\section{Introduction \label{sec:introduction}}

Determining when and how reionization occurred is essential for understanding the IGM physics and galaxy formation in the early Universe \citep[e.g.,][]{munoz_impact_2022}. Recent observations have made significant progress on establishing the timing of reionization and largely point to a midpoint of $z\sim 7-8$. These observations include the electron optical depth to the cosmic microwave background (CMB) photons \citep[][see also \citealp{de_belsunce_inference_2021}]{planck_collaboration_planck_2020}, the Lyman-$\alpha$ (Ly$\alpha$) damping wing in $z\gtrsim7$ QSO spectra \citep[e.g.,][]{banados_800-million-solar-mass_2018,davies_quantitative_2018,wang_significantly_2020-1,yang_poniuaena_2020,greig_igm_2021-1}, the decline in observed Ly$\alpha$\ emission from $z>6$ galaxies \citep[e.g.,][and references therein, but see \citealp{jung_texas_2020-1,wold_lager_2021-1}]{mason_universe_2018,mason_inferences_2019,hoag_constraining_2019,hu_ly_2019}, and the IGM thermal state measurements at $z>5$ \citep[e.g.,][]{boera_revealing_2019,gaikwad_consistent_2021}. 

The observations noted above are generally consistent with reionization ending by $z\sim6$, a scenario supported by existing measurement of the fraction of dark pixels in the Lyman series forests \citep[e.g.,][]{mcgreer_first_2011,mcgreer_model-independent_2015}.  Other observations, however, 
suggest a significantly later end of reionization. The large-scale fluctuations in the measured Ly$\alpha$\ effective optical depth, $\tau_{\rm eff} =-\ln{\langle F \rangle}$, where $F$ is the continuum-normalized transmission flux \citep[e.g.,][]{fan_constraining_2006,becker_evidence_2015,eilers_opacity_2018,bosman_hydrogen_2021,yang_measurements_2020-1}, together with long troughs extending to or below $z\simeq5.5$ in the Ly$\alpha$\ forest \citep[e.g.,][]{becker_evidence_2015,zhu_chasing_2021} potentially indicate the existence of large neutral IGM islands \citep[e.g.,][]{kulkarni_large_2019,keating_long_2020,nasir_observing_2020}.  The underdensities around long dark gaps traced by Ly$\alpha$\ emitting galaxies (LAEs) are also consistent with a late reionization model wherein reionization ends at $z<6$ \citep[][]{becker_evidence_2018,kashino_evidence_2020, christenson_constraints_2021}. 

These Ly$\alpha$\ forest and LAE results are potentially consistent with an alternative scenario wherein the IGM is ionized by $z=6$ but retains large-scale fluctuations in the ionizing UV background down to lower redshifts \citep[][]{davies_determining_2018}. 
On the other hand, recent measurements of the mean free path of ionizing photons measured at $z=5.1$ and 6.0 \citep{becker_mean_2021} are difficult to reconcile with models wherein reionization ends at $z > 6$, and may instead prefer models wherein the IGM is still $20\%$ neutral or more at $z=6$ \citep[][]{becker_mean_2021,cain_short_2021,davies_predicament_2021}.
In addition, a reionization ending at $z < 6$ is consistent with models that reproduce a variety of observations \citep[e.g.,][]{weinberger_modelling_2019,choudhury_studying_2021, qin_reionization_2021}. %

One way of searching for signatures of late ($z_{\rm end} < 6$) reionization in the Ly$\alpha$\ forest is by investigating dark gaps, i.e., contiguous regions of strong absorption  \citep[e.g.,][]{songaila_approaching_2002,furlanetto_constraining_2004,paschos_statistical_2005,fan_constraining_2006,gallerani_glimpsing_2008,gnedin_cosmic_2017,nasir_observing_2020}. In \citet[][hereafter \citetalias{zhu_chasing_2021}]{zhu_chasing_2021}, we explored long dark gaps in the Ly$\alpha$\ forest and found that a fully ionized IGM with a homogeneous UVB is strongly ruled out down to $z\simeq5.3$. In contrast, late reionization models and a model wherein reionization ends by $z\sim6$ but retains large-scale UVB fluctuations are  consistent with the observations. Predictions for the Ly$\alpha$\ dark gap statistics are similar between the two types of models.  This is largely because realistic late reionization models also include UVB fluctuations, which are often associated with the neutral islands.  Ly$\alpha$\ also tends to saturate at a relatively low ($x_{\rm HI} \sim 10^{-3}$) neutral fraction, limiting its sensitivity to neutral gas. 

Given its lower optical depth \footnote{Ly${\beta}$\ absorption has a shorter wavelength ($\lambda_{\rm Ly \beta} = 1025.72$\,\AA) and a lower oscillator strength ($f_{\rm Ly \beta} = 0.0791$) compared to those of Ly$\alpha$\ absorption ($\lambda_{\rm Ly \alpha} = 1215.67$\,\AA, $f_{\rm Ly \alpha} = 0.4164$). The ratio of optical depth is given by $\tau_{\rm Ly\beta}/\tau_{\rm Ly\alpha}=(f_{\rm Ly \beta} \lambda_{\rm Ly\beta}) / (f_{\rm Ly\alpha} \lambda_{\rm Ly\alpha})\simeq 0.16$. }, Ly${\beta}$\ should be a more sensitive probe of neutral gas in the $z \lesssim 6$ IGM. As a result, ultra-late reionization models wherein neutral islands persist down to $z<5.5$ may produce more long Ly${\beta}$\ dark gaps than can be produced by UVB fluctuations alone. Based on this feature, we can potentially use dark gaps in the Ly${\beta}$\ forest to place stronger constraints on the timing of reionization and distinguish the late reionization models from the pure fluctuating UVB models. As presented in \citet{nasir_observing_2020}, distributions of dark gaps in the Ly${\beta}$\ forest differ between these models most strongly on scales of $L\gtrsim10\cmpch$. We are therefore particularly interested in these long dark gaps. 

In this work, we use 42 high-quality QSO spectra that allow us to search for dark gaps in the Ly${\beta}$\ forest over the redshift range $5.5 < z < 6.0$.  The sample includes high-quality X-Shooter spectra from the XQR-30 VLT large program \footnote{\href{https://xqr30.inaf.it}{https://xqr30.inaf.it}} (D'Odorico et al., in prep.). In addition to comparing the results to model predictions, we also constrain $ \langle x_{\rm HI} \rangle $ based on a conservative relationship between dark gap length and neutral fraction derived from simulations.  

This paper is organized as follows. In {Section~\ref{sec:data}} we describe the data and results from the observations. \mbox{Section~\ref{sec:models}} compares our results to model predictions, discusses the implications for reionization, and infers constraints on $x_{\rm HI}$.  Finally, we conclude the findings in \mbox{Section~\ref{sec:conslusion}}. Throughout this paper, we assume a $\Lambda$CDM cosmology with $\Omega_{\rm m}=0.308$, $\Omega_{\Lambda}=0.692$, and $h=0.678$ \citep[][]{planck_collaboration_planck_2014}. Distances are quoted in comoving units unless otherwise noted.


