously, $S\\_1 \\leq \\frac{2(5+\\mu/L\\_{\\v})}{\\mu/L\\_{\\v}} \\log(K\\sqrt{\\eta\\_0}) + 2$. The number of iterations from *s* = 1 to *S*1 is\n \n$$\\begin{split}\n& \\sum\\limits\\_{s=1}^{S\\_1} T\\_s = \\sum\\limits\\_{s=1}^{S\\_1} \\frac{\\max\\{8, 8 G\\_h^2\\}}{\\eta\\_0 L\\_{\\v} K} \\exp(c(s-1))\\\\\n& = \\frac{\\max\\{8, 8 G\\_h^2\\}}{\\eta\\_0 L\\_{\\v} K} \\frac{\\exp(c S\\_1)-1}{\\exp(c) - 1}\\\\\n& \\leq c^{-1} \\frac{\\max\\{8, 8 G\\_h^2\\}}{\\eta\\_0 L\\_{\\v} K} \\exp\\left( 2\\log(K\\sqrt{\\eta\\_0}) + 2c \\right) \\\\\n& = c^{-1} \\frac{\\max\\{8, 8 G\\_h^2\\}}{\\eta\\_0 L\\_{\\v} K} K^2 \\eta\\_0 \\exp\\left(\\frac{2\\mu/L\\_{\\v}}{5+\\mu/L\\_{\\v}}\\right) \\\\\n& \\leq c^{-1} \\max\\{8, 8 G\\_h^2\\} K \\exp\\left(2\\right). \\\\\n\\end{split}$$ \nThus, the total number of communications is\n \n$$\\begin{split}\n&\\sum\\limits\\_{s=1}^{S\\_1} T\\_s + \\sum\\limits\\_{s=S\\_1+1}^{S} \\frac{T\\_s}{I\\_s}\\\\\n& = c^{-1} \\max\\{8, 8G\\_h^2\\} K \\exp\\left(2\\right) + \\sum\\limits\\_{s=S\\_1+1}^{S} \\frac{\\max(8, 8G\\_h^2)}{L\\_{\\v} \\eta\\_0^{1/2}} \\exp\\left( \\frac{s-1}{2} \\frac{\\mu/L\\_{\\v}}{5+\\mu/L\\_{\\v}} \\right)\\\\\n&\\leq c^{-1} \\max\\{8, 8G\\_h^2\\} K \\exp\\left(2\\right) + \\sum\\limits\\_{s=1}^{S} \\frac{\\max(8, 8G\\_h^2)}{L\\_{\\v} \\eta\\_0^{1/2}} \\exp\\left( \\frac{s-1}{2} \\frac{\\mu/L\\_{\\v}}{5+\\mu/L\\_{\\v}} \\right)\\\\\n& \\leq c^{-1} \\max\\{8, 8G\\_h^2\\} K \\exp\\left(2\\right) + \\frac{\\max(8, 8G\\_h^2)}{L\\_{\\v} \\eta\\_0^{1/2}} \\frac{\\exp(\\frac{S}{2}\\frac{\\mu/L\\_{\\v}}{5+\\mu/L\\_{\\v}}) - 1} {\\exp(\\frac{\\mu/L\\_{\\v}}{2(5+\\mu/L\\_{\\v}})) - 1}\\\\\n& \\in O\\left(\\max \\left(\\frac{K}{\\mu} + \\frac{\\Delta\\_0}{\\mu \\eta\\_0^{1/2}\\epsilon^{1/2}}, \\frac{K}{\\mu} + \\frac{L\\_{\\v}^{1/2}}{\\mu^{3/2} \\epsilon^{1/2}}\\right)\\right).\n\\end{split}$$ \nProof of Lemma [properties]\n===========================\nTo prove Lemma [properties], we need the following Lemma [lem:alphabounded] and Lemma [lem:abbounded] to show that the trajectories of *α*, *a* and *b* are constrained in closed sets in Algorithm [innerloop].\n[lem:alphabounded] Suppose Assumption ([assumption1]) holds and $\\eta \\leq \\frac{1}{2p(1-p)}$. Running Algorithm [innerloop] with the input given by Algorithm [outerloop], we have $|\\alpha\\_t^k| \\leq \\frac{\\max\\{p, (1-p)\\}}{p(1-p)}$ for any iteration *t* and any machine *k*. [alphabounded]\n*Proof.* Firstly, we need to show that the input for any call of Algorithm ([innerloop]) satisfies $|\\alpha\\_0| \\leq \\frac{\\max\\{p, (1-p)\\}}{p(1-p)}$. If Algorithm [innerloop] is called by Algorithm [outerloop] for the first time, we know $|\\alpha\\_0| = 0 \\leq \\frac{\\max\\{p, (1-p)\\}}{p(1-p)}$. Otherwise, by the update of *a**l**p**h**a**s* in Algorithm ([outerloop]) (lines 4-7), we know that the input for Algorithm ([innerloop]) satisfies $|\\alpha\\_0| \\leq 2 \\leq \\frac{\\max\\{p, (1-p)\\}}{p(1-p)}$ since $h(\\w; \\x^k) \\in [0, 1]$ by Assumption [assumption1](*i**v*).\nNext, we will show by induction that $|\\alpha\\_t^k| \\leq \\frac{\\max\\{p, (1-p)\\}}{p(1-p)}$ for any iteration *t* and any machine *k* in Algorithm [innerloop]. Obviously, $|a\\_0^k| \\leq 2 \\leq \\frac{\\max\\{p, (1-p)\\}}{p(1-p)}$ for any k.\nAssume $|a\\_{t}^k| \\leq \\frac{\\max\\{p, (1-p)\\}}{p(1-p)}$ for any *k*.\n(1) If *t* + 1 mod *I* ≠ 0, then we have\n \n$$\\begin{split}\n|\\alpha\\_{t+1}^k| &= \\bigg|\\alpha\\_{t}^k + \\eta (2(ph(\\w\\_{t}^k; \\x)\\I\\_{[y=-1]} - (1-p)h(\\w\\_{t}^k; \\x)\\I\\_{[y=1]}) - 2p(1-p)\\alpha\\_{t})\\bigg|\\\\\n&\\leq \\bigg| (1-2\\eta p(1-p))\\alpha\\_{t}^k \\bigg| + \\bigg|2\\eta(ph(\\w\\_{t}^k; \\x)\\I\\_{[y=-1]} - (1-p)h(\\w\\_{t}^k;\\x)\\I\\_{[y=1]}) \\bigg|\\\\\n&\\leq (1-2\\eta p(1-p)) \\frac{\\max\\{p, (1-p)\\}}{p(1-p)} + 2\\eta \\max \\{p, (1-p)\\}\\\\\n& = (1-2\\eta p(1-p) +2\\eta p(1-p)) \\frac{\\max\\{p, (1-p)\\}}{p(1-p)}\\\\\n&= \\frac{\\max\\{p, (1-p)\\}}{p(1-p)}.\n\\end{split}$$ \n(2) If *t* + 1 mod *I* = 0, then by same analysis as above, we know that $|\\alpha\\_{t+1}^k| \\leq \\frac{\\max\\{p, (1-p)\\}}{p(1-p)}$ before being averaged across machines. Therefore, after being averaged across machines, it still holds that $|\\alpha\\_{t+1}^k| \\leq \\frac{\\max\\{p, (1-p)\\}}{p(1-p)}$.\nTherefore, $|\\alpha\\_t^k| \\leq \\frac{\\max\\{p, (1-p)\\}}{p(1-p)}$ holds for any iteration *t* and any machine *k* at any call of Algorithm ([innerloop]). □\n[lem:abbounded] Suppose Assumption ([assumption1]) (1) holds and $\\eta \\leq \\min(\\frac{1}{2(1-p)}, \\frac{1}{2p})$. Running Algorithm [innerloop] with the input given by Algorithm ([outerloop]), we have that ∣*a**t**k*∣ ≤ 1 and ∣*b**t**k*∣ ≤ 1 for any iteration *t* and any machine *k*.\n*Proof.* At the first call of Algorithm ([innerloop]), the input satisfies ∣*a*0∣ ≤ 1 and ∣*b*0∣ ≤ 1. Thus ∣*a*0*k*∣ ≤ 1 and ∣*b*0*k*∣ ≤ 1 for any machine *k*.\nAssume ∣*a**t**k*∣ ≤ 1 and ∣*b**t**k*∣ ≤ 1. Then:\n(1) *t* + 1 mod *I* ≠ 0, then we have\n \n$$\\begin{split}\n|a\\_t^k| &= \\bigg|\\frac{\\gamma}{\\eta + \\gamma}a\\_{t-1}^k + \\frac{\\eta}{\\eta+\\gamma}a\\_0 - \\frac{\\eta\\gamma}{\\eta+\\gamma}\\nabla\\_{a} F\\_k(\\v\\_{t-1}^k, \\alpha\\_{t-1}^k, \\z\\_{t-1}^k)\\bigg| \\\\\n&= \\bigg| \\frac{\\gamma}{\\eta + \\gamma}a\\_{t-1}^k + \\frac{\\eta}{\\eta+\\gamma}a\\_0 + \\frac{\\eta\\gamma}{\\eta+\\gamma} (2(1-p)(h(\\w\\_{t-1}^k; \\x^k\\_{t-1})-a\\_{t-1}^k))\\I\\_{y^k=1} \\bigg|\\\\\n&=\\bigg|\\frac{\\eta}{\\eta+\\gamma}a\\_0 + \\frac{\\gamma}{\\eta+\\gamma}a\\_{t-1}^{k}(1 - 2\\eta(1-p))\\I\\_{y^k=1} + \\frac{\\eta\\gamma}{\\eta+\\gamma}2(1-p)h(\\w\\_{t-1}^k; \\x\\_{t-1}^k)\\I\\_{y^k=1}\\bigg|\\\\\n&\\leq \\bigg|\\frac{\\eta}{\\eta+\\gamma}a\\_0\\bigg| + \\bigg|\\frac{\\gamma}{\\eta+\\gamma}a\\_{t-1}^{k}(1 - 2\\eta(1-p))\\I\\_{y^k=1} \\bigg| + \\bigg| \\frac{\\eta\\gamma}{\\eta+\\gamma}2(1-p)h(\\w\\_{t-1}^k; \\x\\_{t-1}^k)\\I\\_{y^k=1}\\bigg| \\\\\n&\\leq \\frac{\\eta}{\\eta+\\gamma} + \\frac{\\gamma}{\\eta+\\gamma}(1-2\\eta(1-p)) + \\frac{\\eta\\gamma}{\\eta+\\gamma}2(1-p) \\\\\n& = 1.\n\\end{split}$$ \n(2) If *t* + 1 mod *I* = 0, then by the same analysis as above, we have that ∣*a**t* + 1*k*∣ ≤ 1 before being averaged across machines. Therefore, after being averaged across machines, it still holds that ∣*a**t* + 1*k*∣ ≤ 1.\nThus, we can see that ∣*a**t**k*∣ ≤ 1 holds for any iteration *t* and any machine *k* in this call of Algorithm [innerloop]. Therefore, the output of the stage also has ∣*ã*∣ ≤ 1.\nThen we know that in the next call of Algorithm ([innerloop]), the input satisfies ∣*a*0∣ ≤ 1, by the same proof, we can see that ∣*a**t**k*∣ ≤ 1 holds for any iteration *t* and any machine *k* in any call of Algorithm ([innerloop]). With the same techniques, we can prove that ∣*b**t**k*∣ holds for any iteration *t* and any machine *k* in any call of Algorithm ([innerloop]). □\nWith the above lemmas, we are ready to prove Lemma [properties] and derive the claimed constants.