\section{Introduction and comparison of bounds on the effective noise coupling}


There is increasing interest in testing wave function collapse models \cite{collapse}, and in particular the continuous spontaneous localization (CSL) model,  by searching for effects associated with the small noise which drives wave function collapse when nonlinearly coupled in the Schr\"odinger
equation.  The original proposals for the noise coupling strength were so small that devising
suitable experiments was problematic, but the situation has changed with the suggestion \cite{adler1}   that latent image formation, such as deposition of a developable track in an emulsion or in an etched track detector, already constitutes a measurement embodying wave function collapse and so requires an enhanced noise coupling. A recent
cantilever experiment of Vinante et al. \cite{vin}  has set bounds consistent with the  parameters suggested in \cite{adler1}, and reports a possible noise signal. Thus, it is timely
to consider further experiments  \cite{alternative} which could detect or rule out a noise coupling with
the strength suggested by \cite{vin}.


For a body comprised of a group of particles of total mass $M$, the center-of-mass energy gain for white noise with mass-proportional coupling is given by the standard formula \cite{gainref}
\begin{equation}\label{engain}
\frac{dE}{dt}=\frac{3}{4}\lambda \frac{\hbar^2}{r_C^2} \frac{M}{m_N^2}~~~,
\end{equation}
with $m_N$ the nucleon mass and $\lambda$ the coupling parameter for white noise with noise correlation length $r_C$ and no frequency spectrum cutoff.  Dividing by $M$, Eq. \eqref{engain}
can be rewritten as a formula for the energy gain rate per unit mass,
\begin{equation}\label{engain1}
\frac{dE}{dt\,dM}=\frac{3}{4}\lambda \frac{\hbar^2}{r_C^2} \frac{1}{m_N^2}~~~.
\end{equation}
For the noise coupling parameter $\lambda=10^{-7.7} {\rm s}^{-1}$ suggested in \cite{vin},  and the conventional value $r_C=10^{-5} {\rm cm}$,
Eq. \eqref{engain1} corresponds to
\begin{equation}\label{engain3}
\frac{dE}{dt\,dM}\simeq 40\frac{ {\rm MeV}}{{\rm g ~s}} \simeq 0.64  \times 10^{-8}  \frac{{\rm W}}{{\rm kg}}~~~.
\end{equation}
Since heating rates of $100 {\rm pW}/{\rm kg}= 10^{-10} {\rm W}/{\rm kg}$ are attained in low temperature experiments \cite{pobell}, with the limit accounted for by modeling
energy deposition from radioactive decays and penetrating muons \cite{nazar}, the residual heating from unknown sources is limited to roughly $10^{-11} {\rm W}/{\rm kg}$.
This means that the effective $\lambda$ for bulk heating is at most $3.1 \times 10^{-11} {\rm s}^{-1}$, ruling out a collapse model white noise interpretation of the
excess noise reported in \cite{vin}.

A similar bound on the bulk heating rate is given by Earth's energy balance.  Table 6.3 of de Pater and Lissauer \cite{depater} gives the luminosity to mass ratio for solar system objects,
with a value $6.4 \times 10^{-12} {\rm W}/{\rm kg}$ for Earth.  Estimates of primordial and radiogenic sources of Earth heat roughly account for this, but have
uncertainties that could allow   $\sim 3\times 10^{-12} {\rm W} /{\rm kg}$ to come from unknown heating sources.  This places a limit of $ \sim 10^{-11} {\rm s}^{-1}$ on the
effective $\lambda$ for bulk heating.   There is a caveat here, because
as noted in \cite{adler1}, when the effects of dissipation are included, as in the model of Bassi, Ippoliti, and Vacchini \cite{ippoliti}, the rate of heat production can vanish at large times where a limiting temperature is reached. For example, with the parameters of \cite{ippoliti},  a limiting temperature of $0.1$ K is reached on a time scale of billions of years, giving a current noise-induced Earth  heat production rate smaller than given by Eq. \eqref{engain1}, permitting a larger effective $\lambda$.

