aainstitutetext: Department of Physics, Sam Houston State University,
Huntsville, TX 77341, USAbbinstitutetext: Mitchell Institute for Fundamental
Physics and Astronomy, Department of Physics and Astronomy, Texas A$\&$M
University, College Station, Texas 77843, USAccinstitutetext: School of
Physics, Korea Institute for Advanced Study, Seoul 02455,
Koreaddinstitutetext: Department of Physics and Astronomy, University of
Hawaii, Honolulu, Hawaii 96822, USA

# Sensitivity to Dark Sector Scales from Gravitational Wave Signatures

James B. Dent b Bhaskar Dutta b,c Sumit Ghosh d Jason Kumar d and Jack Runburg
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We consider gravitational sound wave signals produced by a first-order phase
transition in a theory with a generic renormalizable thermal effective
potential of power law form. We find the frequency and amplitude of the
gravitational wave signal can be related in a straightforward manner to the
parameters of the thermal effective potential. This leads to a general
conclusion; if the mass of the dark Higgs is less than 1% of the dark Higgs
vacuum expectation value, then the gravitational wave signal will be
unobservable at all upcoming and planned gravitational wave observatories.
Although the understanding of gravitational wave production at cosmological
phase transitions is still evolving, we expect this result to be robust.

###### Keywords:

Gravitational Wave, Cosmological phase transition, Dark Higgs

††preprint: MI-HET-774, KIAS-P22016††arxiv: 2203.11736

##  1 Introduction

With the advent of gravitational wave (GW) detection LIGOScientific:2016aoc ,
and the development of new instruments (see Table 1, and references therein),
gravitational wave signals have become among the most promising signatures for
new physics beyond the Standard Model (SM) in the early Universe. In
particular, cosmological first order phase transitions in the early Universe
can generate gravitational wave signatures Kosowsky:1992rz ; Kosowsky:1991ua ;
Apreda:2001us ; Grojean:2006bp ; Huber:2008hg  which can be observed at
current and upcoming experiments (for reviews, see Cai:2017cbj ; Weir:2017wfa
; Caprini:2015zlo ; Mazumdar:2018dfl ; Caprini:2018mtu ; Caprini:2019egz ;
Caldwell:2022qsj ). Considerable work has focused on determining if particular
models can produce GW signals which are observable at upcoming experiments.
There has been a recent flurry of works examining extensions of the Standard
Model which could provide gravitational wave signatures Ashoorioon:2009nf ;
Alves:2018jsw ; Balazs:2016tbi ; Kang:2017mkl ; Matsui:2017ggm ;
Vaskonen:2016yiu ; Lewicki:2021pgr ; Croon:2018new ; Beniwal:2018hyi ;
Hashino:2018wee ; Ahriche:2018rao ; Shajiee:2018jdq ; Vieu:2018nfq ;
Alves:2018oct ; Chala:2018opy ; Fujikura:2018duw ; Graf:2021xku ; Li:2020eun ;
Ghorbani:2017lyk ; Marzo:2018nov ; Marzola:2017jzl ; Verweij:2019xch ;
Prokopec:2018tnq ; Zhou:2018zli ; Archer-Smith:2019gzq ; Cai:2022bcf ;
Paul:2020wbz ; Chao:2017ilw ; Pimentel:2020rkw ; Blinov:2015sna ;
Inoue:2015pza ; Greljo:2019xan ; Croon:2018kqn ; Demidov:2017lzf ;
Bian:2017wfv ; Eichhorn:2020upj ; Miura:2018dsy ; Azatov:2019png ;
Aoki:2017aws ; Chen:2017cyc ; Croon:2019ugf ; Croon:2019iuh ; Costa:2022oaa ;
Chao:2017vrq ; Ghosh:2020ipy ; Chao:2021xqv ; Romero:2021kby ; Alves:2020bpi ;
Hall:2019ank ; Hall:2019rld , including models with a focus on dark sectors
and dark matter Madge:2018gfl ; Croon:2018erz ; Hashino:2018zsi ;
Schwaller:2015tja ; Alanne:2014bra ; Addazi:2017gpt ; Fairbairn:2019xog ;
Bertone:2019irm ; Huang:2021rrk ; Halverson:2020xpg ; Nakai:2020oit ;
Ratzinger:2020koh ; Bhoonah:2020oov ; Helmboldt:2019pan ; Beniwal:2017eik ;
Baldes:2017ygu ; Baldes:2018emh ; Huang:2017kzu ; Tsumura:2017knk ;
Croon:2019rqu ; Addazi:2017nmg ; Imtiaz:2018dfn ; Addazi:2017oge ; Bai:2018dxf
; Bian:2018bxr ; Pandey:2020hoq ; Davoudiasl:2021ijv ; Huang:2017rzf ;
Paul:2019pgt ; Hektor:2018esx ; Kannike:2019mzk ; Kannike:2019wsn ;
Mohamadnejad:2019vzg ; Breitbach:2018ddu ; Arakawa:2021wgz ; Azatov:2021ifm ;
Azatov:2020ufh .

In this work, we approach this problem from a different angle. We scan the
parameter space of a simple and well-motivated thermal effective potential
with a single dark Higgs field, which can arise from a wide class of UV-
complete models. We correlate the properties of the GW signal arising from a
first-order phase transition with the parameters of the effective potential
(for other work examining general models, as well as model and signal features
and classes, see for example, Chung:2012vg ; Dev:2016feu ; Jinno:2017ixd ;
Chala:2019rfk ; Croon:2019kpe ; Ellis:2019flb ; Ellis:2020awk ;
Megevand:2021llq ; Schmitz:2020syl ; Alanne:2019bsm , including cosmological
contraints Bai:2021ibt , non-gaussianities Kumar:2021ffi , and circular
polarization Ellis:2020uid ).

The result of this analysis is a characterization of the sound wave GW signal
which can arise from any UV-complete model whose thermal effective potential
is polynomial and renormalizable. Interestingly, we find that dimensional
analysis and scaling relations control much of the analysis, allowing us to
relate specific features of the GW signal to specific parameters of the
effective potential.