\nBy definition of $F(\\v, \\alpha; \\z)$ and noting that $\\v = (\\w, a, b)$, we have\n \n$$\\begin{split}\n&\\nabla\\_{\\v} F\\_k(\\v, \\alpha; \\z) = [\\nabla\\_{\\w} F\\_k(\\v, \\alpha; \\z)^T, \\nabla\\_{a} F\\_k(\\v, \\alpha; \\z), \\nabla\\_{b} F\\_k(\\v, \\alpha; \\z)]^T.\n\\end{split}\n\\label{decompose\\_nabla\\_v}$$ \nAddressing each of the three terms on RHS, it follows that\n \n$$\\begin{split}\n&\\nabla\\_{\\w} F\\_k(\\v, \\alpha; \\z) = \\bigg[2(1-p)(h(\\w;\\x^k)-a) - 2(1+\\alpha)(1-p)\\bigg] \\nabla h(\\w; \\x^k)\\I\\_{[y^k=1]} \\\\\n&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n+ \\bigg[2p(h(\\w; \\x^k)-b) + 2(1+\\alpha)p\\bigg] \\nabla h(\\w; \\x^k)\\I\\_{[y^k=-1]}, \\\\\n&\\nabla\\_{a} F\\_k(\\v, \\alpha; \\z) = -2(1-p)(h(\\w;x^k)-a)\\I\\_{[y^k=1]},\\\\\n&\\nabla\\_{b} F\\_k(\\v, \\alpha; \\z) = -2p(h(\\w; \\x^k)-b).\n\\end{split}\n\\label{sub\\_nabla\\_v}$$ \nSince $|h(\\w; \\x^k)| \\in [0, 1]$, $\\|\\nabla h(\\w; \\x^k)\\| \\leq G\\_h$, $|\\alpha| \\leq \\frac{\\max\\{p, 1-p\\}}{p(1-p)}$, ∣*a*∣ ≤ 1 and *b* ≤ 1, we have\n \n$$\\begin{split}\n\\|\\nabla\\_{\\w} F\\_k(\\v, \\alpha; \\z) \\| &\\leq \\|2(1-p)(h(\\w; \\x^k)-a) - 2(1+\\alpha)(1-p)\\|G\\_h + \\|2p(h(\\w; \\x^k)-b) + 2(1+\\alpha)p\\|G\\_h\\\\\n&\\leq |6+2\\alpha|(1-p)G\\_h + |6+2\\alpha| p G\\_h\\\\\n&\\leq \\left(6 + 2\\frac{\\max\\{p, 1-p\\}}{p(1-p)}\\right) G\\_h,\n\\end{split}$$ \n \n$$\\begin{split}\n\\|\\nabla\\_{a} F\\_k(\\v, \\alpha; \\z)\\| \\leq 4(1-p),\n\\end{split}$$ \n \n$$\\begin{split}\n\\|\\nabla\\_{b} F\\_k(\\v, \\alpha; \\z)\\| \\leq 4p.\n\\end{split}$$ \nThus,\n \n$$\\begin{split}\n\\|\\nabla\\_{\\v} F\\_k(\\v, \\alpha; \\z)\\|^2 &= \\|\\nabla\\_{\\w} F\\_k(\\v, \\alpha; \\z)\\|^2 + \\|\\nabla\\_{a} F\\_k(\\v, \\alpha; \\z)\\|^2 + \\|\\nabla\\_{b} F\\_k(\\v, \\alpha; \\z)\\|^2 \\\\\n&\\leq \\left(6+\\frac{2\\max\\{p, 1-p\\}}{p(1-p)}\\right)^2 G\\_h^2 +16(1-p)^2 + 16p^2.\n\\end{split}$$ \n \n$$\\begin{split}\n\\|\\nabla\\_{\\alpha} F\\_k(\\v, \\alpha; \\z)\\|^2 &= \\|2p h(\\w; \\x^k)\\I\\_{y^k=-1} - 2(1-p)h(\\w; \\x^k)\\I\\_{y^k=1} - 2p(1-p)\\alpha\\|^2 \\\\\n&\\leq (2p + 2(1-p) + 4\\max\\{p, 1-p\\})^2 = (2+4\\max\\{p, 1-p\\})^2.\n\\end{split}$$ \nThus, $B\\_{\\v}^2 = \\left(6+\\frac{2\\max\\{p, 1-p\\}}{p(1-p)}\\right)^2 G\\_h^2 +16(1-p)^2 + 16p^2$ and *B**α*2 = (2 + 4max{*p*, 1 − *p*})2.\nIt follow that\n \n$$\\begin{split}\n|\\nabla\\_{\\v} f\\_k(\\v, \\alpha)| = |E[\\nabla\\_{\\alpha} F\\_k (\\v, \\alpha; \\z^k)]| \\leq B\\_{\\v}.\n\\end{split}$$ \nTherefore,\n \n$$\\begin{split}\nE[\\|\\nabla\\_{\\v} f\\_k(\\v, \\alpha) - \\nabla\\_{\\v} F\\_k(\\v, \\alpha; \\z^k)\\|^2] \\leq [2|\\nabla\\_{\\v} f\\_k(\\v, \\alpha)|^2 + 2|E[\\nabla\\_{\\v} F\\_k(\\v, \\alpha; \\z^k)]|^2] \\leq 4B\\_{\\v}^2.\n\\end{split}$$ \nSimilarly,\n \n$$\\begin{split}\n|\\nabla\\_{\\alpha} f\\_k(\\w, a, b, \\alpha)| = |E[\\nabla\\_{\\alpha} F\\_k (\\w, a, b, \\alpha; \\z^k)]| \\leq B\\_{\\alpha}.\n\\end{split}$$ \nTherefore,\n \n$$\\begin{split}\nE[\\|\\nabla\\_{\\alpha} f\\_k(\\v, \\alpha) - \\nabla\\_{\\alpha} F\\_k(\\v, \\alpha; \\z^k)\\|^2] \\leq 2|\\nabla\\_{\\alpha} f\\_k(\\v, \\alpha)|^2 + 2E[F\\_k(\\v, \\alpha; \\z^k)]|^2 \\leq 4B\\_{\\alpha}^2.\n\\end{split}$$ \nThus, $\\sigma\\_{\\v}^2 = 4B\\_{\\v}^2$ and *σ**α*2 = 4*B**α*2.\nNow, it remains to derive the constant *L*2 such that $\\|\\nabla\\_{\\v}F\\_k(\\v\\_1, \\alpha; \\z) - \\nabla\\_{\\v} F\\_k(\\v\\_2, \\alpha; \\z)\\| \\leq L\\_2 \\|\\v\\_1 - \\v\\_2\\|$.\nBy ([subnablav]), we get\n \n$$\\begin{split}\n&\\|\\nabla\\_{\\w} F\\_k(\\v\\_1,\\alpha;\\z) - \\nabla\\_{\\w}F\\_k(\\v\\_2, \\alpha; \\z)\\|\\\\\n&= \\bigg\\|\\bigg[2(1-p)(h(\\w\\_1;\\x^k)-a\\_1) - 2(1+\\alpha)(1-p)\\bigg] \\nabla h(\\w\\_1; \\x^k)\\I\\_{[y^k=1]} \n+ \\bigg[2p(h(\\w\\_1; \\x^k)-b\\_1) + 2(1+\\alpha)p\\bigg] \\nabla h(\\w\\_1; x^k)\\I\\_{[y^k=-1]} \\\\\n&~~~~~\n- \\bigg[2(1-p)(h(\\w\\_2;\\x^k)-a\\_2) - 2(1+\\alpha)(1-p)\\bigg] \\nabla h(\\w\\_2; \\x^k)\\I\\_{[y^k=1]}\n- \\bigg[2p(h(\\w\\_2; \\x^k)-b\\_2) + 2(1+\\alpha)p\\bigg] \\nabla h(\\w\\_2; \\x^k)\\I\\_{[y^k=-1]}\n\\bigg] \\bigg\\| \\\\\n&= \\Bigg\\| 2(1-p)\\bigg[h(\\w\\_1; \\x^k)\\nabla h(\\w\\_1;\\x^k)-h(\\w\\_2;\\x^k)\\nabla h(\\w\\_2;\\x^k)\\bigg] \\I\\_{[y^k=1]}\n+ 2p \\bigg[ h(\\w\\_1; \\x^k)\\nabla h(\\w\\_1; \\x^k)-h(\\w\\_2; \\x^k)\\nabla h(\\w\\_2;\\x^k) \\bigg]\\I\\_{[y^k=-1]}\\\\\n& - (2(1+\\alpha))(1-p)(\\nabla h(\\w\\_1; \\x^k) - \\nabla h(\\w\\_2;\\x^k))\\I\\_{[y^k=1]}\n+(2(1+\\alpha)p)(\\nabla h(\\w\\_1; \\x^k) - \\nabla h(\\w\\_2; \\x^k))\\I\\_{[y^k=-1]}\\\\\n&- 2(1-p)(a\\_1 \\nabla h(\\w\\_1; \\x^k) - a\\_2 \\nabla h(\\w\\_2; \\x^k))\\I\\_{y^k=1}\n-2p(b\\_1 \\nabla h(\\w\\_1; \\x^k) - b\\_2 \\nabla h(\\w\\_2;\\x^k))\\I\\_{[y^k=-1]}\n\\Bigg \\| \\\\\n&\\leq 2(1-p)\\|h(\\w\\_1; \\x^k)\\nabla h(\\w\\_1;\\x^k) - h(\\w\\_2; \\x^k)\\nabla h(\\w\\_2;\\x^k)\\| + 2p \\|h(\\w\\_1; \\x^k)\\nabla h(\\w\\_1;\\x^k) - h(\\w\\_2; \\x^k)\\nabla h(\\w\\_2;\\x^k)\\| \\\\\n&~~~~~+ \\|2(1+\\alpha)(1-p)\\|\\|\\nabla h(\\w\\_1; \\x^k) - \\nabla h(\\w\\_2; \\x^k)\\| + \\|2(1+\\alpha)p\\| \\|\\nabla h(\\w\\_1; \\x^k) - \\nabla h(\\w\\_2; \\x^k)\\|\\\\\n&~~~~~+2(1-p)\\|a\\_1 \\nabla h(\\w\\_1; \\x^k) - a\\_2 \\nabla h(\\w\\_2; \\x^k)\\| + 2p\\|b\\_1 \\nabla h(\\w\\_1; \\x^k) - b\\_2 \\nabla h(\\w\\_2; \\x^k)\\|.\n\\end{split}\n\\label{H\\_grad}$$ \nDenoting $\\Gamma\\_1(\\w; \\x^k) = h(\\w; \\x^k) \\nabla h(\\w; \\x^k)$,\n \n$$\\begin{split}\n\\|\\nabla \\Gamma\\_1(\\w; \\x^k)\\| &= \\|\\nabla h(\\w; \\x^k) \\nabla h(\\w; \\x^k)^T + h(\\w; \\x^k)\\nabla^2 h(\\w; \\x^k)\\| \\\\\n& \\leq \\|\\nabla h(\\w; \\x^k)\\nabla h(\\w; \\x^k)^T\\| + \\|h(\\w; \\x^k) \\nabla^2 h(\\w; \\x^k)\\| \\\\\n& \\leq G\\_h^2 + L\\_h.\n\\end{split}$$ \nThus, $\\|\\Gamma\\_1(\\w\\_1; \\x^k) - \\Gamma\\_1(\\w\\_2; \\x^k)\\| =\\|h(\\w\\_1;\\x^k) h'(\\w\\_1; \\x^k) - h(\\w\\_2;\\x^k) h'(\\w\\_2; \\x^k)\\| \\leq (G\\_h^2 + L\\_h)\\|\\w\\_1 - \\w\\_2\\|$. Define $\\Gamma\\_2(\\w, \\alpha; \\x^k) = a \\nabla h(\\w; \\x^k)$. By Lemma [lem:abbounded] and Assumption [assumption1], we have\n \n$$\\begin{split}\n\\nabla\\_{\\w, a} \\Gamma\\_2 (\\w, a; \\x^k) \\leq \\|\\nabla\\_{\\w} \\Gamma\\_2(\\w, a; \\z^k)\\| + \\|\\nabla\\_{a} \\Gamma\\_2(\\w, a; \\z^k)\\| = \\|a\\nabla^2 h(\\w; \\x^k)\\| + \\|\\nabla h(\\w; \\x^k)\\| \\leq L\\_h + G\\_h.\n\\end{split}$$ \nTherefore,\n \n$$\\begin{split}\n \\|\\! \\Gamma\\_2 (\\!\\w\\_1, \\! a\\_1;\\! \\x^k)\\! -\\! \\Gamma\\_2 (\\!\\w\\_2,\\! a\\_2;\\! \\x^k\\!)\\! \\|\\! =\\! \\| a\\_1\\! \\nabla\\! h(\\w\\_1;\\! \\x^k)\\! -\\! a\\_2\\! \\nabla\\! h(\\w\\_2;\\! \\x^k\\!)\\!\\|\\! \\leq\\! (L\\_h\\! +\\! G\\_h\\!)\\!\\sqrt{\\!