\section{The Data and Results \label{sec:data}}

\subsection{QSO Spectra}

\begin{figure*}[!t]
    \centering \includegraphics[width=7in]{10-PSOJ025-11.pdf}
    \caption{
        \textbf{{Top panel:}} 
        Spectrum and continuum fits for the $z_{\rm em} = 5.849$ QSO PSO J025$-$11. 
        The light blue and dark red lines represents flux and flux error in the original binning. Dashed curves redward and blueward of the Ly$\alpha$\ peak show the best-fitting and predicted QSO continuum in absence of absorption, respectively. The fitting and prediction are based on Principal Component Analysis (PCA). The continuity between the Ly$\alpha$\ forest and the Ly${\beta}$\ forest continuum is broken intentionally near 1020 \AA\ in the rest frame. We label the wavelength range over which we search for dark gaps in the Ly${\beta}$\ forest and its corresponding Ly$\alpha$\ forest in redshift.
        \textbf{{Bottom panel:}} 
        Ly${\beta}$\ forest and dark gaps detected. The dashed black line labels the flux threshold of 0.02. The thick black line displays the flux binned to $1\cmpch$. Light blue and dark red lines show the flux and flux error in the original binning. Dark gaps detected are shaded with gray. We also label the redshift range and length of each long ($L\geq10\cmpch$) dark gap. \\
        (The complete figure set [42 images] is available online at \url{https://ydzhuastro.github.io/lyb.html}.)
        }
        \label{fig:spectra}
\end{figure*}

Our sample includes the 42 out of 43 spectra of QSOs at $5.77 \lesssim z_{\rm em} \lesssim 6.31$ that were used for \citetalias{zhu_chasing_2021}.  The exception is PSO J004$+$17, whose spectrum has lower S/N that does not meet the requirement of our flux threshold for the Ly${\beta}$\ forest (Section \ref{sec:detection-method}).  The spectra are taken with the Echellette Spectrograph and Imager (ESI) on Keck \citep{sheinis_esi_2002} and the X-Shooter spectrograph on the Very Large Telescope (VLT; \citealp{vernet_x-shooter_2011}). Among these, 19 X-Shooter spectra are from the XQR-30 VLT large program (D'Odorico et al., in prep). The details of the data reduction procedures are given in \citetalias{zhu_chasing_2021} and \citet{becker_evolution_2019}. We note that the targets are selected without foreknowledge of dark gaps in the Ly${\beta}$\ forest. Figure Set~\ref{fig:spectra} displays the spectra, continuum fits, and dark gaps detected in the Ly${\beta}$\ forest for each QSO (see details below).  An example is given in Figure~\ref{fig:spectra}.


\subsection{Continuum Fitting}
We use Principal Component Analysis (PCA) to predict the unabsorbed QSO continuum and normalize the transmission in the Ly${\beta}$\ forest. We follow a similar method as described in \citetalias{zhu_chasing_2021} to fit and predict the continuum. Briefly, we implement and apply the log-PCA method of \cite{davies_predicting_2018} in the Ly$\alpha$\ and Ly${\beta}$\ forest portion of the spectrum following \citet{bosman_hydrogen_2021}. The continuity between the Ly$\alpha$\ forest and the Ly${\beta}$\ forest continuum is broken intentionally to correct for the effect of overlapping Ly$\alpha$\ absorption in the Ly${\beta}$\ forest in the PCA training sample. We fit the red-side (rest-frame wavelength $\lambda_0>1230$ \AA) continuum up to 2000 \AA\ in the rest frame for X-Shooter spectra with NIR observations. The ESI spectra are fit using an optical-only PCA, which is presented in \citet{bosman_comparison_2021}. The Ly${\beta}$\ dark gap detection is not very sensitive to the continuum, and we also test that using a power-law continuum does not significantly change the dark gap results in this work.

\subsection{Dark Gap Detection \label{sec:detection-method}}
Similar to the definition of a dark gap in the Ly$\alpha$\ forest in \citetalias{zhu_chasing_2021}, we define a dark gap in the Ly${\beta}$\ forest to be a continuous spectral region in which all pixels binned to $1\cmpch$ have an observed normalized flux $F = F_{\rm obs}/F_{\rm c} < 0.02$, where $F_{\rm obs}$ is the observed flux and $F_{\rm c}$ is the continuum flux. The minimum length of a dark gap is $1\cmpch$ following \citetalias{zhu_chasing_2021}. 
\add{A bin size of $1\cmpch$ enables us to retain significant transmission profiles while reducing the influence of very small  peaks on dark gap detection.  The precise bin size should have relatively little impact on our results provided that the observations and mock spectra are treated consistently. We have verified that using bin sizes of $0.5\cmpch$ or $1.5\cmpch$ does not change our major conclusions, although the lengths of some dark gaps would change.}
A flux threshold of 0.02 is used here instead of 0.05, which we used for the Ly$\alpha$\ gaps, because spectra in this sub-sample have higher signal-to-noise (S/N) levels. In addition, the Ly${\beta}$\ forest at the redshifts that we are interested in is less contaminated by sky lines than the Ly$\alpha$\ forest. We have tested that using a threshold of 0.05 will not change our results fundamentally, although the difference between the models (Section~\ref{sec:models}) may become less significant. 
In order to reduce false detections caused by foreground Ly$\alpha$\ absorption, we further require that all Ly${\beta}$\ dark gaps correspond to Ly$\alpha$\ dark gaps as defined in \citetalias{zhu_chasing_2021} over the same redshifts for both the observed and mock (Section \ref{sec:mock_spectra}) spectra. That is to say, each $1\cmpch$ bin in the Ly${\beta}$\ dark gap also has a normalized flux less than 0.05 in the Ly$\alpha$\ forest at the same redshift.
\footnote{Based on our test, whether requiring gaps to be also dark in the Ly$\alpha$\ forest or not only affects a small fraction of gaps and does not change the results in this paper significantly. Although this requirement may not remove all false detection, it partially avoids contamination from random foreground density fluctuations.}
For reference, we present the relationship between Ly${\beta}$\ dark gaps and Ly$\alpha$\ dark gaps in Appendix \ref{app:Lyb_Lya}.