As a specific example, we find that amplitude of the gravitational wave signal
is strongly related to the ratio of the dark Higgs mass to the dark Higgs
vacuum expectation value (vev) at zero temperature. If the dark Higgs mass is
less than ${\cal O}(1\%)$ of the dark Higgs vev, then the GW signal will be
too small to be observed at any upcoming experiments. Because the amplitude of
the signal scales as several powers of ratio of dark Higgs mass to vev, our
result is robust. Indeed, theoretical work connecting the amplitude of a
gravitational wave signal to the underlying parameters of the phase transition
is constantly evolving (for example, more refined calculations for determining
the kinetic energy fraction Espinosa:2010hh ; Giese:2020rtr ; Giese:2020znk ,
the wall velocity Bodeker:2009qy ; Bodeker:2017cim ; Hoche:2020ysm , and
adoption of more general parameterized forms for the frequency spectra
Schmitz:2020rag  which can account for the variations from different studies,
such as Hindmarsh:2013xza ; Hindmarsh:2015qta ; Hindmarsh:2017gnf ;
Jinno:2017fby ; Cutting:2018tjt ; Cutting:2020nla ; Jinno:2020eqg ;
Dahl:2021wyk ; Auclair:2022jod ), leading one to wonder how well one can trust
any even state-of-the-art result. But in order to change our bound by an
order-of-magnitude, one would need a correction to the gravitational wave
signal of many orders of magnitude, giving our result some robustness to
future developments.

We also find a general relationship between the symmetry-breaking vev and peak
frequency of the GW signal. We then characterize the optimal instruments for
probing generic first-order phase transitions, in terms of the symmetry-
breaking scale at zero temperature.

But it is also important to point out what we do not do – we do not provide a
complete analysis of any particular UV-complete model. We assume a thermal
effective potential for a single scalar field which is renormalizable and of
the polynomial form. Although well-motivated, for any particular model, this
may or may not be a good approximation to the thermal effective potential. Our
results apply directly to models for which this thermal effective potential is
a good approximation, while for other models, our results are at best
indicative.

The plan of this paper is as follows. In Section 2, we parameterize the
thermal effective potential, and characterize the allowed phase transitions
and resulting gravitational wave signals. In Section 3, we describe the
results of our scans over the parameter space of the thermal effective
potential. We conclude in Section 4 with a discussion of our results.

##  2 First-Order Phase Transitions with a Renormalizable Scalar Potential

We consider a thermal effective potential for a single real scalar field
$\phi$. We consider a renormalizable polynomial potential Croon:2018erz  of
the form

$\displaystyle V(T,\phi)$ $\displaystyle=$
$\displaystyle\Lambda^{4}\left[\left(-\frac{1}{2}+c\left(\frac{T}{v}\right)^{2}\right)\left(\frac{\phi}{v}\right)^{2}+b\frac{T}{v}\left(\frac{\phi}{v}\right)^{3}+\frac{1}{4}\left(\frac{\phi}{v}\right)^{4}\right],$
(1)

where $V(T=0,\phi)$ is minimized at $\phi=\pm v$, and where the mass of the
dark Higgs (the excitation of $\phi$ about this minimum) is given by
$m^{2}/v^{2}=2(\Lambda/v)^{4}$. This form of the potential arises, for
example, from thermal corrections in the high temperature limit, where the
dimensionless coefficients $b$ and $c$ depend on the details of the UV model.
We assume $\Lambda/v\lesssim 1$, ensuring that effective potential has a
perturbative quartic coupling. We also take $b<0$, as this coefficient arises
in the high temperature limit from fermion loops which contribute with a
negative sign to the effective potential.

It is convenient to express the potential in terms of the scale-free
quantities $\tilde{\phi}=\phi/v$ and $\tilde{T}=T/v$. Essentially, the
symmetry-breaking scale $v$ is used as the scale against which all other
energies are measured. Defining
$\tilde{V}(\tilde{T},\tilde{\phi})=\Lambda^{-4}V(T,\phi)$, we have

$\displaystyle\tilde{V}(\tilde{T},\tilde{\phi})$ $\displaystyle=$
$\displaystyle\left(-\frac{1}{2}+c\tilde{T}^{2}\right)\tilde{\phi}^{2}+b\tilde{T}\tilde{\phi}^{3}+\frac{1}{4}\tilde{\phi}^{4}.$
(2)

We can now consider the generic behavior of this thermal effective potential.
At finite temperature, the potential can have up to three extrema, at $\phi=0$
and at factor of 2 under the radical is fixed

$\displaystyle\tilde{\phi}$ $\displaystyle=$
$\displaystyle\frac{-3b\tilde{T}\pm\sqrt{9b^{2}\tilde{T}^{2}+4(1-2c\tilde{T}^{2})}}{2}.$
(3)

If $c\tilde{T}^{2}<(1/2)+(9/8)b^{2}\tilde{T}^{2}$, then the discriminant is
positive, and three distinct extrema exist. In that case, one extremum is a
local maximum, while the other two are local minima, and at least one is a
symmetry-breaking minimum.

$V(T,\phi=0)=0$, where this extremum at $\phi=0$ is a local minimum for
$c\tilde{T}^{2}>1/2$ and a local maximum for $c\tilde{T}^{2}<1/2$. The
condition for $V(T,\phi)=0$ for $\phi>0$ is

$\displaystyle c\tilde{T}^{2}\leq\frac{1}{2}+b^{2}\tilde{T}^{2}.$ (4)

We can thus classify the behavior of the potential as the Universe cools from
high temperature. At sufficiently high temperature, the symmetry-preserving
extremum is always a local minimum at $V=0$. But for $c/b^{2}<1$, the
potential has two other zeros, with a global minimum between them, even at
high temperature. In this case, there is no phase transition, since the
Universe is always in the symmetry-breaking phase. We thus restrict ourselves
to the case $c/b^{2}>1$.

For $c/b^{2}>1$, then at high temperature there is a global minimum at
$\phi=0$ and a local minimum at $\phi_{+}>0$, with a local maximum at
$\phi_{-}$ between them. But at low enough temperature
($\tilde{T}_{C}=1/\sqrt{2(c-b^{2})}$), we find $V(\phi_{+})=0$.

For $T<T_{C}$, $\phi_{+}$ is now the global minimum, and a first-order phase
transition is possible. The transition will not occur until the nucleation
temperature, $T_{N}$, at which time the bubble nucleation rate is larger than
the Hubble expansion rate. But at sufficiently low temperature
($\tilde{T}<1/\sqrt{2c}$), the potential barrier goes to zero; the symmetry-
preserving extremum is now a local maximum, and if the phase transition has
not yet completed, then $\phi$ can simply roll to the global minimum, yielding
a second-order phase transition.