\\|\\w\\_1\\!-\\!\\w\\_2\\|^2\\! +\\! \\|a\\_1\\! -\\! a\\_2\\|^2}.\n\\end{split}\n\\label{local\\_ah\\_lip}$$ \nSimilarly, we can prove that\n \n$$\\begin{split}\n\\| b\\_1 \\nabla h(\\w\\_1; \\x^k) - b\\_2 \\nabla h(\\w\\_2; \\x^k)\\| \\leq (L\\_h+G\\_h)\\sqrt{\\|\\w\\_1-\\w\\_2\\|^2+\\|b\\_1-b\\_2\\|^2}.\n\\end{split}\n\\label{local\\_bh\\_lip}$$ \nThen plugging ([localahlip]), ([localbhlip]) and Assumption [assumption1] into ([Hgrad]), we have\n \n$$\\begin{split}\n&\\|\\nabla\\_{\\w} F\\_k(\\v\\_1,\\alpha;\\z) - \\nabla\\_{\\w}F\\_k(\\v\\_2, \\alpha; \\z)\\| \\\\\n&\\leq 2(G\\_h^2+L\\_h)\\|\\w\\_1-\\w\\_2\\| + 2|1+\\alpha|G\\_h\\|\\w\\_1 - \\w\\_2\\| \\\\\n&~~~~~+(L\\_h + G\\_h)\\sqrt{\\|\\w\\_1 - \\w\\_2\\|^2 + \\|a\\_1-a\\_2\\|^2} + (L\\_h + G\\_h)\\sqrt{\\|\\w\\_1-\\w\\_2\\|^2 + \\|b\\_1-b\\_2\\|^2}\\\\\n&\\leq (2(G\\_h^2+L\\_h) + |2(1+\\alpha)|G\\_h + 2L\\_h + 2G\\_h)\\sqrt{\\|\\w\\_1 - \\w\\_2\\|^2 + \\|a\\_1-a\\_2\\|^2 + \\|b\\_1-b\\_2\\|^2}\\\\\n&\\leq \\left( 2 G\\_h^2 + 4L\\_h + \\left(4 + \\frac{2\\max\\{p, 1-p\\}}{p(1-p)}\\right) G\\_h\\right) \\|\\v\\_1-\\v\\_2\\|.\\\\\n\\end{split}$$ \nBy ([subnablav]), we also have\n \n$$\\begin{split}\n&\\|\\nabla\\_{a} F\\_k(\\v\\_1, \\alpha; \\z) - \\nabla\\_{a} F\\_k(\\v\\_2, \\alpha; \\z)\\|^2 \\leq 4(1-p)^2(\\|h(\\w\\_1; \\x^k) - h(\\w\\_2; \\x^k)\\|^2+\\|a\\_1 - a\\_2\\|^2)\\\\\n&\\leq 4(1-p)^2(G\\_h^2 \\|\\w\\_1-\\w\\_2\\|^2 + \\|a\\_1 - a\\_2\\|^2 + \\|b\\_1 - b\\_2\\|^2)\n\\leq 4(1-p)^2(G\\_h^2 + 1)\\|\\v\\_1 - \\v\\_2\\|^2,\n\\end{split}$$ \nand\n \n$$\\begin{split}\n&\\|\\nabla\\_{b} F\\_k(\\v\\_1, \\alpha; \\z) - \\nabla\\_{b} F\\_k(\\v\\_2, \\alpha; \\z)\\|^2 \\leq 4(1-p)^2(\\|h(\\w\\_1; \\x^k) - h(\\w\\_2; \\x^k)\\|^2+\\|b\\_1 - b\\_2\\|^2)\\\\\n&\\leq 4(1-p)^2(G\\_h^2 \\|\\w\\_1-\\w\\_2\\|^2 + \\|a\\_1 - a\\_2\\|^2 + \\|b\\_1 - b\\_2\\|^2)\\leq 4(1-p)^2(G\\_h^2 + 1)\\|\\v\\_1 - \\v\\_2\\|^2.\n\\end{split}$$ \n \n$$\\begin{split}\n\\|\\nabla\\_{\\v} F\\_k (\\v\\_1, \\alpha; \\z) - \\nabla\\_{\\v} F\\_k(\\v\\_2, \\alpha; \\z)\\|^2 &= \\|\\nabla\\_{\\w}F\\_k(\\v\\_1, \\alpha; \\z) - \\nabla\\_{\\w}F\\_k(\\v\\_2, \\alpha; \\z)\\|^2 \\\\\n&~~~~~\n+ \\|\\nabla\\_{a}F\\_k(\\v\\_1, \\alpha; \\z) - \\nabla\\_{b}F\\_k(\\v\\_2, \\alpha; \\z)\\|^2 + \\|\\nabla\\_{b} F\\_k(\\v\\_1, \\alpha; \\z) - \\nabla\\_{b}F\\_k(\\v\\_1, \\alpha; \\z)\\|^2\\\\\n&\\leq \\left( G\\_h^2 + L\\_h + 4 + \\frac{2\\max\\{p, 1-p\\}}{p(1-p)} 8(1-p)^2 (G\\_h^2 + 1)\\right) \\|\\v\\_1 - \\v\\_2\\|^2.\n\\end{split}$$ \nThus, we get $L\\_2 = \\left( G\\_h^2 + L\\_h + 4 + \\frac{2\\max\\{p, 1-p\\}}{p(1-p)} 8(1-p)^2 (G\\_h^2 + 1)\\right)^{1/2}$.\nProof of Lemma [lemmaonestage]\n==============================\n*Proof.* Plugging Lemma [lem:var1] and Lemma [lem:var2] into Lemma [lem:objgap], we get\n \n$$\\begin{split}\n&\\psi(\\tilde{\\v}) - \\psi(\\v\\_{\\psi}^\\*) \\\\\n&\\leq \\frac{1}{T}\\sum\\limits\\_{t=1}^{T} \\Bigg[ \\underbrace{ \\left(\\frac{L\\_{\\v}+3G\\_{\\alpha}^2/\\mu\\_{\\alpha}}{2} - \\frac{1}{2\\eta}\\right) \\|\\bar{\\v}\\_{t-1} - \\bar{\\v}\\_t\\|^2 + \\left(\\frac{L\\_{\\alpha}+3G\\_{\\v}^2/L\\_{\\v}}{2} - \\frac{1}{2\\eta}\\right)(\\bar{\\alpha}\\_t - \\bar{\\alpha}\\_{t-1})^2}\\_{C\\_1}\\\\\n&+\\underbrace{\\left(\\frac{1}{2\\eta} - \\frac{\\mu\\_{\\alpha}}{3} \\right) (\\bar{\\alpha}\\_{t-1} - \\alpha^\\*(\\tilde{\\v}))^2 - \\left(\\frac{1}{2\\eta} - \\frac{\\mu\\_{\\alpha}}{3}\\right)(\\bar{\\alpha}\\_t-\\alpha^\\*(\\tilde{\\v}))^2}\\_{C\\_2}\n+\\underbrace{\\left(\\frac{2L\\_{\\v}}{3} + \\frac{1}{2\\eta}\\right) \\|\\bar{\\v}\\_{t-1} - \\v\\_{\\psi}^\\*\\|^2 - \\left(\\frac{1}{2\\eta} + \\frac{2L\\_{\\v}}{3}\\right)\\|\\bar{\\v}\\_t - \\v\\_\n{\\psi}^\\*\\|^2}\\_{C\\_3}\\\\\n& +\\underbrace{\\frac{1}{2\\eta}((\\alpha^\\*(\\Tilde{\\v}) - \\Tilde{\\alpha}\\_{t-1})^2 - (\\alpha^\\*(\\Tilde{\\v})-\\Tilde{\\alpha}\\_t)^2)}\\_{C\\_4} +\\underbrace{\\left(\\frac{3G\\_{\\v}^2}{2\\mu\\_{\\alpha}} + \\frac{3L\\_{\\v}}{2}\\right)\\frac{1}{K}\\sum\\limits\\_{k=1}^{K}\\|\\bar{\\v}\\_{t-1}-\\v\\_{t-1}^k\\|^2 \n+ \\left(\\frac{3 G\\_{\\alpha}^2}{2L\\_{\\v}} + \\frac{3 L\\_{\\alpha}^2}{2\\mu\\_{\\alpha}}\\right)\\frac{1}{K}\\sum\\limits\\_{k=1}^{K} (\\bar{\\alpha}\\_{t-1} - \\alpha\\_{t-1}^k)^2}\\_{C\\_5}\\\\\n&+\\underbrace{\\eta \\left\\|\\frac{1}{K}\\sum\\limits\\_{k=1}^{K} [\\nabla\\_{\\v} f\\_k(\\v\\_{t-1}^k, \\alpha\\_{t-1}^k) - \\nabla\\_{\\v}F\\_k(\\v\\_{t-1}^k, \\alpha\\_{t-1}^k; \\z\\_{t-1}^k)]\\right\\|^2}\\_{C\\_6} + \\underbrace{\\frac{3\\eta}{2} \\left\\| \\frac{1}{K}\\sum\\limits\\_{k=1}^{K} \\nabla\\_{\\alpha}f\\_k(\\v\\_{t-1}^k, \\alpha\\_{t-1}^k) - \\nabla\\_{\\alpha} F\\_k(\\v\\_{t-1}^k, \\alpha\\_{t-1}^k; \\z\\_{t-1}^k) \\right\\|^2}\\_{C\\_7} \\\\\n&\\!+\\!\\underbrace{\\left\\langle\\! \\frac{1}{K}\\sum\\limits\\_{k=1}^{K}[\\nabla\\_{\\v}f\\_k(\\v\\_{t-1}^k, \\alpha\\_{t-1}^k)\\! -\\! \\nabla\\_{\\v}F\\_k(\\v\\_{t-1}^k, \\alpha\\_{t-1}^k; \\z\\_{t-1}^k)],\\! \\hat{\\v}\\_t\\! -\\! \\v\\_{\\psi}^\\* \\right\\rangle}\\_{C\\_8} \n+\\underbrace{\\left\\langle\\! \\!\\frac{1}{K}\\sum\\limits\\_{k=1}^{K}[\\nabla\\_{\\alpha} f\\_k(\\v\\_{t-1}^k, \\alpha\\_{t-1}^k)\\! -\\! F\\_k(\\v\\_{t-1}^k, \\alpha\\_{t-1}^k; \\z\\_{t-1}^k)], \\tilde{\\alpha}\\_{t-1}\\! -\\! \\hat{\\alpha}\\_t) \\right\\rangle}\\_{C\\_9} \\Bigg].\\\\ \n\\end{split}\n\\label{before\\_summation}$$ \nSince $\\eta\\leq \\min(\\frac{1}{L\\_{\\v} + 3G\\_{\\alpha}^2/\\mu\\_{\\alpha}}, \\frac{1}{L\\_{\\alpha}+3G\\_{\\v}^2/L\\_{\\v}})$, thus in the RHS of ([beforesummation]), *C*1 can be cancelled. *C*2, *C*3 and *C*4 will be handled by telescoping sum. *C*5 can be bounded by Lemma [stichlemma].\nTaking expectation over *C*6,\n \n$$\\begin{split}\n&E\\left[\\eta \\left\\|\\frac{1}{K}\\sum\\limits\\_{k=1}^{K}[\\nabla\\_{\\v} f\\_k(\\v\\_{t-1}^k, \\alpha\\_{t-1}^k) - \\nabla\\_{\\v}F\\_k(\\v\\_{t-1}^k, \\alpha\\_{t-1}^k; \\z\\_{t-1}^k)]\\right\\|^2\\right]\\\\\n&=E\\left[\\frac{\\eta}{K^2} \\left\\|\\sum\\limits\\_{k=1}^{K}[\\nabla\\_{\\v} f\\_k(\\v\\_{t-1}^k, \\alpha\\_{t-1}^k) - \\nabla\\_{\\v}F\\_k(\\v\\_{t-1}^k, \\alpha\\_{t-1}^k; \\z\\_{t-1}^k)]\\right\\|^2\\right]\\\\\n&=E\\left[\\frac{\\eta}{K^2}\\left(\\sum\\limits\\_{k=1}^K \\|\\nabla\\_{\\v} f\\_k(\\v\\_{t-1}^k, \\alpha\\_{t-1}^k) - \\nabla\\_{\\v}F\\_k(\\v\\_{t-1}^k, \\alpha\\_{t-1}^k; \\z\\_{t-1}^k)\\|^2\\right.\\right.\\\\\n&~~~~~~\\left.\\left.+ 2\\sum\\limits\\_{i=1}^{K}\\sum\\limits\\_{j=i+1}^{K} \\left\\langle \\nabla\\_{\\v} f\\_i(\\v\\_{t-1}^i, \\alpha\\_{t-1}^i) \n- \\nabla\\_{\\v} F\\_i(\\v\\_{t-1}^{i}, \\alpha\\_{t-1}^{i}; \\z\\_{t-1}^i), \n\\nabla\\_{\\v} f\\_j(\\v\\_{t-1}^j, \\alpha\\_{t-1}^j) \n- \\nabla\\_{\\v} F\\_j(\\v\\_{t-1}^{j}, \\alpha\\_{t-1}^{j}; \\z\\_{t-1}^j)\n\\right\\rangle \\right) \\right]\\\\\n&\\leq \\frac{\\eta \\sigma\\_{\\v}^2}{K}.