For each QSO sightline, dark gaps are detected from 976 \AA\ in the rest frame to 11 proper-Mpc blueward of the QSO's redshift, which corresponds to approximately 1000 \AA\ in the rest frame. The lower wavelength limit ensures that the detection is not affected by the Ly$\gamma$\ absorption. We use the higher limit to avoid the QSO proximity zone transmission, and the cut is comparable to the choice in, e.g., \citet{bosman_comparison_2021}. Following \citetalias{zhu_chasing_2021}, we also exclude from the statistical analysis an additional $10\cmpch$ buffer zone blueward of the proximity zone cut. This allows the pixel at the red end of each sightline to intersect a {\em possible} long ($L\ge10\cmpch$) dark gap, and hence helps to mitigate potential truncation issues. \footnote{Without this additional buffer zone, it is possible that the $F_{10}$ (Section \ref{sec:gap_statistics}) is underestimated near the red end of a sightline, since there can exist otherwise $\ge10\cmpch$ gaps that are truncated by the edge or peaks in the proximity zone.}

To avoid the contamination from sky line subtraction residuals, we mask out $\pm 75~{\rm km\,s^{-1}}$ intervals of the spectra centered at sharp peaks in the flux error array when searching for dark gaps. The exception is that we do not mask transmission with a $>3\sigma$ detection. In addition, we make no attempts to correct for the effects of damped Ly$\alpha$\ systems (DLAs) or metal-enriched absorbers, although known systems in a sub-sample of the spectra with a relatively complete identification of metal-enriched systems are discussed in   \mbox{Appendix~\ref{app:moreMetal}}. Figure~\ref{fig:all_los} displays an overview of dark gaps detected in the Ly${\beta}$\ forest from our sample. 

\begin{figure*}
    \centering
    \includegraphics[width=6in]{all_los.pdf}
    \caption{Overview of dark gaps identified in the Ly${\beta}$\ forest from our sample of 42 QSO spectra. Black (gray) bars represent dark gaps longer (shorter) than $10\cmpch$. Pink squares label redshifts of XQR-30 targets, and blue dots mark the redshifts of the rest of QSOs.  Light blue shaded regions demonstrate the redshift coverage of the Ly${\beta}$\ forest. We note that the Ly${\beta}$\ forest is truncated at 11 pMpc from the QSO. The Ly${\beta}$\ forest shown in this figure includes the $10\cmpch$ buffer zone labeled with dashed white line at the red end. 
    \label{fig:all_los}}
\end{figure*}

\subsubsection{Dark Gaps Statistics}\label{sec:gap_statistics}

\begin{figure*}
\centering
    \gridline{
    \hspace{-1cm}
        \fig{all_obs_gaps.pdf}{2.6in}{(a)}
    \hspace{-0.3cm}
        \fig{CDF_obs_bdyz.pdf}{2.6in}{(b)}
    \hspace{-0.4cm}
        \fig{F10obs.pdf}{2.6in}{(c)}
    }
    %
        %
        %
    %
    %
    \caption{Observed Ly${\beta}$\ dark gaps. 
    \textbf{{(a)}} Gap length versus central redshift. Dark gaps cut at the red end by the proximity zone cut are labeled with arrows.  
    %
    %
    \textbf{{(b)}} Cumulative distribution of dark gap length for two redshift intervals. The upper/lower bounds of the shaded region are described in Section~\ref{sec:gap_statistics}.
    \textbf{{(c)}} The fraction of QSO spectra showing dark gaps longer than $10 \cmpch$ as a function of redshift, $F_{10}$. $F_{10}$ is plotted with a binning of $\Delta z = 0.025$. See text and Appendix \ref{app:F01} for discussions about the drop of $F_{10}$ near $z=5.9$.
    %
    \\
    (The data used to create this figure are available online at \url{https://ydzhuastro.github.io/lyb.html}.)
    \label{fig:obs}} 
\end{figure*}

Figure~\ref{fig:obs} displays the statistical properties of dark gaps detected in the Ly${\beta}$\ forest from our sample. We detect 195 dark gaps in total, of which 24 have $L \geq 10 \cmpch$. Panel (a) plots length versus central redshift of these dark gaps. Long dark gaps become less common as redshift decreases. Nevertheless, some long gaps still exist down to $z<5.6$.

We calculate the cumulative distribution function of dark gap length, $P(<L)$. Dark gaps are sorted into two redshift bins according to their redshifts at both ends. For distributions that include dark gaps cut at the red end by the proximity zone limit, we calculate a lower bound on $P(<L)$ by assuming a infinite length for these gaps, and an upper bound by assuming the gap length that would appear in the absence of any proximity effect is the same as the one measured. As shown in Figure~\ref{fig:obs} (b), longer dark gaps become more numerous over $5.75 < z < 6.00$ compared to $5.50 < z < 5.75$. This significant evolution of $P(<L)$ is consistent with the results shown in panel (a).

Following \citetalias{zhu_chasing_2021}, we quantify the spatial coverage of large Ly${\beta}$-opaque regions by calculating the fraction of QSO spectra showing long ($L\geq10\cmpch$) dark gaps as a function of redshift, $F_{10}(z)$. Here we use $10\cmpch$ as the threshold because dark gaps longer than this in the late reionization models (see Section~\ref{sec:models}) are dominated by those containing neutral islands. Based on our tests, the number of dark gaps predicted by different models differs the most for $L\gtrsim10\cmpch$, as also suggested by \citet{nasir_observing_2020}.

We calculate $F_{10}$ \add{at each redshift} and average over $\Delta z=0.025$ bins. As shown in Figure~\ref{fig:obs} (c), $F_{10}$ has a significant redshift evolution over $5.5<z<6.0$. It increases rapidly with redshift over $5.5<z<5.8$, from $\sim 10\%$ to $\sim 40\%$, and climbs up to $\sim80\%$ by $z=6.0$ after a drop at $z\sim5.9$. The reason of the drop is unclear,
but the limited number of QSO sightlines may produce large statistical fluctuations (Appendix~\ref{app:F01}), as also shown in the model predictions in Section~\ref{sec:models}. For comparison, we compute $F_{01}$, the fraction of QSO sightlines exhibiting dark gaps with $L\geq 1\cmpch$, and find no significant drop near $z=5.9$ (see Appendix~\ref{app:F01}).

\subsubsection{Long Dark Gap toward PSO J025\texorpdfstring{$-$}{Lg}11}
We find a particularly interesting Ly${\beta}$\ gap toward the $z=5.849$ QSO PSO J025$-$11 (Figure~\ref{fig:spectra}). This dark gap is within the redshift interval of a long ($68\cmpch$) trough in the Ly$\alpha$\ forest. It spans $5.526\leq z \leq 5.613$ with a length of $28\cmpch$. This is longer and extending to a even lower redshift than the 19 and $23\cmpch$ Ly${\beta}$\ troughs with $z_{\rm min}=5.66$ within the extreme ($110\cmpch$) Ly$\alpha$\ trough over $5.523 \leq z \leq 5.879$ toward ULAS J0148$+$0600 \citep{becker_evidence_2015}.  
This dark gap toward J025$-$11 contains no apparent transmission peaks, even in the unbinned data. The $2\sigma$ lower limit of $\tau_{\rm eff}  \geq 6.067$ measured over the complete trough indicates that it is highly opaque. 