Note that our analysis of the minima of the potential depended only on the
parameters $b$ and $c$, as $v$ appears only as a scale parameter, and
$\Lambda$ has factored out. But the dependence on $\Lambda$ returns when we
determine the nucleation temperature. The bubble nucleation rate per unit
volume ($p(T)$) is given by $p(T)=T^{4}\exp[-S_{E}/T]$, where $S_{E}$ is the
Euclidean action for the radial bounce solution between the symmetry-
preserving and symmetry-breaking vacua. The nucleation temperature $T_{N}$ is
the temperature at the nucleation time $t_{N}$, at which a fraction $e^{-1}$
of the Universe has tunnelled to the global minimum of the potential. The
nucleation time is given approximately by the relation $p(t_{N})t_{N}^{4}=1$.
Assuming a radiation-dominated universe with $g_{*}={\cal O}(100)$
relativistic degrees of freedom, we then find Caprini:2019egz ; Quiros:1999jp
; wainwright2012:cosmotransitions ; Ramsey-Musolf:2019lsf

$\displaystyle\frac{S_{E}}{T_{N}}=140+{\cal O}(\log T_{N}/\rm TeV).$ (5)

$S_{E}$ is the Euclidean action evaluated on the bounce solution to the radial
equation Akula:2016gpl ; wainwright2012:cosmotransitions

$\displaystyle\frac{d^{2}\phi}{dr^{2}}+\frac{2}{r}\frac{\partial\phi}{\partial
r}$ $\displaystyle=$ $\displaystyle\frac{\partial
V_{E}(T,\phi)}{\partial\phi}.$ (6)

where $V_{E}(T,\phi)$ is the temperature and field dependent potential in the
Euclidean action. This solution interpolates between $\phi=0$ and
$\phi=\phi_{+}$. We will utilize an analytic approximation to the bounce
solution for this form of the effective potential Dine:1992wr , given by

$\displaystyle\frac{S_{E}}{T}=\frac{4.85M^{3}}{E^{2}T^{3}}\left[1+\frac{\alpha}{4}\left(1+\frac{2.4}{1-\alpha}+\frac{0.26}{(1-\alpha)^{2}}\right)\right],$
(7)

where

$\displaystyle M^{2}\equiv
2\frac{\Lambda^{4}}{v^{2}}\left(\frac{cT^{2}}{v^{2}}-\frac{1}{2}\right),\hskip
28.45274ptE\equiv-\frac{b\Lambda^{4}}{v^{4}},\hskip
28.45274pt\alpha\equiv\frac{M^{2}\Lambda^{4}}{2E^{2}T^{2}v^{4}}.$ (8)

For certain models, this analytic solution was compared to the Euclidean
action computed using CosmoTransitions wainwright2012:cosmotransitions , a
Python package that numerically computes the properties of phase transitions
in scalar field theories, and good agreement was found. A more detailed
comparison was performed in Guo:2020grp , similarly finding good agreement.

To put eqn. 6 in scale-free form, we define $\tilde{r}\equiv
r(\Lambda^{2}/v)$, yielding

$\displaystyle\frac{d^{2}\tilde{\phi}}{d\tilde{r}^{2}}+\frac{2}{\tilde{r}}\frac{\partial\tilde{\phi}}{\partial\tilde{r}}$
$\displaystyle=$
$\displaystyle\frac{\partial\tilde{V}_{E}(\tilde{T},\tilde{\phi})}{\partial\tilde{\phi}}.$
(9)

The scale-free Euclidean action, $\tilde{S}_{E}$, is obtained by integrating
the scale-free Lagrangian, evaluated on the bounce solution of the above
equation, with respect to $d^{3}\tilde{r}~{}d(1/\tilde{T})$, and is related to
$S_{E}$ by

$\displaystyle\frac{S_{E}}{T}$ $\displaystyle=$
$\displaystyle\frac{v^{2}}{\Lambda^{2}}\frac{\tilde{S}_{E}}{\tilde{T}}.$ (10)

We thus find

$\displaystyle\frac{\tilde{S}_{E}}{\tilde{T}_{N}}$ $\displaystyle\sim$
$\displaystyle 140\left(\frac{\Lambda}{v}\right)^{2},$ (11)

where the left-hand side of eqn. 11 depends only on $b$, $c$ and
$\tilde{T}_{N}$.

###  2.1 Thermal Parameters of the Phase Transition

We are now able to determine determine $S_{E}(\tilde{T})$ and $\tilde{T}_{N}$
for any parameters $b$, $c$ and $\Lambda/v$. From these, we can determine the
thermal parameters of the phase transition, which in turn feed into the
gravitational wave signature. These are speed parameter of the phase
transition ($\beta/H$), and the latent heat parameter ($\xi$).

We can write the thermal parameters as

$\displaystyle\left(\frac{\beta}{H}\right)$ $\displaystyle=$
$\displaystyle\left(\frac{\tilde{\beta}}{H}(b,c,\Lambda/v)\right)\times\left(\frac{\Lambda}{v}\right)^{-2},$
$\displaystyle\xi$ $\displaystyle=$
$\displaystyle\tilde{\xi}(b,c,\Lambda/v)\times\left(\frac{\Lambda}{v}\right)^{4}\left(\frac{g_{*}}{100}\right)^{-1},$
(12)

where

$\displaystyle\frac{\tilde{\beta}}{H}(b,c,\Lambda/v)$ $\displaystyle=$
$\displaystyle\left(\tilde{T}\frac{d(\tilde{S}_{E}/\tilde{T})}{d\tilde{T}}\right)_{\tilde{T}=\tilde{T}_{N}},$
$\displaystyle\tilde{\xi}(b,c,\Lambda/v)$ $\displaystyle=$
$\displaystyle\left[\left(\frac{3}{10\pi^{2}\tilde{T}_{N}^{4}}\right)\left(\Delta\tilde{V}-\tilde{T}\Delta\frac{d\tilde{V}}{d\tilde{T}}\right)\right]_{\tilde{T}=\tilde{T}_{N}},$
(13)

and $g_{*}$ is the number of relativistic degrees of freedom. Essentially, the
parameter $\Lambda/v$ controls an overall rescaling of the potential, and the
power-law dependence of the thermal parameters on $\Lambda/v$ in eq. 12
encodes the dependence of these parameters on that scaling.