\n\\end{split} \n\\label{local\\_variance\\_v}$$ \nThe last inequality holds because $\\|\\nabla\\_{\\v}f\\_k(\\v\\_{t-1}^k,\\alpha\\_{t-1}^k) - \\nabla\\_{\\v}F\\_k (\\v\\_{t-1}^k, \\alpha\\_{t-1}^k;\\z\\_{t-1}^k)\\|^2 \\leq \\sigma\\_{\\v}^2$ for any *k* and $E\\left[ \\left\\langle \\nabla\\_{\\v} f\\_i(\\v\\_{t-1}^i, \\alpha\\_{t-1}^i) \n- \\nabla\\_{\\v} F\\_i(\\v\\_{t-1}^{i}, \\alpha\\_{t-1}^{i}; \\z\\_{t-1}^i), \n\\nabla\\_{\\v} f\\_j(\\v\\_{t-1}^j, \\alpha\\_{t-1}^j) \n- \\nabla\\_{\\v} F\\_j(\\v\\_{t-1}^{j}, \\alpha\\_{t-1}^{j}; \\z\\_{t-1}^j)\n\\right\\rangle \\right] = 0$ for any *i* ≠ *j* as each machine draws data independently. Similarly, we take expectation over *C*7 and have\n \n$$\\begin{split}\n&E\\left[\\frac{3\\eta}{2} \\left(\\frac{1}{K}\\sum\\limits\\_{k=1}^{K}[\\nabla\\_{\\alpha} f\\_k(\\v\\_{t-1}^k, \\alpha\\_{t-1}^k) - \\nabla\\_{\\alpha}F\\_k(\\v\\_{t-1}^k, \\alpha\\_{t-1}^k; \\z\\_{t-1}^k)]\\right)^2\\right]\n\\leq \\frac{3\\eta \\sigma\\_{\\alpha}^2}{2K}.\n\\end{split} \n\\label{local\\_variance\\_alpha}$$ \nNote $E\\bigg[\\left\\langle \\frac{1}{K}\\sum\\limits\\_{k=1}^{K}[\\nabla\\_{\\v}f\\_k(\\v\\_{t-1}^k, \\alpha\\_{t-1}^k) - \\nabla\\_{\\v}F\\_k(\\v\\_{t-1}^k, \\alpha\\_{t-1}^k; \\z\\_{t-1}^k)], \\hat{\\v}\\_t - \\v\\_{\\psi}^\\* \\right\\rangle\\bigg] = 0 $ and \n$E\\bigg[\\left\\langle -\\frac{1}{K}\\sum\\limits\\_{k=1}^{K}[\\nabla\\_{\\alpha} f\\_k(\\v\\_{t-1}^k, \\alpha\\_{t-1}^k) - F\\_k(\\v\\_{t-1}^k, \\alpha\\_{t-1}^k; \\z\\_{t-1}^k)], \\tilde{\\alpha}\\_{t-1} - \\hat{\\alpha}\\_t \\right\\rangle\\bigg] = 0$. Therefore, *C*8 and *C*9 will diminish after taking expectation.\nAs $\\eta \\leq \\frac{1}{L\\_{\\v} + 3G\\_{\\alpha}^2/\\mu\\_{\\alpha}}$, we have $L\\_{\\v} \\leq \\frac{1}{\\eta}$. Plugging ([localvariancev]) and ([localvariancealpha]) into ([beforesummation]), and taking expectation, it yields\n \n$$\\begin{split}\nE[\\psi(\\tilde{\\v}) - \\psi(\\v\\_{\\psi}^\\*)]\n&\\leq E\\bigg\\{\\frac{1}{T}\\left(\\frac{2L\\_{\\v}}{3} + \\frac{1}{2\\eta}\\right) \\|\\bar{\\v}\\_0-\\v\\_{\\psi}^\\*\\|^2 + \\frac{1}{T}\\left(\\frac{1}{2\\eta} - \\frac{\\mu\\_{\\alpha}}{3} \\right)(\\bar{\\alpha}\\_0 - \\alpha^\\*(\\tilde{\\v}))^2 \n+ \\frac{1}{2\\eta T} (\\tilde{\\alpha}\\_0 - \\alpha^\\*(\\Tilde{\\v}))^2 \\\\ \n&~~~~~+ \\frac{1}{T}\\sum\\limits\\_{t=1}^{T}\\left(\\frac{3G\\_{\\v}^2}{2 \\mu\\_{\\alpha}} + \\frac{3L\\_{\\v}}{2}\\right)\\frac{1}{K}\\sum\\limits\\_{k=1}^{K}\\|\\bar{\\v}\\_{t-1} - \\v\\_{t-1}^k\\|^2 + \\frac{1}{T}\\sum\\limits\\_{t=1}^{T}\\left(\\frac{3G\\_{\\alpha}^2}{2L\\_{\\v}} + \\frac{3L\\_{\\alpha}^2}{2\\mu\\_{\\alpha}}\\right)\\frac{1}{K}\\sum\\limits\\_{k=1}^{K}\\|\\bar{\\alpha}\\_{t-1} - \\alpha\\_{t-1}^k\\|^2\\\\\n&~~~~~+\\frac{1}{T} \\sum\\limits\\_{t=1}^{T}\\frac{\\eta \\sigma\\_{\\v}^2}{K}\n+\\frac{1}{T} \\sum\\limits\\_{t=1}^{T}\\frac{3\\eta \\sigma\\_{\\alpha}^2}{2 K}\\bigg\\}\\\\ \n&\\leq \\frac{2}{\\eta T} \\|\\v\\_0 - \\v\\_{\\psi}^\\*\\|^2 + \\frac{1}{\\eta T} (\\alpha\\_0 -\\alpha^\\*(\\tilde{\\v}))^2 + \\left(\\frac{6G\\_{\\v}^2}{\\mu\\_{\\alpha}} +6L\\_{\\v} + \\frac{6G\\_{\\alpha}^2}{L\\_{\\v}} + \\frac{6L\\_{\\alpha}^2}{\\mu\\_{\\alpha}}\\right) \\eta^2 I^2 B^2\\I\\_{I>1} + \\frac{\\eta(2\\sigma\\_{\\v}^2 + 3\\sigma\\_{\\alpha}^2)}{2K},\n\\end{split}$$ \nwhere we use Lemma [stichlemma], $\\v\\_0 = \\bar{\\v}\\_0$, *α*0 = *ᾱ*0 and $B^2 = \\max\\{B^2\\_{\\v}, B^2\\_{\\alpha}\\}$ in the last inequality. □\nProof of Lemma [lem:objgap]\n===========================\n*Proof.* Define $\\alpha^\\*(\\tilde{\\v}) = \\arg\\max\\limits\\_{\\alpha} f(\\tilde{\\v}, \\alpha)$ and $\\tilde{\\alpha} = \\frac{1}{K}\\sum\\limits\\_{k=1}^{K} \\frac{1}{T}\\sum\\limits\\_{t=1}^{T}\\alpha\\_t^k$.\n \n$$\\begin{split}\n\\psi(\\tilde{\\v}) - \\min\\limits\\_{\\v}\\psi (\\v) &= \\max\\limits\\_{\\alpha}\\left[f(\\tilde{\\v}, \\alpha) + \\frac{1}{2\\gamma} \\|\\tilde{\\v} - \\v\\_{0}\\|^2\\right] - \\min\\limits\\_{\\v} \\max\\limits\\_{\\alpha}\\left[f(\\v, \\alpha)+\\frac{1}{2\\gamma}\\|\\v-\\v\\_{0}\\|^2\\right]\\\\\n&= \\left[f(\\tilde{\\v}, \\alpha^\\*(\\tilde{\\v})) + \\frac{1}{2\\gamma} \\|\\tilde{\\v} - \\v\\_{0}\\|^2 \\right] - \\max\\limits\\_{\\alpha}\\left[f(\\v\\_{\\psi}^\\*, \\alpha) + \\frac{1}{2\\gamma}\\|\\v\\_{\\psi}^\\*-\\v\\_{0}\\|^2\\right]\\\\\n&\\leq \\left[f(\\tilde{\\v}, \\alpha^\\*(\\tilde{\\v})) + \\frac{1}{2\\gamma} \\|\\tilde{\\v} - \\v\\_{0}\\|^2 \\right] - \\left[f(\\v\\_{\\psi}^\\*, \\tilde{\\alpha}) + \\frac{1}{2\\gamma}\\|\\v\\_{\\psi}^\\*-\\v\\_{0}\\|^2\\right]\\\\\n&\\leq \\frac{1}{T}\\sum\\limits\\_{t=1}^{T} \\left[\\left(f(\\bar{\\v}\\_t, \\alpha^\\*(\\tilde{\\v})) +\\frac{1}{2\\gamma} \\|\\bar{\\v}\\_t - \\v\\_{0}\\|^2\\right) - \\left(f(\\v\\_{\\psi}^\\*, \\bar{\\alpha}\\_t) + \\frac{1}{2\\gamma}\\|\\v\\_{\\psi}^\\* - \\v\\_{0}\\|^2\\right)\\right],\\\\\n\\end{split}\n\\label{objgap\\_relax}$$ \nwhere the last inequality uses Jensen’s inequality and the fact that $f(\\v, \\alpha) + \\frac{1}{2\\gamma}\\|\\v-\\v\\_{0}\\|^2$ is convex w.r.t. $\\v$ and concave w.r.t. *α*.\nBy $L\\_{\\v}$-weakly convexity of *f*( ⋅ ) w.r.t. $\\v$, we have\n \n$$\\begin{split}\n f(\\bar{\\v}\\_{t-1}, \\bar{\\alpha}\\_{t-1}) + \\langle \\nabla\\_{\\v}f(\\bar{\\v}\\_{t-1}, \\bar{\\alpha}\\_{t-1}), \\v\\_{\\psi}^\\* - \\bar{\\v}\\_{t-1}\\rangle - \\frac{L\\_{\\v}}{2}\\|\\bar{\\v}\\_{t-1}-\\v\\_{\\psi}^\\*\\|^2 \\leq f(\\v\\_{\\psi}^\\*, \\bar{\\alpha}\\_{t-1}),\n\\end{split}\n\\label{weakly\\_convex\\_v}$$ \nand by $L\\_{\\v}$-smoothness of *f*( ⋅ ) w.r.t. $\\v$, we have\n \n$$\\begin{split}\nf(\\bar{\\v}\\_t, \\alpha^\\*(\\Tilde{\\v})) &\\leq f(\\bar{\\v}\\_{t-1}, \\alpha^\\*(\\Tilde{\\v})) \n+ \\langle \\nabla\\_{\\v}f(\\bar{\\v}\\_{t-1}, \\alpha^\\*(\\Tilde{\\v})), \\bar{\\v}\\_t - \\bar{\\v}\\_{t-1}\\rangle + \\frac{L\\_{\\v}}{2} \\|\\bar{\\v}\\_t - \\bar{\\v}\\_{t-1}\\|^2\\\\\n&= f(\\bar{\\v}\\_{t-1}, \\alpha^\\*(\\Tilde{\\v})) + \\langle \\nabla\\_{\\v}f(\\bar{\\v}\\_{t-1}, \\alpha^\\*(\\Tilde{\\v})), \\bar{\\v}\\_t - \\bar{\\v}\\_{t-1}\\rangle + \\frac{L\\_{\\v}}{2}\\|\\bar{\\v}\\_t - \\bar{\\v}\\_{t-1}\\|^2 \\\\\n&~~~~ + \\langle \\nabla\\_{\\v}f(\\bar{\\v}\\_{t-1}, \\bar{\\alpha}\\_{t-1}), \\bar{\\v}\\_t - \\bar{\\v}\\_{t-1}\\rangle - \\langle \\nabla\\_{\\v}f(\\bar{\\v}\\_{t-1}, \\bar{\\alpha}\\_{t-1}), \\bar{\\v}\\_t - \\bar{\\v}\\_{t-1}\\rangle \\\\\n&= f(\\bar{\\v}\\_{t-1}, \\alpha^\\*(\\Tilde{\\v})) + \\langle \\nabla\\_{\\v}f(\\bar{\\v}\\_{t-1}, \\bar{\\alpha}\\_{t-1}), \\bar{\\v}\\_t - \\bar{\\v}\\_{t-1}\\rangle + \\frac{L\\_{\\v}}{2}\\|\\bar{\\v}\\_t - \\bar{\\v}\\_{t-1}\\|^2 \\\\\n&~~~~ + \\langle\\nabla\\_{\\v} f(\\bar{\\v}\\_{t-1}, \\alpha^\\*(\\Tilde{\\v}))-\\nabla\\_{\\v}f(\\bar{\\v}\\_{t-1}, \\bar{\\alpha}\\_{t-1}), \\bar{\\v}\\_t - \\bar{\\v}\\_{t-1} \\rangle\\\\\n&\\overset{(a)}\\leq f(\\bar{\\v}\\_{t-1}, \\alpha^\\*(\\Tilde{\\v})) + \\langle \\nabla\\_{\\v}f(\\bar{\\v}\\_{t-1}, \\bar{\\alpha}\\_{t-1}), \\bar{\\v}\\_t - \\bar{\\v}\\_{t-1}\\rangle + \\frac{L\\_{\\v}}{2}\\|\\bar{\\v}\\_t - \\bar{\\v}\\_{t-1}\\|^2 \\\\\n&~~~~ + G\\_{\\alpha} |\\bar{\\alpha}\\_{t-1} - \\alpha^\\*(\\Tilde{\\v})| \\|\\bar{\\v}\\_t - \\bar{\\v}\\_{t-1}\\| \\\\\n&\\overset{(b)}\\leq f(\\bar{\\v}\\_{t-1}, \\alpha^\\*(\\Tilde{\\v})) + \\langle \\nabla\\_{\\v}f(\\bar{\\v}\\_{t-1}, \\bar{\\alpha}\\_{t-1}), \\bar{\\v}\\_t - \\bar{\\v}\\_{t-1}\\rangle + \\frac{L\\_{\\v}}{2}\\|\\bar{\\v}\\_t - \\bar{\\v}\\_{t-1}\\|^2 \\\\\n&~~~~ + \\frac{\\mu\\_{\\alpha}}{6} |\\bar{\\alpha}\\_{t-1} - \\alpha^\\*(\\Tilde{\\v})|^2 + \\frac{3G\\_{\\alpha}^2}{2\\mu\\_{\\alpha}} \\|\\bar{\\v}\\_t - \\bar{\\v}\\_{t-1}\\|^2,\\\\ \n\\end{split}\n\\label{smooth\\_v}$$ \nwhere (*a*) holds because we know that $\\nabla\\_{\\v} f(\\cdot)$ is *G**α* = 2max{*p*, 1 − *p*}-Lipshitz w.r.t. *α* by the definition of *f*( ⋅ ), and (*b*) holds by Young’s inequality.