There is a possibility that part of the trough may be due to either Ly${\beta}$\ or foreground Ly$\alpha$\ absorption from the circum-galacitc medium (CGM) around intervening galaxies. In this case, corresponding metal lines may be present. We check for potential CGM absorption using the XQR-30 metal absorber catalog (Davies et al., in prep; see Appendix \ref{app:moreMetal} for technique details). We find no intervening metal systems within the redshift range of the gap. We note that this line of sight has a DLA near the redshift of the QSO, as evidenced by the damping wing at the blue edge of the Ly$\alpha$\ proximity zone.  The Ly${\beta}$\ gap described here is at a velocity separation of $>$3000$~{\rm km\,s^{-1}}$ from the QSO, however, and is unaffected by the DLA. The XQR-30 catalog does include a \ion{C}{4} system towards J025$-$11 at $z=4.5138$, for which Ly$\alpha$\ would fall at the blue end of the Ly${\beta}$\ trough.
Overall, however, the general lack of metal absorbers associated with this long dark gap may suggest that the gap probes a low-density region. This would favor the association of highly opaque sightlines with galaxy underdensities, as seen in imaging surveys for galaxies surrounding long Ly$\alpha$\ troughs  \citep[][]{becker_evidence_2018,kashino_evidence_2020,christenson_constraints_2021}. 
 

We examine the possible role of metal-enriched absorbers more broadly in Appendix~\ref{app:moreMetal}, finding little evidence for a strong correlation with long Ly${\beta}$\ troughs.  We also examined a sample of lower-redshift lines of sight in Appendix~\ref{app:lowz}, finding that metal-enriched absorbers in the foreground Ly$\alpha$\ alone are unlikely to create such a long dark gap.  We emphasize that this gap falls in redshift within a long Ly$\alpha$\ trough spanning $5.461\leq z \leq5.674$ with $L=68\cmpch$ that does not appear to be affected by metal absorbers (\citetalias{zhu_chasing_2021}).  This combination of factors gives us confidence that the $L=28\cmpch$ dark gap is genuinely caused by IGM opacity \footnote{In an extreme case where this foreground absorber links two shorter dark gaps, although very unlikely, one of these two shorter dark gaps would still have a size of $L\sim25\cmpch$.}.


\section{Models and Discussion\label{sec:models}}
\subsection{Models and Mock Spectra \label{sec:mock_spectra}}
Here we compare our results to predictions from models based on hydrodynamical simulations. 
We use the following models, which were also used in \citetalias{zhu_chasing_2021}: 
\begin{enumerate}
\item the {\tt homogeneous-UVB} model \add{from the Sherwood Simulation Suite} \citep[][]{bolton_sherwood_2017}\add{, which uses a homogeneous \citet{haardt_radiative_2012} UV background}; 
\item late reionization models wherein reionization ends at $z \lesssim 5.3$, including {\tt K20-low-{$\tau_{\rm CMB}$}}, {\tt K20-low-{$\tau_{\rm CMB}$}-hot}, {\tt K20-high-{$\tau_{\rm CMB}$}} models from \citet{keating_constraining_2020}, and {\tt ND20-late-longmfp}, {\tt ND20-late-shortmfp} models from \citet{nasir_observing_2020}; and 
\item an early reionization model wherein the IGM is mostly ionized by $z=6$ but large scale fluctuations in the UVB, which are amplified by a short mean free path of ionizing photons ($\lambda_{\rm mfp}^{912}=10\cmpch$ at $z=5.6$), persist down to lower redshifts ({\tt ND20-early-shortmfp}, \citealp{nasir_observing_2020}).   
\end{enumerate}
\add{These models were chosen, in part, because they reproduce at least some other observations of the Ly$\alpha$\ forest. The {\tt homogeneous-UVB} model agrees well with observations at $z<5$ including the IGM temperature and flux power spectra \citep{bolton_sherwood_2017}, although it fails to predict the Ly$\alpha$\ opacity distribution at $z>5$ \citep[e.g.,][]{bosman_hydrogen_2021}. The late reionization and fluctuating UVB models are broadly consistent with observations of IGM temperature, mean Ly$\alpha$\ transmission, and fluctuations in Ly$\alpha$\ opacity over the redshift range we are interested in \citep{keating_constraining_2020,nasir_observing_2020}. Moreover, these models are able to produce long Ly$\alpha$\ troughs at $z<6$ (e.g., \citetalias{zhu_chasing_2021}). We note that, nevertheless, that none of the models we use can self-consistently predict the mean free path evolution over $5<z<6$ as measured in \citet{becker_mean_2021}.}

\add{In the {\tt homogeneous-UVB} model, the IGM is instantaneously reionized at $z=15$. At $z<6$, therefore, the IGM in this model is fully ionized and the gas is hydrodynamically relaxed.  A homogeneous UVB model that produced a later reionization \citep[e.g.,][]{puchwein_consistent_2019,villasenor_inferring_2021} would mainly alter the temperature and pressure smoothing at $z < 6$.  These are small-scale effects, however, that should only minimally impact our measurements.  We would generally expect that any homogeneous UVB model that reionizes by $z = 6$ would produce similar dark gap statistics as the \citet[][]{haardt_radiative_2012} UVB once the ionization rates at $z < 6$ are rescaled to reproduce the observed mean flux.}

The {\tt K20-low-{$\tau_{\rm CMB}$}-hot} model shares a similar reionization history with the {\tt K20-low-{$\tau_{\rm CMB}$}} model, but it has a different thermal history with a volume-weighted mean temperature at the mean density at $z=6$ of $T_0=10,000$ K compared to that of the latter being 7000 K. The {\tt K20-high-{$\tau_{\rm CMB}$}} model has an earlier mid-point of reionization at $z=8.4$, which is at the upper end of the value suggested by CMB measurements \citep{planck_collaboration_planck_2020}. As for the late reionization models from \citet{nasir_observing_2020}, the main difference is that the {\tt ND20-late-shortmfp} model includes stronger post-reionization UVB fluctuations than the {\tt ND20-late-longmfp} model, while they both have neutral islands surviving at $z<6$. The mean free path of ionizing photons at $z=5.6$ in these two models are $\lambda_{\rm mfp}^{912}=10$ and 30 $\cmpch$, respectively. 

The box sizes we use in this work are $L=160$, 160, and 200 $\cmpch$, for simulations in \citet[][]{bolton_sherwood_2017}, \citet{keating_constraining_2020}, and \citet{nasir_observing_2020}, respectively. We note that the {\tt K20} models are from radiative transfer simulations run 
in post-processing and that the {\tt ND20} models are semi-numeric models. For more details on the models, see \citetalias{zhu_chasing_2021}.