The effective CSL model $\lambda$ for bulk heating experiments can be strongly reduced if the noise is non-white, with a power spectrum cutoff.  A spectral energy cutoff is already suggested by experimental limits on spontaneous gamma ray emission from germanium \cite{germ}, which shows that the noise strength suggested in \cite{adler1} is ruled out unless
the noise power spectrum cuts off at an angular frequency below $\sim 15 \,{\rm keV}/\hbar \sim 2 \times 10^{19} {\rm s}^{-1}$.  For bulk heating of solids the noise
couples through longitudinal acoustic phonon excitation, and with $r_c\sim 10^{-5} {\rm cm}$,  heating  takes place only if the noise has frequency components of at least $\omega_L(|\vec q|=r_c^{-1} )\sim v_s |\vec q| \sim 0.4 \times 10^{11} {\rm s}^{-1}$, with $\omega_L(\vec q)$ the longitudinal phonon frequency at wave number $\vec q$, and $v_s$ the speed of sound (which for this estimate we have taken as $4000 \, {\rm m}/{\rm s}$ characteristic of copper at low temperature).  A  quantitative calculation given in
the Appendix  shows that for non-white noise with power spectrum $\lambda(\omega)$ heating a solid by phonon excitation, Eq. \eqref{engain1} is replaced by
\begin{equation}\label{engain4}
\frac{dE}{dt\,dM}=\frac{3}{4}\lambda_{\rm eff} \frac{\hbar^2}{r_C^2} \frac{1}{m_N^2}~~~,
\end{equation}
with $\lambda_{\rm eff}$ given by
\begin{equation}\label{lameff}
\lambda_{\rm eff}= \frac{2}{3 \pi^{3/2}} \int d^3w e^{-\vec w^2} \vec w^2 \lambda(\omega_L(\vec w/r_c))~~~.
\end{equation}
When $\lambda(\omega)$ is a constant independent of $\omega$, Eq. \eqref{lameff} reduces to $\lambda_{\rm eff}=\lambda$, and Eq. \eqref{engain1} is recovered, but when
there is a frequency cutoff below the phonon excitation frequency, the effective noise coupling is strongly reduced. Thus, if the noise reported in \cite{vin} at the
very low cantilever frequency of $8174 {\rm s}^{-1}$ were due to CSL,
 it would be  further evidence for non-white CSL model noise.

\section{Alternative experiments for bounding $\lambda_{\rm eff}$}
Assuming now a  maximum value $\lambda_{\rm eff}\sim  10^{-11} {\rm s}^{-1}$ consistent with low temperature experiments and non-dissipative Earth heating, let us explore possible
alternative experiments for detecting or further bounding $\lambda_{\rm eff}$.  {\it Feasibility of such experiments assumes (a) that the background heating rate from cosmic ray muons and radioactive decays can be reduced much below $3 \times 10^{-12} {\rm W}/{\rm kg}$ by shielding, underground operation, and careful choice of materials, and (b) all other known sources of heat leaks in ultralow temperature experiments, such as vibrations, relaxation from two-level systems, and hydrogen ortho-para conversion, have been suppressed.} To present estimates we multiply
$3 \times 10^{-12} {\rm W}/{\rm kg}$  by  the solid density $\rho$, so that the maximum allowed  value of $\lambda_{\rm eff}$ corresponds to a volume heating rate of
\begin{equation}\label{heating}
H=3\times 10^{-15} \rho {\rm W}/{\rm cm}^3 ~~~,
\end{equation}
with $\rho$ the density in units ${\rm g}/{\rm cm}^3$.  We consider two geometries for
which the heat transport problem is effectively one dimensional and easily solved.