\nBy $\\frac{1}{\\gamma}$-strong convexity of $\\frac{1}{2\\gamma} \\|\\v - \\v\\_{0}\\|^2$ w.r.t. $\\v$, we have\n \n$$\\begin{split}\n\\frac{1}{2\\gamma}\\|\\bar{\\v}\\_t - \\v\\_{0}\\|^2 + \\frac{1}{\\gamma}\\langle \\bar{\\v}\\_t - \\v\\_{0}, \\v\\_{\\psi}^\\* - \\v\\_t \\rangle + \\frac{1}{2\\gamma}\\|\\v\\_{\\psi}^\\* - \\v\\_t\\|^2 \\leq \\frac{1}{2\\gamma}\\|\\v\\_{\\psi}^\\* - \\v\\_{0}\\|^2.\n\\end{split}\n\\label{convex\\_v}$$ \nAdding ([weaklyconvexv]), ([smoothv]), ([convexv]), and rearranging terms, we have\n \n$$\\begin{split}\n &f(\\bar{\\v}\\_{t-1}, \\bar{\\alpha}\\_{t-1}) \n+ \nf(\\bar{\\v}\\_t, \\alpha^\\*(\\Tilde{\\v})) \n+\n\\frac{1}{2\\gamma}\\|\\bar{\\v}\\_t - \\v\\_{0}\\|^2 \n-\n\\frac{1}{2\\gamma}\\|\\v\\_{\\psi}^\\* - \\v\\_{0}\\|^2\\\\\n\\leq\n&f(\\v\\_{\\psi}^\\*, \\bar{\\alpha}\\_{t-1})\n+\nf(\\bar{\\v}\\_{t-1}, \\alpha^\\*(\\Tilde{\\v})) + \\langle \\nabla\\_{\\v}f(\\bar{\\v}\\_{t-1}, \\bar{\\alpha}\\_{t-1}), \\bar{\\v}\\_t - \\bar{\\v}\\_{\\psi}^\\*\\rangle \n+\n \\frac{L\\_{\\v} + 3G\\_{\\alpha}^2/\\mu\\_{\\alpha}}{2}\\eta^2 \\|\\bar{\\v}\\_t-\\bar{\\v}\\_{t-1}\\|^2 \n+ \\frac{L\\_{\\v}}{2}\\|\\bar{\\v}\\_{t-1}-\\v\\_{\\psi}^\\*\\|^"},"data_id":{"kind":"string","value":"arxiv_0000375"}}}],"truncated":false,"partial":true},"paginationData":{"pageIndex":0,"numItemsPerPage":100,"numTotalItems":46503,"offset":0,"length":100}},"jwt":"eyJhbGciOiJFZERTQSJ9.eyJyZWFkIjp0cnVlLCJwZXJtaXNzaW9ucyI6eyJyZXBvLmNvbnRlbnQucmVhZCI6dHJ1ZX0sImlhdCI6MTc3NTYxOTE5NSwic3ViIjoiL2RhdGFzZXRzL0x5dW4wOTEyL0xvbmdBQkMiLCJleHAiOjE3NzU2MjI3OTUsImlzcyI6Imh0dHBzOi8vaHVnZ2luZ2ZhY2UuY28ifQ.2LoODTZAegpc19ly4kvxUJiEUjDAikRYnp_Qac9ItVXSxCXTV-Q6IKJJPIh9_-ka8nF_L5mAVD9uiq4U1WMSAw","displayUrls":true,"splitSizeSummaries":[{"config":"default","split":"train","numRows":46503,"numBytesParquet":2327876865}]},"discussionsStats":{"closed":0,"open":2,"total":2},"fullWidth":true,"hasGatedAccess":true,"hasFullAccess":true,"isEmbedded":false,"savedQueries":{"community":[],"user":[]}}">
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Noncommutative Wilson lines in higher-spin theory and correlation functions of conserved currents for free conformal fields
Roberto Bonezzi*a*, Nicolas Boulanger*a*, David De Filippi*a*
Per Sundell*b*
**a* -.1truecm Groupe de Mécanique et Gravitation, Unit of Theoretical and Mathematical Physics,
University of M... | arxiv_0000005 |
and knowing the mass in each of these gas reservoirs, we can deduce the various timescales. Evolution with both the redshift and the stellar mass of five main timescales are shown in Fig. [fig:maindisktime scales]. We focus on:
* the orbital time *t**o**r**b* estimated at a radius, 2.2*r**d* ;
* the cooling time-scale... | arxiv_0000008 |
e^{-\Bar{m}\_s / x} \left[ 1 + \Bar{m}\_s \left( 1 - \frac{\nu}{3} \right) + \mathcal{O}(\Bar{m}\_s^2) \mathcal{O}(x) \right] \;.$$
This expansion can generally be written as
$$e^{-\Bar{m}\_s / \gamma\_\text{GR}} = \sum\_{n=0}^\infty f\_{n}(x) \frac{\Bar{m}\_s^n}{n!} e^{-\Bar{m}\_s / x}$$
with *f**n*(*x*) absor... | arxiv_0000012 |
Improved Constraints on the Preferential Heating and Acceleration of Oxygen Ions in the Extended Solar Corona
=============================================================================================================
We present a detailed analysis of oxygen ion velocity distributions in the extended solar corona, ba... | arxiv_0000019 |
description with no problem:
$$\left\{
\begin{array}{l}
p\_1 \in \mathbb R\\
\displaystyle \mbox{Always } \big( (p\_1 > 0) \mbox{ AND } (p\_1 < 0) \big)\\
\end{array}
\right..$$
The PDL language grammar is not a tool with capabilities to perceive logical contradictions which may be contained within statements. Th... | arxiv_0000022 |
Lindblad Equation for the Inelastic Loss of Ultracold Atoms
============================================================
The loss of ultracold trapped atoms due to deeply inelastic reactions has previously been taken into account in effective field theories for low-energy atoms by adding local anti-Hermitian terms to ... | arxiv_0000033 |
Finite presentations for stated skein algebras and lattice gauge field theory
=============================================================================
[section] [theorem]Main Theorem [theorem]Proposition [theorem]Corollary [theorem]Corollaire [theorem]Lemma [theorem]Notations [theorem]Convention [theorem]Problem [... | arxiv_0000036 |
& 0.264 & 0.336 & ∞ & 0.867 & 1.05 & ∞ & 3.23 & 3.95
108 & 3 ⋅ 107 & ∞ & 3.19 & 3.56 & ∞ & 9.75 & 10.3 & ∞ & 145 & 160
[!ht]
|c|c|c|c|c|c|c|c|c|c|c| *m*1[*M*⊙] & *m*2[*M*⊙] &
& & & &
& & RWF & SWF & FWF & RWF & SWF & FWF & RWF & SWF & FWF
3 ⋅ 105 & 105 & 4.95 ⋅ 10− 4 & 4.11 ⋅ 10− 4 & 2.94 ⋅ 10− 4 & 1.47 ⋅ 10... | arxiv_0000039 |
**T**R* = 0, and the equilibrium is as proposed.
It is immediately apparent that while the LERGM rate function resembles the transition function of a Gibbs sampler, the change inhibition process resembles the corresponding Metropolis algorithm. However, it is again important to note that in this process all change even... | arxiv_0000040 |
26 & 6.38 & 2.13 & 4.79 & 4.26
`Clean All X` & 0.91 & 0.00 & 4.55 & 3.64 & 9.09 & 3.64 & 6.36 & 4.55 & 2.73
`Put All X In One Y` & 1.02 & 0.00 & 2.04 & 6.12 & 1.02 & 4.08 & 3.06 & 3.06 & 3.06
`Put All X On Y` & 0.00 & 2.94 & 0.00 & 0.00 & 5.88 & 4.41 & 4.41 & 0.00 & 4.41
`Prepare Salad` & 0.75 & 0.38 & 1.13 & 6... | arxiv_0000048 |
6614) – (-2.859999947028949,2.6303203984981516); (-2.859999947028949,2.6303203984981516) – (-2.849999947824581,2.6270104970150245); (-2.849999947824581,2.6270104970150245) – (-2.8399999486202128,2.6232331667569087); (-2.8399999486202128,2.6232331667569087) – (-2.8299999494158445,2.6189978490807517); (-2.829999949415844... | arxiv_0000051 |