\add{We rescale the optical depths in the simulations as needed in order to match the observed mean flux in the Ly$\alpha$\ forest \citep[see, e.g., \S2.2 in][and references therein]{bolton_sherwood_2017}.  We scale to the measurements of \cite{bosman_new_2018}, which are consistent with the mean Ly$\alpha$\ fluxes obtained from our sample.  The same rescaling factor is then applied to both the Ly$\alpha$\ and corresponding Ly${\beta}$\ optical depths.  We note that this rescaling mainly applies to the {\tt homogeneous-UVB} model, for which scaling by factors of $\sim$0.4$-$0.6 is required over $5<z<6$.  We are therefore explicitly testing only a {\tt homogeneous-UVB} model that also matches the observed mean Ly$\alpha$\ flux.}  The \citet{keating_constraining_2020} models already produce a mean Ly$\alpha$\ transmission consistent with the measurements of \citet{bosman_new_2018}.  The mock spectra from this simulation are continuous in redshift, with a smoothly evolving mean flux.   \add{\citet[][]{nasir_observing_2020} also calibrated their simulations to the observed $\tau_{\rm eff} $ from \citet[][]{bosman_new_2018} but provide one-dimensional skewers at discrete redshifts. For these simulations we therefore only need to rescale the optical depths such that the mock spectra described below have a mean flux that evolves smoothly with redshift.}

We derive dark gap predictions from forward-modeled mock spectra that are matched to the observed sample in QSO redshift, resolution, and S/N.  \citet{keating_constraining_2020} provide mock spectra of the Ly${\beta}$\ forest including the foreground Ly$\alpha$\ absorption.  \add{For the {\tt homogeneous-UVB} model and models from \citet{nasir_observing_2020}} we follow the methods described in \citetalias{zhu_chasing_2021} to build the mock Ly${\beta}$\ forest and foreground Ly$\alpha$\ forest.  In all cases we re-bin the mock spectra and apply the noise arrays according to each observed spectrum. 


\subsection{Model Comparisons}
\subsubsection{Comparisons of \texorpdfstring{$F_{10}$}{Lg}}

\begin{figure*}[htb!]
    \centering \includegraphics[width=6.6in]{F10.pdf}
    \caption{Fraction of QSO spectra showing long ($L\geq 10\cmpch$) Ly${\beta}$\ dark gaps as a function of redshift predicted by different models. The colored lines, dark-shaded regions, and light-shaded regions show the mean, 68\%, and 95\% limits of $F_{10}$ predicted by models based on 10,000 realizations. The black lines plot $F_{10}$ from the observations. For comparison, we also overplot the mean $F_{10}$ predicted by the {\tt ND20-early-shortmfp} in the panel of the {\tt ND20-late-longmfp} model with a gray line.
    }
    \label{fig:F10}
\end{figure*}

We compute the predicted $F_{10}$ of each model based on 10,000 randomly selected sets of mock spectra of the same size as the observed sample.  Their mean, 68\%, and 95 \% limits are plotted in Figure~\ref{fig:F10}, along with the observations. Similar to $F_{30}$ 
\footnote{$F_{30}$ is defined as the fraction of QSO spectra exhibiting gaps longer than $30\cmpch$ as a function of redshift.}
for the Ly$\alpha$\ forest in \citetalias{zhu_chasing_2021}, $F_{10}$ shows jagged features due to the combined effects of step changes in the number of sightlines with redshift and the quantization of $F_{10}$ for a finite sample size. We note that 68 and 95 percentile limits can share their upper and/or lower bounds at some redshifts, for the same reason. These features, on the other hand, show the constraining ability of the current sample size and data quality. The drop at $z \sim 5.9$ seen in the observed $F_{10}$ is also broadly included in the 95\% limits for most of the models. 

The {\tt homogeneous-UVB} model is not supported by the data at the $\geq95\%$ level. 
This is consistent with the conclusion based on the Ly$\alpha$\ forest in \citetalias{zhu_chasing_2021} that a fully ionized IGM with a homogeneous UVB scenario is disfavored by the data at $z<6$ down to $z\simeq5.3$. In contrast, the late reionization models are still consistent with the data, except for the {\tt K20-high-$\tau_{\rm CMB}$} model, which covers the observed $F_{10}$ just within its 95\% upper limit. This supports the conclusion of \citetalias{zhu_chasing_2021} that this extended reionization model is less favored by the data due to its insufficient neutral hydrogen and/or UVB fluctuations at $z<6$.

Our results further show that dark gaps in Ly${\beta}$\ are more sensitive probes of neutral regions than gaps in Ly$\alpha$.
For dark gaps in the Ly$\alpha$\ forest, we see little difference between the {\tt ND20-early-shortmfp} model and the {\tt ND20-late} models (\citetalias{zhu_chasing_2021}). In the Ly${\beta}$\ forest, however, the former predicts a smaller $F_{10}$ than the latter by $\sim 0.05$ at most redshifts. This difference is not enough for us to distinguish them based on the current sample, although the Ly${\beta}$\ gaps put some pressure on the early reionization model. \citet{nasir_observing_2020} note that these models are different in their Ly${\beta}$\ dark gap length distributions while they cannot be distinguished in the Ly$\alpha$\ forest. Nevertheless, we compute the differential dark gap length distribution for individual $\Delta L$ bins, $L\Delta P(L)/\Delta L$, and find their differences are minor compared to the scatter of the data. Looking ahead to the era of ELTs, we forecast that $\sim 100$ lines of sight with the Ly${\beta}$\ forest covering $z\sim 5.8$ would be needed to distinguish the two models at $\sim$95\% confidence based on $F_{10}$. A signal-to-noise ratio of 50 per $10\,\rm km\,s^{-1}$ for the spectra would be adequate according to our tests using mock spectra.

\subsubsection{Total Number of Long Dark Gaps at \texorpdfstring{$z \leq 5.8$}{Lg}}
\begin{figure*}[htb!]
    \begin{center}
    \gridline{
    \fig{ccount_10_zlt58.pdf}{2.6in}{(a)}
    \fig{ccount_10_zlt58gt56.pdf}{2.6in}{(b)}
    }
        \caption{
        \textbf{(a) }
        Number of long ($L\ge10\cmpch$) Ly${\beta}$\ dark gaps that lie entirely below $z=5.8$ in the mock sample from each model.
        \textbf{(b) }
        Number of long dark gaps entirely over $5.6\le z \le 5.8$. 
        In both panels the center lines, boxes, and error bars show the mean, 68\% limits, and 95\% limits, respectively. The observed numbers of long dark gaps in each redshift range are indicated by dashed green lines.
        \label{fig:ccount_10_zlt58}}
    \end{center}
\end{figure*}
To further illustrate the differences between models, we use our mock data to calculate the total number of long dark gaps predicted to lie entirely at $z < 5.8$. Figure~\ref{fig:ccount_10_zlt58} compares the model results to the observations. Given that the {\tt ND20} models only have outputs down to $z=5.6$, we exclude these models when counting the total number of long dark gaps below $z = 5.8$. We nevertheless include the {\tt ND20} models for dark gaps that fall entirely over $5.6 \le z \le 5.8$ for reference.