Note that the scale-free thermal parameters $(\tilde{\beta}/H)$ and
$\tilde{\xi}$ depend on $\Lambda/v$ only through the determination of the
scale-free nucleation temperature, $\tilde{T}_{N}$. Thus, the actual thermal
parameters $(\beta/H)$ and $\xi$ have a dependence on $\Lambda/v$ which is
nearly power-law. This dependence would be exactly power law if the nucleation
temperature were exactly the same as the critical temperature. In fact, the
nucleation temperature will be less than the critical temperature, leading to
a deviation from power-law behavior. But we will see that this power-law
behavior largely determines the maximum allowed gravitational wave signal.
Moreover, this form of the thermal effective potential is a good approximation
in the high temperature limit; if the nucleation temperature is significantly
smaller than the critical temperature, this may not be a good approximation. A
typical scale where the high-temperature approximation remains valid is given
by $T\gtrsim m$ (for example, in Croon:2018erz  the criteria $T^{2}>2m^{2}$
was employed). This can be translated to the constraint
$\tilde{T}\gtrsim\sqrt{2}(\Lambda/v)^{2}$. However, the size of the effects of
non-power law terms that would be generated if this criteria is not met would
be model dependent, thus we do not pursue that line of inquiry further in this
work.

###  2.2 The Gravitational Wave Signal

First order phase transitions can produce gravitational waves via three
different mechanisms: i) bubble collisions, ii) sound waves from bubble
expansion in the fluid, and iii) turbulence in the fluid. In a runaway bubble
expansion, where bubble wall velocities are not sufficiently slowed by
friction from the fluid, bubble collisions can provide the dominant
gravitational wave signal. This situation can arise, for example, for
extremely supercooled systems Kosowsky:1992vn ; Cutting:2018tjt ;
Lewicki:2019gmv . The study of turbulence is ongoing, with the efficiency of
converting kinetic energy into turbulent energy being one source of some
uncertainty Hindmarsh:2015qta ; Schmitz:2020syl ; Auclair:2022jod . Thus, as
we are not examining extremely supercooled situations, we will assume that the
dominant gravitational wave signature arises from sound waves
Hindmarsh:2016lnk . We can express the amplitude ($h^{2}\Omega_{sw}$) and
frequency ($f_{sw}$) of this gravitational wave signal as Weir:2017wfa ;
Guo:2020grp ; Guo:2021qcq

$\displaystyle h^{2}\Omega_{sw}(f)$ $\displaystyle=$ $\displaystyle
h^{2}\Omega_{sw}^{max}\times\left(\frac{f}{f_{sw}}\right)^{3}\left(\frac{7}{4+3(f/f_{sw})^{2}}\right)^{7/2},$
(14)

where

$\displaystyle f_{sw}$ $\displaystyle=$
$\displaystyle\tilde{f}_{sw}(b,c,\Lambda/v)\times\left(\frac{\Lambda}{v}\right)^{-2}\left(\frac{T_{N}}{100~{}\rm
GeV}\right)\left(\frac{g_{*}}{100}\right)^{1/6},$ $\displaystyle
h^{2}\Omega_{sw}^{max}$ $\displaystyle=$ $\displaystyle
h^{2}\tilde{\Omega}_{sw}^{max}(b,c,\Lambda/v)\times\left(\frac{\Lambda}{v}\right)^{10+8n}\left(\frac{g_{*}}{100}\right)^{-5/3-2n}$
(15) $\displaystyle\times\left[1-\frac{1}{\sqrt{1+2\tau_{sh}H_{s}}}\right],$

and Espinosa:2010hh

$\displaystyle\tilde{f}_{sw}(b,c,\Lambda/v)$ $\displaystyle=$ $\displaystyle
8.9\times 10^{-3}{\rm
mHz}\left(\frac{1}{v_{w}}\right)\left(\frac{\tilde{\beta}}{H}\right)\left[\times\left(\frac{z_{p}}{10}\right)\frac{1}{\sqrt{1+2\tau_{sh}H_{s}}}\right],$
$\displaystyle h^{2}\tilde{\Omega}_{sw}^{max}(b,c,\Lambda/v)$ $\displaystyle=$
$\displaystyle 8.5\times
10^{-6}\left(\frac{\Gamma}{4/3}\right)^{2}(\tilde{\kappa}_{f}\tilde{\xi})^{2}\left(\frac{\tilde{\beta}}{H}\right)^{-1}v_{w},$
$\displaystyle\tilde{\kappa}_{f}$ $\displaystyle\sim$
$\displaystyle\tilde{\xi}^{n}.$ (16)

Here, $v_{w}$ is the sound speed, which determines the exponent $n$; for
$v_{w}\sim 1$, we have $n=1$, while for $v_{w}\sim 0.5$, we have $n\sim 2/5$
Espinosa:2010hh . $\Gamma$ is the adiabatic index, which we will take to be
$\Gamma=4/3$ Croon:2018erz , and $z_{p}\sim 10$ Hindmarsh:2017gnf ;
Guo:2020grp .

There are additional factors listed in brackets Guo:2020grp , which depend on
the details of the model, encoded in quantities such as the time scale for
turbulence ($\tau_{sh}$) and the Hubble scale at which the source becomes
active ($H_{S}$). Indeed, as pointed out in Guo:2021qcq , there are a variety
of suppression factors which can arise from a more diligent treatment of
gravitational wave production, which can suppress the signal amplitude by more
than an order of magnitude. As we begin with an effective potential for the
scalar, but no detailed microphysics, it is not possible to attain this level
of precision. But for our purpose, it is not necessary. Since the
gravitational wave amplitude scales as more than 10 powers of $(\Lambda/v)$,
this analysis will be sufficient to determine classes of models which will be
easily inaccessible to all upcoming experiments. Moreover, although future
theoretical developments may uncover additional correction factors, unless
their collective size is several orders of magnitude, they will not change our
results appreciably. For similar reasons, it is sufficient for us define the
thermal parameters at the nucleation temperature, rather than the percolation
temperature.

In a similar vein, there are a variety of theoretical uncertainties with both
the first-order phase transition and the associated gravitational wave
signature, and there has been much recent theoretical progress, including
studies of gauge dependence, renormalization group equations, and the one-loop
effective potential Chiang:2017zbz ; Niemi:2021qvp , discussions within an
effective field theory framework Cai:2017tmh ; Chala:2018ari ; Gould:2021ccf ,
issues related to the perturbative expansion including its scale dependence
Gould:2021oba ; Schicho:2021gca ; Gould:2021dzl ; Postma:2020toi , and non-
perturbative models and resummation and use of numerical simulations (for
example, see Hindmarsh:2013xza ; Hindmarsh:2015qta ; Hindmarsh:2017gnf ;
Cutting:2018tjt ; Cutting:2019zws ; RoperPol:2019wvy ; Giese:2020rtr ;
Cutting:2020nla ; LatticeStrongDynamics:2020jwi ; Lewicki:2020azd ;
Cutting:2021tqt ; Jinno:2021ury ; Schicho:2022wty ). It would be interesting
to incorporate these effects in a more detailed analysis, but it is not
necessary for our purpose here.