representation of manifolds.
To illustrate these ideas, consider the case of *d* = 2. In this case the *d* − 1 manifolds are disjoint unions of circles, and the cobordisms are ``closed string" worldsheets with circular ingoing and outgoing boundaries. All 2d closed, oriented manifolds can be generated by gluing a fini... | arxiv_0000053 |
in Ref. for comparison. Ref. uses a broadening of 0.36 eV in order to reproduce the experimental resolution. We use 0.36 eV in our Fig. [SrVO3mfbsdftvc] as well.). However, in our case the peak between 2 eV and 2.5 eV stems mainly from the *e**g* states, which are not included into the Hubbard model in Ref. . Ref. ... | arxiv_0000058 |
Aperture effects on the oxygen abundance determinations from CALIFA data
========================================================================
This paper aims at providing aperture corrections for emission lines in a sample of spiral galaxies from the Calar Alto Legacy Integral Field Area Survey (CALIFA) database. I... | arxiv_0000059 |
**Gravitational multipole momentsfor asymptotically de Sitter spacetimes**
==========================================================================
We provide a prescription to compute the gravitational multipole moments of compact objects for asymptotically de Sitter spacetimes. Our prescription builds upon a recent... | arxiv_0000062 |
. However, for a resonant frequency response of a floating structure, estimating some tail aspects may be less critical. Other responses such as fatigue are not governed by extremes of the environment in general.
### Linearising the failure surface
Due to the complexity of solving Equation [E:pF], approximate approache... | arxiv_0000063 |
Parametrized Homology via Zigzag Persistence
============================================
This paper introduces parametrized homology, a continuous-parameter generalization of levelset zigzag persistent homology that captures the behavior of the homology of the fibers of a real-valued function on a topological space. T... | arxiv_0000064 |
})\right) + \text{div\,}\alpha(\mathbf{x}))/2 + C,\end{aligned}$$
where $C := \| \nabla\log\pi(\hat{\mathbf{x}})\|^2 /2
+ \Delta \log\pi(\hat{\mathbf{x}})/2$ is a constant. Letting A now be the law of $I,J\overset{\text{iid}}{\sim} {\text{U}}\{0, \ldots,n\}$ the following unbiased estimator of *ϕ* can be constructed,... | arxiv_0000068 |
– a computer programme for the dynamical processing of the initial binary star population
==========================================================================================
[firstpage]
The first version of the Binary Population Synthesizer () is made publicly available. It allows to efficiently calculate binar... | arxiv_0000074 |
In this article, we combine Bhargava’s geometry-of-numbers methods with the dynamical point-counting methods of Eskin–McMullen and Benoist–Oh to develop a new technique for counting integral points on symmetric varieties lying within fundamental domains for coregular representations. As applications, we study the distr... | arxiv_0000076 |
723944 & 3.6427 & 0.1949
15 & rs16832560 & 3 & 181564785 & SOX2-OT & 3.6316 & 0.1060
16 & rs818055 & 9 & 131092123 & LAMC3 & 3.6123 & 0.2010
17 & rs10018350 & 4 & 6320374 & PPP2R2C & 3.5524 & 0.1383
18 & rs949292 & 18 & 60501574 & & 3.5417 & 0.3216
19 & rs6878253 & 5 & 132277600 & LOC124901063 & 3.5211 & 0.08... | arxiv_0000077 |
On the Windfall and Price of Friendship:Inoculation Strategies on Social Networks
=================================================================================
This article investigates selfish behavior in games where players are embedded in a social context. A framework is presented which allows us to measure the ... | arxiv_0000079 |
Charged Particle and Photon Multiplicity, and Transverse Energy Production in High-Energy Heavy-Ion Collisions
==============================================================================================================
We review the charged particle and photon multiplicity, and transverse energy production in heavy-... | arxiv_0000080 |
magnitude gravitational wave inspiral time. Gravitational radiation is the relevant loss term (as opposed to, for example, tides) because the orbits limited by this criteria are in the gravitational wave dominated regime of pericenter distance (Figure [fig:terms]).
The combination of these limits on the captured WD po... | arxiv_0000085 |
Cutoff for the Glauber dynamics of the lattice free field
=========================================================
The Gaussian Free Field (GFF) is a canonical random surface in probability theory generalizing Brownian motion to higher dimensions. In two dimensions, it is critical in several senses, and is expected to... | arxiv_0000086 |
1}, while *S* = *T**H**b**h**v*, so [*T* : *S*] = 2. Since *p*, *q* and *r* have odd order, they must be elements of *S*. It is routine to verify that the involution *t*− 1*h*2 ∈ *S* and ⟨*p*, *q*, *r*, *t*− 1*h*2⟩ ≃ *C*33 : *C*2, so *S* = ⟨*p*, *q*, *r*, *t*− 1*h*2⟩. Now *S* fixes the vertex *H**b**h**b**h**v* ∈ Γ(*H... | arxiv_0000094 |
Rank Bounds for Design Matrices with Applications to Combinatorial Geometry and Locally Correctable Codes
=========================================================================================================
A (*q*, *k*, *t*)-design matrix is an *m* × *n* matrix whose pattern of zeros/non-zeros satisfies the follow... | arxiv_0000099 |
Asymptotic Existence of Fair Divisions for Groups
=================================================
The problem of dividing resources fairly occurs in many practical situations and is therefore an important topic of study in economics. In this paper, we investigate envy-free divisions in the setting where there are mul... | arxiv_0000100 |
masses of magnetars.
140mm
ccccccc *q* & $Bp[\rm{G}]$ & $Bc[\rm{G}]$ & *M*[*M*⊙] & $P[\rm{ms}]$ & $R[\rm{km}]$ & h
0.90&6.73E+16&2.63E+17&1.99E+01&1.25E+00&13.1E+00&0.367E+00
0.90&7.32E+16&2.80E+17&3.01E+01&1.06E+00&14.9E+00&0.326E+00
| | |
| --- | --- |
| | |
| | |
| | |
| --- | --- |
| | |
| |... | arxiv_0000101 |
Bachelor Thesis
**Engineering Faster Sorters**
**for Small Sets of Items**
Jasper Anton Marianczuk
Date: May 09, 2019
| | |
| --- | --- |
| Supervisors: | Prof. Dr. Peter Sanders |
| | Dr. Timo Bingmann |
Institute of Theoretical Informatics, Algorithmics
Department of Informatics
Karlsruhe Institute of Tech... | arxiv_0000102 |
Quantum and classical registers
===============================
We present a generic theory of “registers” in imperative programs and instantiate it in the classical and quantum setting. Roughly speaking, a register is some mutable part of the program state. Mutable classical variables and quantum registers and wires i... | arxiv_0000116 |
/9606222].