The results are consistent with those from $F_{10}$. As shown in Figure~\ref{fig:ccount_10_zlt58} (a), the 95\% upper limit of the predicted number of long dark gaps by the {\tt homogeneous-UVB} model is 2. This is a factor of 4 smaller than the observed value, which is 8.  The {\tt K20-high-$\tau_{\rm CMB}$} model is also disfavored by the data at $>95\%$ confidence given its deficit of long dark gaps. 
In contrast, the rapid late reionization models, i.e. {\tt K20-low-$\tau_{\rm CMB}$} models, agree with the observations within their $95\%$ limits. 

Over $5.6 \le z \le 5.8$ the observed number of long dark gaps decreases by one while the simulation predictions have little change. In this case, rapid late reionization models from \citet[][]{keating_constraining_2020} are still consistent with the data. The observations also support both fluctuating UVB and late reionization models from \citet[][]{nasir_observing_2020}. We note that the difference between the predicted mean numbers and the observed value is smallest for the {\tt ND20-late} models, wherein $ \langle x_{\rm HI} \rangle$ is still higher than $\sim5\%$ by $z=5.6$. On the other hand, the {\tt K20-high-$\tau_{\rm CMB}$} model is disfavored by the data also in this redshift range. 
This would suggest that very extended reionization scenarios in which insufficient neutral hydrogen and/or UVB fluctuations remain at $z<6$ may be disfavored.

\subsubsection{Detection Rate of an \texorpdfstring{$L\geq 28\cmpch$}{Lg} Dark Gap \label{sec:detection-rate}}

\begin{figure}[htb!]
\begin{center}
    %
    %
    %
    %
        %
    \includegraphics[width=0.45\textwidth]{detection-CDF.pdf}
    \caption{
    %
    Cumulative distribution of dark gaps that are entirely below $z=5.8$. Vertical colored lines indicate the 97.5 percentile limit for each model.
    %
    The color shaded regions plot the 68\% limit for each model. From left to right are the {\tt homogeneous-UVB} model, the {\tt K20-high-$\tau_{\rm CMB}$} model, and the {\tt K20-low-$\tau_{\rm CMB}$-hot} model (almost completely overlapped with the {\tt K20-low-$\tau_{\rm CMB}$} model), respectively.
    %
    \label{fig:detectionrate}}
    \end{center}
\end{figure}
Perhaps the most conspicuous feature in the observations is the $L=28\cmpch$ dark gap toward PSO J025$-$11 that extends down to $z \simeq 5.5$.  The appearance of this gap may indicate that significant neutral hydrogen islands and/or UVB fluctuations persist down to $z \simeq 5.5$, and provide further leverage for discriminating between models. As the outputs of the {\tt ND20} models have no redshift coverage for this dark gap, we only compare the {\tt K20} models and the {\tt homogeneous-UVB} model for this section. 

For each model, we use 10,000 bootstrapping realizations to calculate the cumulative distribution function (CDF) of dark gap length, $P(<L)$. Figure~\ref{fig:detectionrate} compares the observed and predicted $P(<L)$ for dark gaps that are entirely below $z=5.8$. As indicated by the vertical lines, the observed dark gap with $L=28\cmpch$ is well beyond the $95\%$ limits of all the models shown here. These results suggest that the $L=28\cmpch$ gap we observed in the Ly${\beta}$\ forest toward PSO J025$-$11 is extremely rare in these models. We perform Mann-Whitey U tests \citep[][]{mann_test_1947} for the hypotheses that the distributions of $L$ in the data and predicted by models are equal, for each model respectively. 
The hypothesis is rejected with $p$-values $<0.05$ for the {\tt homogeneous-UVB} model. 


We further calculate the detection rate of at least one $L=28\cmpch$ gap entirely below $z=5.61$ in the mock samples from each model with the required redshift coverage. We note that in the data there are 10 QSO spectra where the Ly${\beta}$\ forest (excluding the proximity zone) covers the full central redshift range of the $L=28\cmpch$ dark gap. We find zero detections of such long dark gaps in the {\tt homogeneous-UVB} model out of 10,000 trails. The {\tt K20-high-$\tau_{\rm CMB}$} model yields a detection rate of $4\%$. Both the {\tt K20-low-$\tau_{\rm CMB}$} and {\tt K20-low-$\tau_{\rm CMB}$-hot} models give higher detection rate of $10\%$. These results suggest that in the case of a late reionization, models with a volume-weighted average neutral hydrogen fraction $\langle x_{\rm HI}\rangle \gtrsim 5\% $ at $z=5.6$ are more consistent with the observations. In addition, the relatively rare presence of $L\geq 28 \cmpch$ gaps in the models could also be related to the size of the simulation volume ($160 \cmpch$ for the {\tt K20} simulations). Simulations run in larger volumes may be needed to compute more accurate statistics on the incidence of these rare, long troughs in late reionization models. %
 

\subsection{Neutral Hydrogen Fraction \label{sec:fHI}}


\begin{figure*}
    \centering
    \gridline{
        \fig{xHIvsL.pdf}{3in}{(a)}
        \fig{xHIdist.pdf}{3.1in}{(b)}
    }
    \caption{(a) The mean volume-weighted neutral fraction ($ \langle x_{\rm HI} \rangle $) over a Ly${\beta}$\ dark gap for a given dark gap length in the mock data. In this figure, ``ND20'' and ``K20'' refer to the {\tt ND20-late-longmfp} model and the {\tt K20-low-$\tau_{\rm CMB}$} model, respectively.
    (b) Distribution of $ \langle x_{\rm HI} \rangle $ for a given Ly${\beta}$\ dark gap length, $f(x_{\rm HI}  | L)$, in the {\tt ND20-late-longmfp} model at $z=5.6$. $f(x_{\rm HI}  | L)$ is normalized for each $L$ interval.
    }
    \label{fig:xHIvsL}
\end{figure*}



\begin{figure}
    \centering
    \includegraphics[width=3.3in]{FHI_sum.pdf}
    \caption{Inference on the neutral hydrogen fraction ($\langle x_{\rm HI} \rangle$) from Ly${\beta}$\ dark gaps, which are also dark in the Ly$\alpha$\ forest by definition. We show constraints based on Ly$\alpha$\ dark gaps from \citetalias{zhu_chasing_2021} with red error bars, which are shifted by $-0.02$ in redshift for display. Gray markers plot the constraints based of fraction of pixels that are dark in both the Ly$\alpha$\ and Ly${\beta}$\ forests from \citet[][]{mcgreer_model-independent_2015}. The vertical error bars show the $68\%$ ($1\sigma$) limits. The horizontal error bars indicate the $\Delta z=0.2$ redshift bins. \add{For reference, colored lines plot the redshift evolution of $\langle x_{\rm HI} \rangle$ for the reionization models used in this work.}}
    \label{fig:fHI}
\end{figure}

We can further use dark gaps to infer an upper limit on $\langle x_{\rm HI} \rangle$.  One can set a strict upper limit on the neutral fraction by assuming that all dark gaps correspond to neutral gas \citep[e.g.,][]{mcgreer_first_2011,mcgreer_model-independent_2015}.  At the end of reionization, however, a combination of density and UVB fluctuations will tend to produce dark gaps even once the gas is ionized.  We therefore wish to use insights from reionization models to derive a more physically motivated but still conservative upper limit on $\langle x_{\rm HI} \rangle$ from the covering fraction of dark gaps.  As described below, we use the fact that dark gaps in the late reionization models tend to show a correlation between the volume-averaged neutral fraction within a gap and the gap length.  By applying this relationship to the observed gap length distribution we can set constraints on $\langle x_{\rm HI} \rangle$.