The main quantities we are interested in are the peak amplitude of the GW
signal ($h^{2}\Omega_{sw}^{max}$), and the frequency at which that peak is
obtained ($f_{sw}$). If the nucleation temperature were equal to the critical
temperature, then $h^{2}\tilde{\Omega}_{sw}^{max}$ and $\tilde{f}_{sw}$ would
be independent of $\Lambda/v$, the dependence of the GW signal parameters on
$\Lambda/v$ would be entirely determined by the scaling relations in eq. 12.
We would then find

$\displaystyle\left(h^{2}\Omega_{sw}^{max}\right)_{T_{N}=T_{C}}$
$\displaystyle\propto$
$\displaystyle\xi^{2(n+2)}\left(\frac{\beta}{H}\right)^{-1}\propto\left(\frac{\Lambda}{v}\right)^{10+8n},$
$\displaystyle\left(f_{sw}\right)_{T_{N}=T_{C}}$ $\displaystyle\propto$
$\displaystyle\frac{\beta}{H}v\propto\left(\frac{\Lambda}{v}\right)^{-2}v,$
(17)

where $(\Lambda/v)^{4}=m^{2}/2v^{2}$, and the power-law dependence on
$\Lambda/v$ is induced by the dependence of the thermal parameters on the
overall scale of the potential.

Reducing $m/v$ by an order of magnitude would reduce the amplitude of the GW
signal by a factor of at least $10^{5}$ (taking $n=0$). We thus expect to find
a sharp minimum for $m/v$; for dark Higgs masses which are sufficiently
smaller than the symmetry-breaking scale, the amplitude of the GW signal will
be unobservable.

Note, however that although the amplitude of the GW signal depends on $m/v$,
it has no actual dependence on $v$ itself, a result which could be anticipated
from dimensional analysis. But the symmetry-breaking scale does enter into
$f_{sw}$, to which it is directly proportional. Thus far, our analysis has
been completely scale-free. We now see that the energy scale of the phase
transition only enters in the last step, setting the frequency scale of the GW
signal.

Some of these considerations will be modified once we include the dependence
of the nucleation temperature on $(\Lambda/v)$. But we expect that this
modification should not change the result substantially; if $T_{N}\ll T_{C}$,
then the phase transition is very supercooled, and this form of the thermal
effective potential may in any case not be a good approximation to the thermal
potential of a UV complete model. We will see from more detailed numerical
calculation that this intuition is correct.

##  3 Results

In this section, we describe our strategy for scanning over the parameter
space of the thermal effective potential and then present the results of the
scan. First, we briefly specify the conditions for a parameter space point to
be of interest.

  * •

$\Lambda/v<1$: This condition ensures that the zero-temperature quartic
coupling is perturbative.

  * •

$c/b^{2}>1$: This condition is required for a phase transition to occur at
all. If it is not satisfied, then the symmetry-breaking minimum is the global
minimum even at high temperature.

  * •

$c\tilde{T}_{N}^{2}>1/2$: This is necessary in order for the transition to be
first-order. If this is not satisfied, then the barrier disappears before the
nucleation temperature is reached, leading to a smooth phase transition.

  * •

$\beta/H>1$: This condition is needed for the bubble growth rate to exceed the
Hubble expansion, so that bubbles grow. This is equivalent to
$\tilde{\beta}/H>(\Lambda/v)^{2}$.

  * •

$\xi>0$: a first order transition will only proceed if the latent heat
parameter is positive at the nucleation temperature. This is equivalent to
$\tilde{\xi}>0$.

It may seem unusual that the latent heat parameter can be negative. It is easy
to demonstrate that, if the phase transition occurs at the critical
temperature, then the latent heat parameter must be positive. But it may
become negative if the phase transition is sufficiently supercooled. As we
have already mentioned, such a transition would potentially stretch the
effective potential beyond its regime of validity. But in any case, if the
latent heat parameter is negative, no GW signal will be produced.

Figure 1: Contours of $\tilde{S}_{E}/140\tilde{T}$ are shown in the $(b,c)$
plane for smaller values of $\tilde{T}$: $\tilde{T}=0.15$ (upper left), $0.20$
(upper right), $0.25$ (lower left), and $0.30$ (lower right). The contours
also satisfy the conditions: $c\geq b^{2}$, $c\tilde{T}\geq 1/2$, $\alpha\leq
1$, $0\leq\tilde{S}_{E}/140\tilde{T}\leq 1$,
$\tilde{\beta}/H\geq\tilde{S}_{E}/140\tilde{T}$ and $\tilde{\xi}\geq 0$.

The scale-free Euclidean action, $\tilde{S}_{E}$, is determined by $\tilde{T}$
and the parameters $b$ and $c$. For any choice of parameters $b$, $c$ and
$\Lambda/v$, the nucleation temperature $\tilde{T}_{N}$ is approximately
determined by the solution of the equation

$\displaystyle\frac{\tilde{S}_{E}(b,c,\tilde{T}_{N})}{140\tilde{T}_{N}}$
$\displaystyle=$ $\displaystyle\left(\frac{\Lambda}{v}\right)^{2}.$ (18)

In Figures 1 and 2, we plot contours of
$\tilde{S}_{E}(b,c,\tilde{T})/140\tilde{T}$ in the $(b,c)$-plane, for eight
different choices of $\tilde{T}$. We can then solve eqn. 18 by setting
$(\Lambda/v)^{2}$ equal to the $\tilde{S}_{E}/140\tilde{T}$, for a value of
$\tilde{T}$ which is taken as the nucleation temperature. These figures thus
encode all choices of the parameters $(b,c,\Lambda/v)$ for which the scale-
free nucleation temperature is given by a particular choice $\tilde{T}_{N}$.
Note, we only plot parameters points which are of interest, as defined by the
criteria described at the beginning of this section.