G. D’Ambrosio, G. F. Giudice, G. Isidori and A. Strumia, “Minimal flavor violation: An Effective field theory approach,” Nucl. Phys. B **645**, 155 (2002) [hep-ph/0207036].
I. Z. Rothstein, “TASI lectures on effective field theories,” hep-ph/0308266.
A. Rajaraman, W. Shepherd, T. M. P. Tait and A. M. Wijangc... | arxiv_0000120 |
* axis, thus increasing *α*R. On the contrary, for a given pressure, temperature increases the complex interplay between the electrons and the lattice, yielding a reduction of *α*R. These are the behaviors observed in the trivial phase [Fig. [FigRashbaPT](c), diamond markers].
The TI phase displays the opposite behavio... | arxiv_0000123 |
Propagating left/right asymmetry in the zebrafish embryo: one-dimensional model
===============================================================================
During embryonic development in vertebrates, left-right (L/R) asymmetry is reliably generated by a conserved mechanism: a L/R asymmetric signal is transmitted f... | arxiv_0000124 |
Orbital fitting of imaged planetary companions with high eccentricities and unbound orbits
==========================================================================================
Application to Fomalhaut b and PZ Telecopii B[1](#fn1)
======================================================
### Received....; Accepted..... | arxiv_0000125 |
10+12/10-10/10+0.01) rectangle +(0.08, 0.08); (4/10+3/10, 1/10+12/10-11/10+0.01) rectangle +(0.08, 0.08); (1/10+3/10, 1/10+12/10-12/10+0.01) rectangle +(0.08, 0.08);
[shift=(1.4,0)] (1/10+3/10, 1/10+12/10-1/10+0.01) rectangle +(0.08, 0.08); (1/10+3/10, 1/10+12/10-2/10+0.01) rectangle +(0.08, 0.08); (1/10+3/10... | arxiv_0000126 |
Shape asymmetry: a morphological indicator for automatic detection of galaxies in the post-coalescence merger stages.
=====================================================================================================================
[firstpage]
We present a new morphological indicator designed for automated recognit... | arxiv_0000132 |
An Introduction to Langer’s Theory
==================================
This note provides a pedagogical introduction to Langer’s theory for activated rate processes in multiple dimensions at the high friction limit, with an emphasis on the connection between the theory and the property of the backward committor/splittin... | arxiv_0000146 |
Extractor-Based Time-Space Lower Bounds for Learning
====================================================
A matrix $M: \AA \times \XX \rightarrow \{-1,1\}$ corresponds to the following learning problem: An unknown element $x \in \XX$ is chosen uniformly at random. A learner tries to learn *x* from a stream of samples, ... | arxiv_0000147 |
The Orthologic of Epistemic Modals
==================================
### Forthcoming in *Journal of Philosophical Logic*.
Epistemic modals have peculiar logical features that are challenging to account for in a broadly classical framework. For instance, while a sentence of the form *p* ∧ ◇¬*p* (‘*p*, but it might be t... | arxiv_0000150 |
Heat fluctuations in a harmonic chain of active particles
=========================================================
One of the major challenges in stochastic thermodynamics is to compute the distributions of stochastic observables for small-scale systems for which fluctuations play a significant role. Hitherto much the... | arxiv_0000154 |
the Higgs mechanism). In a lattice model, short-range couplings are of the form *e**α*(*x*)*e*− *α*(*x* + *δ*) where *δ* is of the order of the lattice constant. In a continuum field theory, these terms are represented by polynomials of spatial gradients ∼ ∇*e**α*(*x*). In particular, the NG modes arise from these te... | arxiv_0000157 |
Combinatorics for certain Skew Tableaux, Dyck Paths, Triangulations, and Dissections
====================================================================================
We present combinatorial bijections and identities between certain skew Young tableaux, Dyck paths, triangulations, and dissections.
Introduction
====... | arxiv_0000163 |
Scheduling and dimensioning of heterogeneous energy stores, with applications to future GB storage needs
========================================================================================================
Future “net-zero” electricity systems in which all or most generation is renewable may require very high volum... | arxiv_0000168 |
Onsager’s “Ideal Turbulence” Theory
===================================
Lars Onsager in 1945-1949 made an exact analysis of the high Reynolds-number limit for individual turbulent flow realizations modeled by incompressible Navier-Stokes equations, motivated by experimental observations that dissipation of kinetic ener... | arxiv_0000169 |
the table.
In the case of velocity isotropy (*β* = 0), it requires the excision of both And XII and And XIV from the datasets for the mass estimate of M31 to become comparable to or smaller than the Milky Way. For example, the mass of M31 with And XII and And XIV both removed is $0.85 \pm 0.24 \times 10^{12} \Msun$, a... | arxiv_0000178 |
Unified *k*-space theory of optical coherence tomography
========================================================
We present a general theory of optical coherence tomography (OCT), which synthesizes the fundamental concepts and implementations of OCT under a common 3D *k*-space framework. At the heart of this analysis ... | arxiv_0000181 |
A product formula for certain Littlewood-Richardson coefficients for Jack and Macdonald polynomials
===================================================================================================
Jack polynomials generalize several classical families of symmetric polynomials, including Schur polynomials, and are fu... | arxiv_0000182 |
$0.7043 & +*118%*
& *σ* = 20% & $0.6362 & $1.4534 & +*128%*
& *σ* = 25% & $1.0060 & $2.4051 & +*139%*
The above table illustrates the impact of investment (RPI) volatility (*σ*) on both the RCLA and S-RCLA value, assuming the same Gompertz mortality with parameters *λ* = 0, *m* = 86.3 and *b* = 9.5. Note that bo... | arxiv_0000183 |
experiments have been conducted 100 times for each type of generation of *A* and *B* and the reported results (both in terms of execution time and number of enabling decompositions) correspond to their average. The experiments have been executed on a desktop with medium capabilities (Intel Core i7-4790 CPU at 3.60 GHz... | arxiv_0000184 |
loss cone predicts a flatter distribution than the tide-only simulation does, the stellar encounters cannot entirely smooth out the peaks in the latitude distribution. This is consistent with the results in. Thus the observed non-uniform latitude distribution does not indicate that the Galactic tide dominates at the p... | arxiv_0000187 |
Lyapunov characterization of uniform exponential stability for nonlinear infinite-dimensional systems
=====================================================================================================
In this paper we deal with infinite-dimensional nonlinear forward complete dynamical systems which are subject to ex... | arxiv_0000191 |
Breaking and trapping Cooper pairs by Rydberg-molecule spectroscopyin atomic Fermi superfluids
==============================================================================================
We propose a spectroscopic probe of the breaking and localization of Cooper pairs in an atomic Fermi superfluid interacting with a... | arxiv_0000198 |
above, and shows how the design of an algorithm is strictly related to the problem under investigation. As already discussed in Section [acsop], our algorithm is optimized to exploit time windows. The hierarchic approach we adopt goes into this direction by preparing promising solution fragments that can be exchanged ... | arxiv_0000201 |
Using rotation, magnetic activity and lithium to estimate the ages of low mass stars
====================================================================================
The rotation rate, level of magnetic activity and surface lithium abundance are age-dependent quantities in stars of about a solar mass and below. The... | arxiv_0000204 |
Hybrid Recommender Systems: A Systematic Literature Review
==========================================================
Recommender systems are software tools used to generate and provide suggestions for items and other entities to the users by exploiting various strategies. Hybrid recommender systems combine two or more... | arxiv_0000206 |
Non-integrable dimer models: universality and scaling relations
===============================================================
In the last few years, the methods of constructive Fermionic Renormalization Group have been successfully applied to the study of the scaling limit of several two-dimensional statistical mecha... | arxiv_0000212 |
Improved bounds for the Erdos-Rogers function
=============================================
[classification=text]
The Erdos-Rogers function *f**s*, *t* measures how large a *K**s*-free induced subgraph there must be in a *K**t*-free graph on *n* vertices. While good estimates for *f**s*, *t* are known for some pairs (*... | arxiv_0000214 |
enzeller, I. 1992,, 92, 481 Brown, S. F., Donati, J.-F., Rees, D. E., & Semel, M. 1991,, 250, 463 Browning, M. K. 2008,, 676, 1262 Calvet, N., Muzerolle, J., Briceño, C., et al. 2004,, 128, 1294 Calvet, N., & Gullbring, E. 1998,, 509, 802 Carpenter, J. M., Mamajek, E. E., Hillenbrand, L. A., & Meyer, M. R. 2006,, 651, ... | arxiv_0000221 |
The mass-*L**x* relation for moderate luminosity X-ray clusters
===============================================================
We present measurements of the masses of a sample of 25 moderate X-ray luminosity clusters of galaxies from the 160 square degree ROSAT survey. The masses were obtained from a weak lensing ana... | arxiv_0000224 |
to the high propagation loss of mmWave signals, beamforming techniques have to be employed to achieve sufficiently high signal-to-noise ratios (SNRs) in mmWave communications. Fortunately, benefiting from the small wavelength of mmWave signals, a large number of antennas can be equipped in a small area to realize high... | arxiv_0000235 |
Sparse Regression at Scale: Branch-and-Bound rooted in First-Order Optimization
===============================================================================
### April, 2021
We consider the least squares regression problem, penalized with a combination of the ℓ0 and squared ℓ2 penalty functions (a.k.a. ℓ0ℓ2 regulariz... | arxiv_0000244 |
.16 ± 0.540.54
44 & PSZ2 G150.56+58.32 & CLGJ1115+5319 & 0.47 & S & 12.77 ± 2.40 & 10.18 ± 13.31 & 34.06 ± 18.57 & 0.13 ± 0.02 & 8.14 ± 1.49 & 10.04 ± 1.61 & 7.44 ± 0.530.50
45 & PSZ2 G170.98+39.45 & [SPD2011] 16774 & 0.5131 & S & 10.11 ± 1.38 & 31.48 ± 10.20 & − 30.87 ± 12.67 & 0.12 ± 0.02 & 6.43 ± 0.86 & 8.24 ± ... | arxiv_0000246 |
Monte Carlo on compact complex manifoldsusing Bergman kernels
=============================================================
In this paper, we propose a new randomized method for numerical integration on a compact complex manifold with respect to a continuous volume form. Taking for quadrature nodes a suitable determina... | arxiv_0000247 |
3/ASY-1994-9301>.
P. Azérad, F. Guillén; Mathematical justification of the hydrostatic approximation in the primitive equations of geophysical fluid dynamics, SIAM J. Math. Anal. 33(4) (2001), pp. 847–859, <https://doi.org/10.1137/S0036141000375962>.