Our goal is to set physically reasonable constraints on $\langle x_{\rm HI} \rangle$ while minimizing the model dependency. We therefore wish to identify the maximum average neutral fraction for a given gap length that is allowed by the models.  We first explore the distribution of neutral fractions for a given dark gap length, $f(x_{\rm HI}  | L)$. %
We focus on two models wherein neutral regions contribute significantly to forming dark gaps, the {\tt ND20-late-longmfp} model and the {\tt K20-low-$\tau_{\rm CMB}$} model. Using the mock data, we calculate $x_{\rm HI}$ for each dark gap by averaging the neutral fraction pixel-wise. Figure~\ref{fig:xHIvsL} (a) plots the mean neutral fraction of dark gaps as a function of length, $ \langle x_{\rm HI}\rangle _L $, at different redshifts.  It is related to $f(x_{\rm HI}  | L)$ by
\begin{equation}
    \langle x_{\rm HI} \rangle _L = \int_0^1 x_{\rm HI} f(x_{\rm HI} | L)  d x_{\rm HI}.
    \label{eq:<xHI>}
\end{equation}
As shown in the figure, dark gaps of a given length tend to be more neutral as redshift decreases. 
This is largely because the opacity of the ionized IGM tends to decrease with decreasing redshift, making it more difficult to produce long gaps through density and/or UVB fluctuations alone. 
In order to set conservative upper limits of $\langle x_{\rm HI} \rangle$ we adopt the $\langle x_{\rm HI} \rangle_L$ relationship from {\tt ND20-late-longmfp} at $z = 5.6$. The normalized $f(x_{\rm HI}  | L)$ for each dark gap length interval is plotted in Figure~\ref{fig:xHIvsL} (b).  This is similar to but slightly higher than the relationship from {\tt K20-low-$\tau_{\rm CMB}$} at the same redshift.  We also note that the redshift evolution in $\langle x_{\rm HI} \rangle_L$ in these models is relatively modest, up to a factor of $\sim$2 in the {\tt K20-low-$\tau_{\rm CMB}$} model between $z \sim 6$ and 5.6.


In order to translate the observed gap length distribution into a $\langle x_{\rm HI} \rangle$ constraint, we calculate $\mathscr{F}_L$, the fraction of QSO spectra showing dark gaps with length $L$ as a function of redshift. At a certain redshift, the total mean neutral hydrogen fraction is then given by
\begin{equation}
    \langle x_{\rm HI} \rangle = \sum_{L=1}^{\infty} \mathscr{F}_L \langle x_{\rm HI} \rangle _L.
    \label{eq:fHI}
\end{equation}
Here we use a sum for $L$ instead of an integral because we measure dark gap lengths in increments of $1\cmpch$. 
To estimate the uncertainty in $\langle x_{\rm HI} \rangle$, we randomly select the observed sightlines with replacement and calculate the corresponding $\mathscr{F}_L$. We use bootstrapping to randomly sample the neutral hydrogen fraction from $f( x_{\rm HI} | L)$ given by models and multiply by the observed $\mathscr{F}_L$ of this sample, then sum up for all dark gap lengths. The final uncertainty in $\langle x_{\rm HI} \rangle$ is calculated by repeating this process 10,000 times.

The results are shown in Figure~\ref{fig:fHI}. We calculate $\langle x_{\rm HI} \rangle$ in \add{Equation} (\ref{eq:fHI}) over $\Delta z=0.2$ bins. The inferred upper limits on $\langle x_{\rm HI} \rangle$ are $0.05_{-0.04}^{+0.04}$, $0.17_{-0.05}^{+0.05}$, and $0.29_{-0.10}^{+0.09}$ at $z\simeq 5.55$, 5.75, and 5.95, respectively. We also calculate $\langle x_{\rm HI} \rangle$ following the same method based on Ly$\alpha$\ dark gaps presented in \citetalias{zhu_chasing_2021},  as shown with red symbols in Figure~\ref{fig:fHI}. 
The Ly$\alpha$\ dark gaps yield $\langle x_{\rm HI} \rangle \leq$ $0.05$, $0.17$, and $0.26$ at $z\simeq 5.55$, 5.75, and 5.95, respectively. 
The measurements based on Ly$\alpha$\ and Ly${\beta}$\ dark gaps are highly consistent with each other. Compared to the measurements based the fraction of dark pixels by \citet{mcgreer_model-independent_2015}, our results potentially allow a higher neutral fraction over $5.6 \lesssim z \lesssim 6.0$ and a later reionization. The difference in $\langle x_{\rm HI} \rangle$ might be due to cosmic variance and/or the different definitions of dark gaps and dark pixels used in these works. The $\langle x_{\rm HI} \rangle$ measurement at $z\sim 5.9$ in \citet[][]{mcgreer_model-independent_2015}, moreover, may be biased by transmission peaks in the QSO proximity zone given that their wavelength range for both the Ly$\alpha$\ and Ly${\beta}$\ forests ends at $z_{\rm QSO}-0.1$, which is less than 6.5 pMpc from the QSO at $z\sim 6$ (see proximity zone size measurements in e.g., \citealp{eilers_implications_2017,eilers_detecting_2020}). 



\section{Conclusion \label{sec:conslusion}}
In this work, we explore the IGM near the end of reionization using dark gaps in the Ly${\beta}$\ forest over $5.5\lesssim z \lesssim 6.0$. We show that about 10\%, 40\%, and 80\% of QSO spectra exhibit long ($L\geq 10\cmpch$) dark gaps in their Ly${\beta}$\ forest at $z\simeq5.6$, 5.8, and 6.0, respectively. Among these gaps, we detect a very long ($L=28\cmpch$) and dark ($\tau_{\rm eff} \gtrsim6$) Ly${\beta}$\ gap extending down to $z\sim 5.5$ toward the $z_{\rm em}=5.85$ QSO PSO J025$-$11. 