Figure 2: Contours of $\tilde{S}_{E}/140\tilde{T}$ are shown in the $(b,c)$
plane for larger values of $\tilde{T}$: $\tilde{T}=0.40$ (upper left), $0.50$
(upper right), $0.60$ (lower left), and $0.80$ (lower right). The contours
also satisfy the conditions: $c\geq b^{2}$, $c\tilde{T}\geq 1/2$, $\alpha\leq
1$, $0\leq\tilde{S}_{E}/140\tilde{T}\leq 1$,
$\tilde{\beta}/H\geq\tilde{S}_{E}/140\tilde{T}$ and $\tilde{\xi}\geq 0$.

Note that the range of parameter space which is of interest is very small for
$\tilde{T}_{N}\sim 1$, near $(b,c)\sim(-\sqrt{2},2)$. More parameter space is
available for $\tilde{T}_{N}<1$, and for relatively large $\tilde{T}_{N}$, the
available parameter space is focused near the curve $c=b^{2}$. But as
$\tilde{T}_{N}$ becomes small, the high temperature approximation becomes less
valid; in that case, it is not clear if this effective potential can be
realized from a UV complete model. In particular, note that, although we do
not consider the range $c>25$ for reasons of computational ease, that range in
any case corresponds to small $\tilde{T}_{N}$.

In Figures 3 and 4, we plot contours of $\tilde{\beta}/H$ in the
$(b,c)$-plane, for different choices of $\tilde{T}_{N}$. Only parameter points
satisfying the criteria described in the beginning of this section are
considered. In particular, for any choice of $\tilde{T}_{N}$, only points in
the $(b,c)$-plane are shown for which there exists some choice of $\Lambda/v$
consistent with the given $\tilde{T}_{N}$.

Figure 3: Contours of $\tilde{\beta}/H$ are shown in the $(b,c)$ plane for the
allowed parameter space for lower values of $\tilde{T}_{N}$:
$\tilde{T}_{N}=0.15$ (upper left), $0.20$ (upper right), $0.25$ (lower left),
and $0.30$ (lower right). The contours also satisfy the conditions: $c\geq
b^{2}$, $c\tilde{T}_{N}\geq 1/2$, $\alpha\leq 1$,
$0\leq\tilde{S}_{E}/140\tilde{T}_{N}\leq 1$,
$\tilde{\beta}/H\geq\tilde{S}_{E}/140\tilde{T}_{N}$ and $\tilde{\xi}\geq 0$.

Figure 4:  Contours of $\tilde{\beta}/H$ are shown in the $(b,c)$ plane for
the allowed parameter space for higher values of $\tilde{T}_{N}$:
$\tilde{T}_{N}=0.40$ (upper left), $0.50$ (upper right), $0.60$ (lower left),
and $0.80$ (lower right). The contours also satisfy the conditions: $c\geq
b^{2}$, $c\tilde{T}_{N}\geq 1/2$, $\alpha\leq 1$,
$0\leq\tilde{S}_{E}/140\tilde{T}_{N}\leq 1$,
$\tilde{\beta}/H\geq\tilde{S}_{E}/140\tilde{T}_{N}$ and $\tilde{\xi}\geq 0$.

Similarly, in Figures 5 and 6, we plot contours of $\tilde{\xi}$ in the
$(b,c)$-plane for different choices of $\tilde{T}_{N}$. Note that unless
$\tilde{T}_{N}\ll 1$, we find $\tilde{\xi}\lesssim{\cal O}(10)$ throughout the
entire parameter space. This implies that, unless the phase transition occurs
well below the symmetry breaking scale, we will find

$\displaystyle h^{2}\tilde{\Omega}_{sw}^{max}$ $\displaystyle<$
$\displaystyle{\cal O}(0.1),$ $\displaystyle h^{2}\Omega_{sw}^{max}$
$\displaystyle<$ $\displaystyle{\cal
O}(0.1)\left(\frac{\Lambda}{v}\right)^{10+8n}$ (19)

The maximum sensitivity of LISA is roughly $h^{2}\tilde{\Omega}_{sw}\sim{\cal
O}(10^{-13})$, implying that LISA will be insensitive to any model for which
$\Lambda/v\lesssim{\cal O}(10^{-1})$. Note that we can directly relate the
parameters of the thermal effective potential to the mass and vev of the dark
Higgs at zero temperature through the relation $(\Lambda/v)^{2}=m/\sqrt{2}v$.
We thus see that these simple scaling arguments imply a powerful connection
between gravitational wave observations of a cosmological phase transition and
laboratory probes of the hidden sector at zero temperature. In particular, if
the mass of the dark Higgs particle excitation is less than ${\cal O}(1\%)$ of
the dark Higgs vev, then the gravitational wave signal arising from
condensation of the dark Higgs in the early Universe would have an amplitude
too small to be observed at LISA. If the hidden sector is coupled to the
Standard Model, then the dark Higgs mass and vev can be probed at fixed target
or beam dump experiments, thus determining if a gravitational wave signal can
be seen. Note that, although the maximum sensitivity of BBO will be 4-5 orders
of magnitude better than LISA this leads to much less than an order of
magnitude improvement in minimum value of $m/v$ for models which can be probed
with gravitational waves.

Figure 5: Contours of $\tilde{\xi}$ are shown in the $(b,c)$ plane for the
allowed parameter space for lower values of $\tilde{T}_{N}$:
$\tilde{T}_{N}=0.15$ (upper left), $0.20$ (upper right), $0.25$ (lower left),
and $0.30$ (lower right). The contours also satisfy the conditions: $c\geq
b^{2}$, $c\tilde{T}_{N}\geq 1/2$, $\alpha\leq 1$,
$0\leq\tilde{S}_{E}/140\tilde{T}_{N}\leq 1$,
$\tilde{\beta}/H\geq\tilde{S}_{E}/140\tilde{T}_{N}$ and $\tilde{\xi}\geq 0$.

Figure 6:  Contours of $\tilde{\xi}$ are shown in the $(b,c)$ plane for the
allowed parameter space for higher values of $\tilde{T}_{N}$:
$\tilde{T}_{N}=0.40$ (upper left), $0.50$ (upper right), $0.60$ (lower left),
and $0.80$ (lower right). The contours also satisfy the conditions: $c\geq
b^{2}$, $c\tilde{T}_{N}\geq 1/2$, $\alpha\leq 1$,
$0\leq\tilde{S}_{E}/140\tilde{T}_{N}\leq 1$,
$\tilde{\beta}/H\geq\tilde{S}_{E}/140\tilde{T}_{N}$ and $\tilde{\xi}\geq 0$.