G. Bayada, M. Chambat; The Transition Between the Stokes Equations an... | arxiv_0000248 |
resolution, so the greatest effect would be on the Sersic indices of small, high $\mathrm{n\_{s}}$ ellipticals.
The greater concern with these results is the relaxation that occurs in the bulges of progenitor spirals. The effective radius of the M31 model is 1.5 kpc, but most galaxies are scaled to smaller sizes than ... | arxiv_0000250 |
Measuring Metallicities with HST/WFC3 photometry
================================================
We quantified and calibrated the metallicity and temperature sensitivities of colors derived from nine Wide Field Camera 3 (WFC3) filters aboard the Hubble Space Telescope (HST) using Dartmouth isochrones and Kurucz atmosp... | arxiv_0000251 |
Coherence for adjunctions in a 3-category via string diagrams
=============================================================
We construct a 3-categorical presentation Adj(3, 1) and define a coherent adjunction in a strict 3-category C as a map Adj(3, 1) → C. We use string diagrams to show that any adjunction in C can be... | arxiv_0000257 |
Security of round-robin differential-phase-shift quantum key distribution protocol with correlated light sources
================================================================================================================
Among various quantum key distribution (QKD) protocols, the round-robin differential-phase- s... | arxiv_0000261 |
mathscr{l} }\! {\boldsymbol{E}}}_i \right)}_{i\in\mathcal{I}}, \ell\in\mathcal{J}. The component representation of {\left( {{}^{ 1 }\! {\boldsymbol{E}}}_i \right)}_{i\in\mathcal{I}} w.r.t. {\left( {\boldsymbol{E}}_i \right)}_{i\in\mathcal{I}} is {\left( {\left( 0,1,0 \right)},{\left( 1,0,0 \right)},{\left( 0,0,1 \right... | arxiv_0000279 |
large team projects in capstone courses & ASEE Annual Conference and Exposition
S31 & Engelsma, J. R. & 2014 & Best practices for industry-sponsored CS capstone courses & Journal of Computing Sciences in Colleges
S32 & Matthies, C., Teusner, R., Hesse, G. & 2019 & Beyond Surveys: Analyzing Software Development Art... | arxiv_0000284 |
the factorization structure of soft-collinear divergences leads to more powerful identities. Recall that the box coefficients are so-called ‘quad-cuts’ (co-dimension four residues) of the loop integrand; let us now consider ‘triple-cuts’—co-dimension three residues. Because the loop integral is four-dimensional, a tri... | arxiv_0000291 |
Rossini, G. M. Andolina, D. Rosa, M. Carrega, M. Polini, Phys. Rev. Lett. 125, 236402 (2020).
rosaarxiv07247 D. Rosa, D. Rossini, G. M. Andolina, M. Polini, M. Carrega, J. High Energ. Phys. 2020, 67 (2020).
carregaarxiv03551 M. Carrega, J. Kim, D. Rosa, Entropy 2021, 23(5), 587 (2021).
caravelliprr023095 F. Caravelli,... | arxiv_0000293 |
4 & 2.08 & -1.89 & 1.56 & …& v2
J231433.38-523439.3 & -82.1 & 4925 & 1.90 & -2.11 & 2.71 & …& v2
J231529.71-462235.1 & -52.6 & 4622 & 0.95 & -2.57 & 3.27 & -0.20 & v2
J231645.30-404728.2 & …& …& …& …& …& …& v2 & 1
J231756.18-555332.4 & 267.3 & 4649 & 0.77 & -2.10 & 3.86 & …& v2
J231845.22-280744.8 & -38.1 & 4... | arxiv_0000294 |
The 4d superconformal index near roots of unity and3d Chern-Simons theory
=========================================================================
Introduction
============
In the last few years there has been a renewed interest in the study of the superconformal index of 4d N = 1 superconformal field theories (SCFTs)... | arxiv_0000295 |
Purification in Rapid Repeated Interaction Systems
==================================================
We investigate the open dynamics of a quantum system when it is rapidly repeatedly updated by a quantum channel. Specifically, we analyze when this dynamics can purify the system. We develop a necessary and sufficient ... | arxiv_0000298 |
variable accretion rate. If the mass accretion rate increases, the radiated energy will increase, thus, the intrinsic luminosity. Hence, $\dot{m} \propto L\_{in}$, or *λ**E**d**d* ∝ *L**i**n*. Again, if the luminosity increases, it will cool down the Compton cloud more efficiently implying a steeper photon index, i.e.... | arxiv_0000299 |
because we want to obtain the best spatial resolution using SHARP (see section 2), and because 350 *μ*m emission is weighted towards the warmer emission of the cores immediately surrounding a YSO rather than towards the extended cooler core envelope. This paper gives a summary of our results on three of the YSOs in ou... | arxiv_0000300 |
*Monthly Notices of the Royal Astronomical Society*: LaTeX guide for authors
============================================================================
### Last updated 2020 June 10; in original form 2013 September 5
[firstpage]
This is a guide for preparing papers for *Monthly Notices of the Royal Astronomical Socie... | arxiv_0000302 |
Adaptive Equi-Energy Sampler: Convergence and Illustration
==========================================================
Abstract
========
Markov chain Monte Carlo (MCMC) methods allow to sample a distribution known up to a multiplicative constant. Classical MCMC samplers are known to have very poor mixing properties when... | arxiv_0000309 |
A diffuse interface model for the analysis of propagating bulges in cylindrical balloons
========================================================================================
Axisymmetric membranes Bifurcation Asymptotic analysis Strain gradient elasticity
With the aim to characterize the formation and propagation o... | arxiv_0000310 |
Kinematic structures of the Solar neighbourhoodrevealed by ***Gaia*** DR1/TGAS and RAVE
=======================================================================================
### Received 10 May 2017 / Accepted 19 September 2017
We aim to study the velocity distribution of stars of the Solar neighbourhood and investig... | arxiv_0000315 |
On a Heath-Jarrow-Morton approach for stock options
===================================================
This paper aims at transferring the philosophy behind Heath-Jarrow-Morton to the modelling of call options with all strikes and maturities. Contrary to the approach by Carmona and Nadtochiy and related to the recent... | arxiv_0000316 |
-rate control algorithm for variable bit-rate applications of versatile video coding standard,” *Signal Processing: Image Communication*, vol. 96, p. 116317, 2021.
M. Wang, J. Zhang, L. Huang, and J. Xiong, “Machine learning-based rate distortion modeling for vvc/h. 266 intra-frame,” in *2021 IEEE International Confere... | arxiv_0000319 |
.868756
0.498006 0.868701
0.49792 0.868647
0.497834 0.868592
0.497749 0.868538
0.497663 0.868483
0.497578 0.868429
0.497492 0.868374
0.497407 0.86832
0.497321 0.868265
0.497236 0.86821
0.49715 0.868156
0.497065 0.868101
0.49698 0.868046
0.496894 0.867991
0.496809 0.867936
0.496724 0.8678... | arxiv_0000322 |
Cosmology in generalized Proca theories
=======================================
We consider a massive vector field with derivative interactions that propagates only the 3 desired polarizations (besides two tensor polarizations from gravity) with second-order equations of motion in curved space-time. The cosmological im... | arxiv_0000330 |
Topological Defects on the Lattice I: The Ising model
=====================================================
In this paper and its sequel, we construct topologically invariant defects in two-dimensional classical lattice models and quantum spin chains. We show how defect lines commute with the transfer matrix/Hamiltonia... | arxiv_0000331 |
Moduli space of *J*-holomorphic subvarieties
============================================
We study the moduli space of *J*-holomorphic subvarieties in a 4-dimensional symplectic manifold. For an arbitrary tamed almost complex structure, we show that the moduli space of a sphere class is formed by a family of linear sys... | arxiv_0000333 |
On permutations with decidable cycles
=====================================
Recursive permutations whose cycles are the classes of a decidable equivalence relation are studied; the set of these permutations is called Perm, the group of all recursive permutations G. Multiple equivalent computable representations of deci... | arxiv_0000334 |
Message-Passing on Hypergraphs: Detectability, Phase Transitions and Higher-Order Information
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[1](#fn1)
[2](#fn2)
Hypergraphs are widely adopted tools to examine systems with higher-order interactions. Despite recent advanceme... | arxiv_0000341 |
*t*th test bag. In the inductive setting, the predicted label maximizes the instance label probability given feature vector **x***t**i* and parameter **w** is
*ŷ**t**i* = arg max*k**p*(*y**t**i* = *k*∣**x***t**i*, **w**),
where *p*(*y**t**i* = *k*∣**x***t**i*, **w**) is given in. Note that in the absence of bag... | arxiv_0000345 |
by and. [f12]
In Fig. [f12] we compare the derived effective temperatures with the spectral types determined with the HPI in Sect. [s32]. The expected trend – cooler temperatures correspond to later spectral types – is visible, with only three exceptions. For two of these exceptions (SONYC-NGC1333-17 and -31), the si... | arxiv_0000350 |
the subsonic and remnant phases are unchanged from those given in and respectively.
Isothermal evolution of the shocked shell
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As first pointed out by, the final state of the shocked gas lying between the lobe surface and the bow shock depends critically on whether it expands... | arxiv_0000355 |
velocity during the period Jan.31-Feb. 6 determined from all measurable absorption lines was -1810 ± 40 km s− 1.
Outflow velocities of the order of 1500 - 2200 kms− 1 were also observed in previous outbursts. Adams and Joy (1920) reported the presence of ``dark" components of radial velocity up to -2100 km s− 1 a fe... | arxiv_0000356 |
Schottky Signal Modification as a Diagnostic Tool for Coherent Electron Cooling
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Coherent electron cooling is a promising technique to cool high-intensity hadron bunches by imprinting the noise in the hadron beam on a beam of electrons, ampl... | arxiv_0000358 |
We introduce a model of a controlled random graph process. In this model, the edges of the complete graph *K**n* are ordered randomly and then revealed, one by one, to a player called Builder. He must decide, immediately and irrevocably, whether to purchase each observed edge. The observation time is bounded by paramet... | arxiv_0000359 |
A. Palma and S. P. Patil, Phys. Rev. D **84**, 043502 (2011) [arXiv:1005.3848 [hep-th]].
R. Saito, M. Nakashima, Y. i. Takamizu and J. Yokoyama, JCAP **1211**, 036 (2012) [arXiv:1206.2164 [astro-ph.CO]].
R. Saito and Y. i. Takamizu, JCAP **1306**, 031 (2013) [arXiv:1303.3839, arXiv:1303.3839 [astro-ph.CO]].
S. Renaux-... | arxiv_0000360 |
Unconventional Fusion and Braiding of Topological Defects in a Lattice Model
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We examine non-abelian topological defects in an abelian lattice model in two dimensions. We first construct an exact solvable lattice model that exhibits coexisting ... | arxiv_0000362 |
the harvested energy be stored in the battery before it can be used for transmission. An alternative energy storage model, considered in, is to allow the harvested energy *E**t* to be used instantaneously at time *t* and the remaining amount be stored in the battery. The capacities under the two energy storage models ... | arxiv_0000373 |
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In this paper, we study distributed algorithms for large-scale AUC maximization with a deep neural network as a predictive model. Although distributed learning techniques have been investigated extensively in deep learning, they are not directly applicable to stochastic AUC maximization with deep neural netwo... | arxiv_0000375 |
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