\begin{enumerate}
\item  For a sphere of radius $R$ cm and density $\rho$, the total heating rate  is $4\pi R^3 H/3$, and so the steady state rate of escape of heat from unit area
of the surface will be
\begin{equation}\label{spherheat}
\dot Q_{\rm sphere}= R H/3
\end{equation}
in units ${\rm W}/{\rm cm}^2$.  This must balance the rate of transport of heat per unit area from the surface of the sphere, at temperature $T_1$, to the surrounding cryostat surfaces, at temperature $T_2$.  This is given \cite{pobell} in units ${\rm W}/{\rm cm}^2$ by the formula
\begin{equation}\label{transport}
\dot{Q}_{\rm transport} = 5.67 \times 10^{-12}\epsilon  (T_1^4-T_2^4)[K]^4+ 0.02 a P[{\rm mbar}] (T_1-T_2)[K]~~~,
\end{equation}
with the first term the Stefan-Boltzmann equation for radiative heat transfer, and the second term, which is linear in the pressure $P$,  coming from gas particle conduction in the cryostat.  Here  $a \leq 1$ is an ``accommodation coefficient'' for gas particles on the cryostat walls, which can be as small as 0.02 for a clean metal surface in contact with helium gas, and $\epsilon \leq 1$ is the emissivity for radiative transfer.   The steady state  surface temperature of the sphere $T_1$ is determined by equating $\dot{Q}_{\rm sphere}$ to $\dot{Q}_{\rm transport}$.
Taking as an example
the density of lead $\rho = 11.4 \,{\rm g}/{\rm cm}^3$, and a sphere of radius $R=50\, {\rm cm}$ (which would fit in the CUORE underground experiment cryostat \cite{cuore}) , we have
\begin{equation}\label{sphere1}
\dot{Q}_{\rm sphere}=5.7 \times 10^{-13}~~~,
\end{equation}
while taking $a=0.02$ and $P=10^{-6}{\rm mbar}$ gives
\begin{equation}\label{transp}
\dot{Q}_{\rm transport} = 5.67 \times 10^{-24} \epsilon (T_1^4-T_2^4)[{\rm mK}]^4+4 \times 10^{-13} (T_1-T_2) [{\rm mK}]~~~,
\end{equation}
both in units $ {\rm W}/{\rm cm}^2$. Evidently, for millikelvin $T_1$ and $T_2$  radiative heat transfer is completely negligible and Eqs. \eqref{sphere1} and \eqref{transp} are of similar size at the relatively high pressure of $10^{-6}$ mbar.
 In real millikelvin cryostats the actual residual pressure is typically orders of magnitude lower than $10^{-6}$ mbar because the vapor pressure of anything, including helium, drops to zero exponentially with decreasing temperature.
 The  heating rate H can therefore be estimated by measuring the steady state surface temperature as a function of the gas pressure.  Fast and sensitive measurements of temperature in the range of a few mK can be readily done using resistive probes. For high accuracy, SQUID-based noise thermometry has been recently demonstrated to be very effective down to 42 $\mu$K \cite{roth}.

Similar reasoning gives the temperature distribution inside a sphere of material with thermal conductivity $k(T)$.  At a given distance $r$ from the center of the
sphere, the energy transport rate through the spherical surface of radius r is equal to
\begin{equation}\label{spherout}
E_{\rm out}= -4 \pi r^2 k(T) \frac{dT}{dr}~~~,
\end{equation}
which in steady state  must balance the heating rate of the volume within radius $r$,
\begin{equation}\label{spherin}
E_{\rm in}=\frac{4 \pi}{3} r^3 H~~~,
\end{equation}
giving the differential equation
\begin{equation}\label{diffeq}
-k(T)\frac{dT}{dr}=\frac{1}{3} r H~~~.
\end{equation}
Integrating from the center of the sphere at radius 0  to radius $R$, with respective temperatures $T_c$ and $T_1$, this gives
\begin{equation}\label{int}
-\int_{T_c}^{T_1} \, k(u)du= \frac{R^2 H}{6}~~~.
\end{equation}
For $k(u)=\hat k_0 u^{\beta}$,
this becomes
\begin{equation}\label{int1}
-\frac{\hat k_0}{1+\beta} (T_1^{1+\beta}-T_c^{1+\beta})=\frac{R^2 H}{6}~~~,
\end{equation}
 which gives
\begin{equation}\label{intt1}
 (T_c^{1+\beta}-T_1^{1+\beta})^{1/(1+\beta)}=\left[\frac{1+\beta}{\hat k_0K^{1+\beta}} \frac{R^2 H}{6}\right]^{1/(1+\beta)}[{\rm K}]~~~.
\end{equation}
 To give a numerical estimate, for the good thermal insulator Torlon 4203 \cite{torlon}, with density $1.42~ {\rm g~cm^{-3}}$ and with $k(T)=6.13 \times 10^{-3} (T/{\rm K})^{2.18} {\rm W/(m~K)}$, so that $\beta=2.18$ and $\hat k_0 {\rm K}^{3.18}= 6.13 \times 10^{-3} {\rm W}/{\rm m}$, the right hand side of  Eq. \eqref{intt1} for $R=50 \,{\rm cm}$  is 6.1 mK.
 \medskip

We have used Torlon as a convenient example for estimates, but probably it would not be the most suitable material for these  experiments. As discussed by Pobell \cite{pobell}, amorphous materials are usually rich in two-level systems leading to slow relaxation processes with time-dependent heat release. Moreover, polymers and plastics like Torlon may easily absorb impurities which can also give unreliable thermal properties.  For realistic
 experiments, it would be better to use crystalline insulators (such as sapphire or silicon) or superconducting metals. In both cases the thermal conductivity is much higher than that of Torlon at $T>100~ {\rm mK}$, but drops as $T^3$ and approaches  the conductivity of plastic materials like Torlon in the mK range.