A comparison between the observed Ly${\beta}$\ dark gap statistics for the whole sample of 42 lines of sight and predictions from multiple reionization models \citep[][]{bolton_sherwood_2017,keating_constraining_2020,nasir_observing_2020} confirms that evidence of reionization in the form of neutral islands and/or a fluctuating UV background persists down to at least $z\sim 5.5$. This supports the conclusions in \citetalias{zhu_chasing_2021} and \citet{bosman_hydrogen_2021}.  In \citetalias{zhu_chasing_2021} we noted a possible tension between Ly$\alpha$\ gap statistics and a model wherein reionization ends by $z<6$ but has a relatively early mid-point of $z=8.4$ \citep{keating_constraining_2020}.  With Ly${\beta}$\ this tension becomes more significant ($>95\%$ level) based on the count of long dark gaps at $z\le5.8$, suggesting that very extended reionization scenarios with insufficient remaining neutral hydrogen and/or UVB fluctuations at $z<6$ may be disfavored. In contrast, rapid late reionization models with $\langle x_{\rm HI} \rangle \gtrsim 5 \%$ at $z=5.6$ \citep[][]{keating_constraining_2020,nasir_observing_2020} are consistent with the observations. A model wherein reionization ends early but retains large-scale fluctuations in the ionizing UV background \citep[][]{nasir_observing_2020} is also permitted by the dark gap data. We note, however, that recent IGM temperature measurements from \citet{gaikwad_probing_2020} disfavor this model.

\add{
A caveat is that we are testing only specific reionization models, including only one with a fluctuating UVB in which reionization ends at $z > 6$.  By comparison, \citet{gnedin_cosmic_2017} showed that their full radiative transfer simulations, which reionized near $z \sim 7$, were able to reproduce the Ly$\alpha$\ dark gap distribution measured from ESI spectra of a set of twelve $z \sim 6$ QSOs. Because Ly${\beta}$\ dark gaps are correlated with Ly$\alpha$\ opacities (Appendix \ref{app:Lyb_Lya}), it is possible that some early reionization scenarios with UVB fluctuations can reproduce our Ly${\beta}$\ dark gap distributions while also matching the observed evolution of the mean Ly$\alpha$\ transmission. 
}

Finally, we use the observed Ly${\beta}$\ gaps to place constraints on the neutral hydrogen fraction based on the association between neutral islands and dark gaps seen in reionization simulations. Our results are broadly consistent with, but more permissive than the constraints from \citet{mcgreer_first_2011,mcgreer_model-independent_2015} that are based on the dark pixel fraction.  Notably, we find an upper limit at $z\simeq5.75$ of $\langle x_{\rm HI} \rangle \leq 0.17$.  This constraint is consistent with scenarios wherein reionization extends significantly below $z = 6$. 


\begin{acknowledgments}

    We thank Elisa Boera and Fahad Nasir for their help with simulated data and for useful discussions. \add{We also thank the anonymous referee for their constructive comments. }

    Y.Z., G.D.B., and H.M.C. were supported by the National Science Foundation through grants AST-1615814 and AST-1751404. H.M.C. was also supported by the National Science Foundation Graduate Research Fellowship Program under grant No. DGE-1326120.
    S.E.I.B. and F. Walter acknowledge funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement No. 740246 ``Cosmic Gas'').
    L.C.K. was supported by the European Union's Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No. 885990.
    M.B. acknowledges support from PRIN MIUR project ``Black Hole winds and the Baryon Life Cycle of Galaxies: the stone-guest at the galaxy evolution supper'', contract $\#$2017PH3WAT.
    F.B. acknowledges support from the Australian Research Council through Discovery Projects (award DP190100252) and Chinese Academy of Sciences (CAS) through a China-Chile Joint Research Fund (CCJRF1809) administered by the CAS South America Center for Astronomy (CASSACA).
    H.C. thanks the support by NASA through the NASA FINESST grant NNH19ZDA005K.
    A.-C.E. acknowledges support by NASA through the NASA Hubble Fellowship grant $\#$HF2-51434 awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS5-26555. 
    X.F. and J.Y. acknowledge support from the NSF grants AST 15-15115 and AST 19-08284. 
    M.G.H. acknowledges support from the UKRI STFC (grant Nos. ST/N000927/1 and ST/S000623/1).  
    G.K.'s research is partly supported by the Max Planck Society via a partner group grant.
    A.P. acknowledges support from the ERC Advanced Grant INTERSTELLAR H2020/740120.
    %
    Parts of this work was supported by the Australian Research Council Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), through project $\#$CE170100013.
    %
    F. Wang thanks the support provided by NASA through the NASA Hubble Fellowship grant \#HST-HF2-51448.001-A awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Incorporated, under NASA contract NAS5-26555.
    
    Based on observations collected at the European Southern Observatory under ESO programmes 060.A-9024(A), 084.A-0360(A), 084.A-0390(A), 084.A-0550(A), 085.A-0299(A), 086.A-0162(A),  
    086.A-0574(A), 087.A-0607(A), 088.A-0897(A), 091.C-0934(B), 096.A-0095(A), 096.A-0418(A), 
    097.B-1070(A), 098.A-0111(A), 098.B-0537(A), 0100.A-0243(A), 0100.A-0625(A), 0101.B-0272(A), 0102.A-0154(A), 0102.A-0478(A), 1103.A-0817(A), and 1103.A-0817(B).
    
    Some of the data presented herein were obtained at the W. M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W. M. Keck Foundation.
    The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Maunakea has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain. Finally, this research has made use of the Keck Observatory Archive (KOA), which is operated by the W.M. Keck Observatory and the NASA Exoplanet Science Institute (NExScI), under contract with the National Aeronautics and Space Administration.
    
    This work was performed using the Cambridge Service for Data Driven Discovery (CSD3), part of which is operated by the University of Cambridge Research Computing on behalf of the STFC DiRAC HPC Facility (\url{www.dirac.ac.uk}). The DiRAC component of CSD3 was funded by BEIS capital funding via STFC capital grants ST/P002307/1 and ST/R002452/1 and STFC operations grant ST/R00689X/1. This work further used the DiRAC@Durham facility managed by the Institute for Computational Cosmology on behalf of the STFC DiRAC HPC Facility. The equipment was funded by BEIS capital funding via STFC capital grants ST/P002293/1 and ST/R002371/1, Durham University and STFC operations grant ST/R000832/1. DiRAC is part of the National e-Infrastructure. 


\facilities{Keck:II (ESI), VLT:Kueyen (X-Shooter)}

\software{
    {\tt Astrocook} \citep[][]{cupani_astrocook_2020}, 
    {\tt Astropy} \citep{astropy_collaboration_astropy_2013},
    {\tt Matplotlib} \citep{hunter_matplotlib_2007},
    {\tt NumPy} \citep{van_der_walt_numpy_2011},
    {\tt SpectRes} \citep{carnall_spectres_2017}
}
\end{acknowledgments}