We can provide an even tighter bound on the amplitude of the gravitational
wave signal by accounting for the dependence of $h^{2}\Omega_{sw}^{max}$ on
$\xi$ and $\tilde{\beta}/H$. In Figure 7(a), we scan over parameters
$(b,c,\Lambda/v)$, plotting each point on the
$(h^{2}\Omega_{sw}^{max},\Lambda/v)$-plane. We have assumed $v_{w}=1$. If we
could ignore the effects of supercooling, then we would expect from Eq. 17
that the set of points with the largest values of $h^{2}\Omega_{sw}^{max}$
correspond to the choice of $(b,c)$ for which $h^{2}\tilde{\Omega}_{sw}^{max}$
is largest, and lie along the line
$h^{2}\Omega_{sw}^{max}\propto(\Lambda/v)^{18}$. In fact, the slope of this
line (plotted as the solid blue line in Fig. 7(a)) is $\sim 17.2$, which is
only slightly different from that given above, indicating that the dependence
of $h^{2}\tilde{\Omega}_{sw}^{max}$ on $\Lambda/v$ induced by the effects of
supercooling is relatively minor. Similarly, in Figure 7(b), we scan over the
parameters $(b,c,\Lambda/v)$, plotting each point in the
$(f_{sw},\Lambda/v)$-plane. Again, we see that most of the points roughly
follow the $f_{sw}/v\propto(\Lambda/v)^{-2}$ relation which we found earlier
(ignoring the effects of supercooling), with a spread of roughly an order of
magnitude in $f_{sw}/v$ for any $\Lambda/v$.

In Figure 8, we plot the set of points in the $(b,c)$-plane for which
$h^{2}\Omega_{sw}^{max}$ is within $2\%$ of the maximum for any given value of
$\Lambda/v$. These points correspond to the $(b,c)$ values of the points near
the solid blue line in Figure 7(a), discussed above. We see that scatter in
these points is small, another indication that the effects of supercooling are
relatively minor (if these effects were negligible, then all parameter points
on this line would have the same values of $b$ and $c$, differing only in the
parameter $\Lambda/v$). We also plot shaded regions indicating values of
$(b,c)$ for which a given nucleation temperature $\tilde{T}_{N}$ can be
realized. We see that these models with maximal signal amplitude can only be
realized with $\tilde{T}_{N}<0.4$.

Finally, in Figure 9, we scan over models, which are plotted in ths
$(f_{sw},h^{2}\Omega_{sw}^{max})$-plane, taking $v_{w}=1$. In particular, we
scan over $b$, $c$, and $\Lambda/v$, with $v=1~{}\rm MeV$ (brown), $100~{}\rm
GeV$ (green), $1~{}\rm TeV$ (pink), and $1000~{}\rm TeV$ (blue). We find
$f_{sw}\propto v$, while for any choice of $v$, we roughly find
$h^{2}\Omega_{sw}^{max}\propto f_{sw}^{-9}$, as one would expect from eqn. 17.
We also plot the sensitivity of a variety of current and upcoming
gravitational wave observatories in Fig. 9.

Note that the most sensitive upcoming experiments (BBO and ALIA) will probe
models with symmetry-breaking scales in the range ${\cal O}(100-1000)~{}\rm
GeV$. For scenarios with new physics appearing at the MeV-scale, experiments
such as THEIA would provide the leading sensitivity, while for scenarios with
new physics appearing at the PeV-scale, experiments such as ET and CE would
provide the leading sensitivity. But the absence of sensitivity for models
with $m/v<{\cal O}(0.01)$ is largely independent of scale; even at the scales
for which future experiments will be most sensitive, this lower bound can be
reduced by no more than a factor of a few. These experiments are described in
Table 1.

(a)

(b)

Figure 7:  Plot of $h^{2}\Omega_{sw}^{max}$ (left) and $f_{sw}/v$ (right)
against $\Lambda/v$ for a set of models obtained by scanning of
$0\leq\Lambda/v\leq 1$, $-5\leq b\leq 0$ and $0\leq c\leq 25$. The dashed
lines in the left panel are the maximum sensitivities of LISA (green), BBO
(beige), SKA (magenta) and CE (purple). The solid blue line in the left panel
presents the maximum value of $h^{2}\Omega_{sw}^{max}$ obtained for a given
$\Lambda/v$, for any $(b,c)$; this line has a slope of $\sim 17.2$.  Figure 8:
Plot in the $(b,c)$-plane of parameter points for which
$h^{2}\Omega_{sw}^{max}$ is within 2% of solid blue line shown in Fig. 7(a).
The shaded regions indicate regions in the $(b,c)$-plane for which various
nucleation temperatures $\tilde{T}_{N}$ can be realized.  Figure 9:  Plot in
the $(h^{2}\Omega_{sw},f_{sw})$-plane of the gravitational wave signal
obtained from a scan of parameter points $(b,c,\Lambda/v)$. The values of $v$
used are 1 MeV, 100 GeV, 1 TeV and 1000 TeV (from left to right). Also plotted
are the sensitivities of the following current and upcoming experiments: 1.
EPTA vanHaasteren:2011ni , 2. NANOGrav NANOGrav:2020bcs ; Lazio:2017fos , 3.
Gaia Gaia:2018ydn , 4. SKA Janssen:2014dka , 5. THEIA 10.3389/fspas.2018.00011
, 6. LISA Caprini:2019egz ; LISA:2017pwj ; Robson:2018ifk , 7. Taiji
Ruan:2018tsw , 8.TianQin TianQin:2015yph , 9. ALIA Gong:2014mca , 10. BBO
Corbin:2005ny ; Yagi:2011wg , 11. DECIGO Kawamura:2020pcg ; Kawamura:2006up ,
12. aLIGO Shoemaker:2019bqt ; LIGOScientific:2014pky , 13. A$+$ QuantumLIGO ,
14. ET Punturo:2010zz , 15. CE LIGOScientific:2016wof .

Note from Figure 7(a) that, although the upper bound on
$h^{2}\Omega_{sw}^{max}$ follows fairly closely the scaling relations we have
derived, and in particular, grows steeply with $\Lambda/v$, the actual
amplitude may be much smaller than this maximum. For example, the behavior of
the SM electroweak phase transition is different, with the amplitude of a
gravitational wave signal decreasing with $m_{h}/v$, and disappearing entirely
for $m_{h}=125~{}\rm GeV$, at which point the phase transition is smooth. This
is because the thermal effective potential for the Standard Model Higgs sector
corresponds to a choice of parameters which lie well below the upper bound on
$h^{2}\Omega_{sw}^{max}$. For sufficiently large $m_{h}/v$, the parameter $c$
becomes small enough that the potential barrier disappears before reaching the
nucleation temperature. This example illustrates the point that, although this
analysis provides an upper limit on the gravitational wave signal, it would be
much more difficult to translate a measurement of $m/v$ into a prediction for
the gravitational wave amplitude, or vice versa.