Concerning the feasibility of this experiment, we note that spheres of meter size have already been suspended and cooled down to millikelvin temperature. An example is Minigrail, a CuAl spherical gravitational wave detector, which  was suspended through a central rod suspension and cooled in vacuum  down to 60 mK within a few days time \cite{waard}. In that case, the dominant residual heating mechanism was likely relaxation of defects and hydrogen ortho-para conversion. Both issues can be avoided in a dedicated experiment by a suitable choice of materials.


\item  Another geometry with easily solvable heat transfer  would use a long cylinder (or parallelepiped, or more generally a rod of uniform cross section) of length $L$  with the ``near'' end fastened to a heat sink that provides  a large enough heat transport rate so
that the heat transport from all other surfaces given by Eq. \eqref{transport} can be ignored.  Then the heat transport problem is one dimensional, and the analog of Eq.
\eqref{diffeq} is the differential equation
\begin{equation}\label{zdiffeq}
k(T)\frac{dT}{dz}=(L-z)H~~~.
\end{equation}
Again taking $k(u)=\hat{k}_0u^{\beta}$ and integrating,  the analog of
Eq. \eqref{intt1} relating the ``far'' to the ``near'' end temperatures at $z=L$ and
$z=0$ respectively is
\begin{equation}\label{intt2}
 (T_{\rm far}^{1+\beta}-T_{\rm near}^{1+\beta})^{1/(1+\beta)}=\left[\frac{1+\beta}{\hat k_0K^{1+\beta}} \frac{L^2 H}{2}\right]^{1/(1+\beta)}[{\rm K}]~~~.
\end{equation}
For a rod  with the parameters quoted above, and $L=50\,{\rm cm}$, the right hand side of Eq. \eqref{intt2} is 8.6 mK.

In a  variant of the rod geometry, one can attach to
the ``far'' end of the rod an object of larger size,  which acts as a ``CSL noise absorber''. The CSL heat released in the absorber can be much larger than that in the rod.   The temperature at the ``far'' end of the rod will then be determined by
matching the heat flow per unit area within the rod with the heat flow per unit area $\dot{Q}_{\rm ABS}$ entering the rod from the absorber, so that Eq. \eqref{zdiffeq} becomes
\begin{equation}\label{zdiffeq1}
k(T)\frac{dT}{dz}=(L-z)H+\dot{Q}_{\rm ABS}~~~,
\end{equation}
which on integration gives
\begin{equation}\label{intt3}
 (T_{\rm far}^{1+\beta}-T_{\rm near}^{1+\beta})^{1/(1+\beta)}=\left[\frac{1+\beta}{\hat k_0K^{1+\beta}}\left( \frac{L^2 H}{2}+\dot{Q}_{\rm ABS}L \right)\right]^{1/(1+\beta)}[{\rm K}]~~~.
\end{equation}
  This provides the freedom of independently tuning the rod thermal conductivity (by making the cross section arbitrarily small) and the heat input from CSL (by choosing the  material and size of the CSL absorber). From an experimental point of view this gives a very flexible design. Of course, one must make sure that the heat transport mechanisms of Eq. \eqref{transport} from the surfaces of both the rod and the absorber are kept negligible, as discussed above following Eq. \eqref{transp}.

\end{enumerate}

To conclude, current bounds on the effective noise coupling $\lambda_{\rm eff}$ for bulk heating of solids show that in designing experiments to test the enhanced rate \cite{adler1}, \cite{vin} that makes latent image formation a measurement, it will be important to take the power spectrum of the noise into account.  Because thermal transport rates vanish at zero temperature, millikelvin and submillikelvin experiments to further improve the bounds on $\lambda_{\rm eff}$ by one or two orders of magnitude may be feasible.  However, underground operation is probably necessary in order to evade the limiting heating rate from cosmic rays.
\section{Acknowledgements}
We wish to thank Angelo Bassi for helpful comments.  A.V. acknowledges support from
EU FET project TEQ (grant agreement 766900). The authors thank the referee for a careful reading of the paper and helpful comments.

The appendix (except for the addition of the paragraph containing
Eq. \eqref{lambdaest}) is based on the
preprint \cite{adler2} by one of the authors (SLA).  The other author
(AV) later noted a preprint by M. Bahrami \cite{bahrami}  in which the calculation is done by a non-perturbative
method.