##  4 Conclusions

We have considered gravitational wave signals from sound waves in the context
of an effective field theory with a renormalizable thermal potential
exhibiting a first-order phase transition in the early Universe. We have
considered a thermal effective potential of power law form, which would arise,
for example, in the high temperature limit. This approach allows us to
consider the gravitational wave signals produced by phase transitions in a
wide class of models.

We find that for this class of models, there is a relatively clean
relationship between the parameters of the gravitational wave signal and the
parameters of the thermal effective potential. The effective potential can be
expressed in terms of one dimensionful parameter, the vacuum expectation value
of the dark Higgs ($v$), which sets the frequency scale of the gravitational
wave signal. The amplitude of the gravitational wave signal is largely set by
one dimensionless parameter, $\Lambda/v$, which sets the ratio of the
potential energy scale to the dark Higgs vev. Since the amplitude of the
gravitational wave signal scales as $\Lambda/v$ to a very high power, the
dependence on the other dimensionless parameters is subdominant.

This approach leads to a striking conclusion. If the ratio of the dark Higgs
mass to the dark Higgs vev is less than ${\cal O}(10^{-2})$, then the
magnitude of the gravitational wave signal will be so small as to be
undetectable by any upcoming observatories. This bound on sensitivity is
nearly independent of the scale of new physics. This result is also robust
against future developments in our understanding of the amplitude of
gravitational sound waves produced by a first-order cosmological phase
transition. Since the amplitude depends on $m_{f}/v$ raised to a high power,
to modify our bound by an order of magnitude would require a new correction to
the sound wave amplitude of several orders of magnitude.

This result leads to several intriguing possibilities. For example, in
scenarios of new MeV-scale physics, one may potentially be able to probe the
symmetry-breaking scale with forward detectors at high-luminosity beam
experiments, which could potentially determine the mass and coupling of the
dark photon. The detection of a gravitational wave signal from the
corresponding first-order phase transition would then provide a lower bound on
the Higgs mass. Similarly, it is possible that future high-energy beam
experiments could produce a heavy ($>{\cal O}(\rm TeV)$) dark Higgs. In this
case, the amplitude of a gravitational wave signal from the phase transition
would correspond to an upper bound on the symmetry-breaking scale.

Classifications based on frequency reach Dark sector scale that can be probed
Type of experiments Experiments Very low frequency $10^{-9}-10^{-8}$ Hz $\leq
1$ MeV Ground-based telescope using pulsar timing observation technique EPTA
(European Pulsar Timing Array) vanHaasteren:2011ni  NANOGrav (North American
Nanohertz Observatory for Gravitational Wave) NANOGrav:2020bcs ; Lazio:2017fos
SKA (Square Kilometer Array) Janssen:2014dka : World’s largest radio telescope
Space-based telescope using astrometry Gaia Gaia:2018ydn  THEIA (Telescope for
Habitable Exoplanets and Intersteller/Intergalactic Astronomy)
10.3389/fspas.2018.00011 : Proposed upgrade of Gaia $\mu$-frequency
$10^{-6}-10^{-5}$ Hz $\sim$ 100 MeV Space-based heliocentric constellation of
satellites using Laser interferometry technique $\mu$-Ares Sesana:2019vho  Low
frequency $10^{-3}-1$ Hz 100 GeV-1 TeV Space-based interferometer using Laser
interferometry technique LISA (Laser Interferometer Space Antenna)
Caprini:2019egz ; LISA:2017pwj ; Robson:2018ifk  ALIA (Advanced Laser
Interferometer Antenna) Gong:2014mca  Taiji Ruan:2018tsw  TianQin
TianQin:2015yph  BBO (Big Bang Observer) Corbin:2005ny ; Yagi:2011wg  DECIGO
(Deci-Hertz Interferometer Gravitational Wave Observatory) Kawamura:2020pcg ;
Kawamura:2006up

Classifications based on frequency reach Dark sector scale that can be probed
Type of experiments Experiments High frequency $10-100$ Hz $\sim$1000 TeV
Second generation Ground-based interferometer using Laser interferometry
technique aLIGO (Advanced LIGO) Shoemaker:2019bqt ; LIGOScientific:2014pky
A$+$ QuantumLIGO  (Quantum LIGO) Third generation Ground-based interferometer
using Laser interferometry technique ET (Einstein Telescope) Punturo:2010zz
CE (Cosmic Explorer) LIGOScientific:2016wof

Table 1:  Description of upcoming gravitational wave observatories, including
the frequency range to which they would be sensitive, and the correspond
symmetry-breaking scale.

There are several avenues for further exploration. We have considered a power-
law form for the thermal effective potential, which is expected to be a good
approximation for renormalizable models, in the high-temperature limit. But
there are a variety of models in which the phase transition is supercooled
(see, for example Kobakhidze:2017mru ; Ellis:2019oqb ; Ellis:2020nnr ;
Wang:2020jrd ), and nucleation temperature is relatively low. In these cases,
although the corrections to the form of the potential may be small, they need
not be. There has been significant recent work regarding theoretical issues in
finite temperature perturbation theory (see, for example, Croon:2020cgk ;
Gould:2021oba ).

Similarly, we have assumed that the thermal effective potential is
renormalizable. There are interesting models of new physics in which the phase
transition can only be seen with the inclusion of non-renormalizable terms. It
would be interesting to consider more general forms of the thermal effective
potential, to better determine the extent to which the lessons found here
generalize.

###### Acknowledgements.

We are grateful to Djuna Croon, Jeremy Sakstein, Tim Tait, and especially
Graham White for useful discussions. The work of JK is supported in part by
DOE grant DE-SC0010504. JR is supported by NSF grant AST-1934744. The work of
BD and SG is supported in part by DOE grant DE-SC0010813. The work of SG is
also supported in part by National Research Foundation of Korea (NRF) Grant
No. NRF-2019R1A2C3005009 (SG). JBD acknowledges support from the National
Science Foundation under grants no. PHY-1820801 and PHY-1748958.

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