††institutetext: Rudolf Peierls Centre for Theoretical Physics, University of
Oxford, Parks Road, Oxford OX1 3PU, United Kingdom

# Avoided Deconfinement in Randall-Sundrum Models

Prateek Agrawal  and Michael Nee<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study first order phase transitions in Randall-Sundrum models in the early
universe dual to confinement in large-$N$ gauge theories. The transition rate
to the confined phase is suppressed by a factor $\exp(-N^{2})$, and may not
complete for $N\gg 1$, instead leading to an eternally inflating phase. To
avoid this fate, the resulting constraint on $N$ makes the RS effective field
theory only marginally under control. We present a mechanism where the IR
brane remains stabilized at very high temperature, so that the theory stays in
the confined phase at all times after inflation and reheating. We call this
mechanism avoided deconfinement. The mechanism involves adding new scalar
fields on the IR brane which provide a stablilizing contribution to the radion
potential at finite temperature, in a spirit similar to Weinberg’s symmetry
non-restoration mechanism. Avoided deconfinement allows for a viable cosmology
for theories with parametrically large $N$. Early universe cosmological
phenomena such as WIMP freeze-out, axion abundance, baryogenesis, phase
transitions, and gravitational wave signatures are qualitatively modified.

##  1 Introduction

Large-$N$ gauge theories are important both theoretically as well as
phenomenologically. The large-$N$ limit makes many problems in strongly-
coupled gauge theories tractable, providing an expansion parameter for non-
Abelian theories Hooft1974a . It remarkably also plays a central role in
gauge-gravity duality Maldacena1999 ; Witten1998k ; Gubser1998  where a
large-$N$ gauge theory is dual to a gravitational theory in higher dimensions.
A major phenomenological application of the duality appears in the context of
the Randall-Sundrum (RS) model Randall1999 . The RS model provides an elegant
solution to the hierarchy problem, with an exponential separation of the
Planck and weak scales that is generated by the warping of the extra
dimension. The ratio of the two scales is set by the size of the extra
dimension, which is fixed by introducing a stabilisation mechanism, generating
a potential for the radion Goldberger1999 . This mechanism is dual to adding
an almost-marginal operator in the gauge theory which explicitly breaks the
scale invariance of the theory. The small anomalous dimension for this
operator, equivalent to a small bulk mass for the stabilising field
Witten1998k , generates an exponentially small scale in the IR in a manner
analogous to dimensional transmutation in QCD.

The RS model also provides an effective description for a number of string
constructions of realistic vacua with reduced supersymmetry Luty:2000ec ;
Verlinde2000 ; Chan2000 ; Brummer2006 ; Randall2019 . In the large-$N$ limit,
these explicit constructions are described by an effective quantum field
theory where gravity is weakly coupled. A prominent example is the KKLT
scenario Kachru:2003aw , which is partly based on the Klebanov-Witten
Klebanov:1998hh  and Klebanov-Strassler Klebanov2000 ; Klebanov:2000hb
constructions. For this construction the stability of the de Sitter vacuum is
often justified in the probe approximation which is valid for parametrically
large $N$ Bena:2018fqc ; Kachru:2019dvo . Therefore, large-$N$ gauge theories
with gravitational duals are expected to be an important part of a UV complete
description of our universe.

Theories which are described by the RS model in an appropriate limit suffer
from a severe cosmological constraint. At high temperature the gauge theory is
in the deconfined phase, and the confined phase becomes thermodynamically
preferred below a critical temperature. However, the confinement phase
transition is first order and exponentially suppressed with a rate
proportional to $\exp(-N^{2}/\lambda)$, where $\lambda$ denotes a possible
weak coupling. The gravitational description of the deconfined phase is the
AdS-Schwarzschild (AdS-S) solution with a UV brane. The confinement transition
corresponds to the appearance of the IR brane from behind the AdS-S horizon
Hawking1983 ; Witten1998 ; Creminelli2002 . The confinement scale in the gauge
theory is dual to the vacuum expectation value of the radion field which sets
the size of the extra dimension in the RS model.

For large-$N$, the suppressed phase transition is much slower than the
expansion rate of the universe, leading to eternal inflation. Requiring the
phase transition to complete leads to a robust upper bound on $N$ Kaplan2006 :

$\displaystyle N\lesssim\sqrt{4\lambda\log\frac{M_{\rm pl}}{\Lambda_{c}}}\sim
12\sqrt{\lambda}\,.$ (1)

where $\Lambda_{c}\simeq 1\ {\rm TeV}$ is the confinement scale for the gauge
theory. This bound follows just from dimensional analysis and is independent
of the details of the RS model. The ratio $N/4\pi$ sets the hierarchy between
the curvature scale $k$ and the 5d reduced Planck scale $M_{5}$,
$N/4\pi\sim(M_{5}/k)^{3/2}$ Gubser2001 . A small value of this ratio, as
implied by the bound (1), then means that corrections due to Planck scale
physics become important, making the EFT control in the RS model delicate.
Gravitational loop corrections can be estimated by the following loop counting
parameter,

$\displaystyle\frac{N_{\rm species}k^{3}}{16\pi^{2}M_{5}^{3}}<1\Rightarrow
N^{2}\gtrsim N_{\rm species}$ (2)

which is in tension with equation (1) even with just the SM degrees of freedom
contributing to $N_{\rm species}\sim 100$.

In fact, the bound is much more stringent within the simplest version of the
RS model. In this setup, the backreaction of the stabilization mechanism and
breaking of scale invariance are assumed to be small even close to the
confinement scale. The gauge theory is an approximately conformal field theory
(CFT), with spontaneously broken conformal invariance in the confined phase.
The approximate conformality suppresses the phase transition further, making
$\lambda\ll 1$, so that the bound on $N$ in equation (1) is impossible to
satisfy.

There is a large body of work devoted to relaxing this more stringent
constraint on the RS model by changing the details of the stabilisation
mechanism Randall2006 ; Nardini2007 ; Konstandin2010 ; Konstandin2011 ;
Dillon2017 ; VonHarling2017 ; Bruggisser2018a ; Bruggisser2018 ; Megias2018a ;
Bunk2018 ; Baratella2019 ; Agashe2019 ; Fujikura2019 ; Azatov:2020nbe ;
Megias2020 ; Agashe2020  in such a way that $\lambda\simeq 1$ and the phase
transition occurs more rapidly. However, the $N^{2}$ dependence of the
tunnelling rate is a generic feature of the confinement phase transition.
While modifying the stabilisation mechanism can change the numerical value of
the bound on $N$, in all these models the phase transition is exponentially
suppressed at large-$N$ and therefore subject to the bound in equation (1). As
pointed out in Hassanain2007 , for Klebanov-Strassler type constructions, the
effective value of $N$ itself varies over the extra-dimensional coordinate –
the relevant $N$ in this case is the value near the confinement scale.

In this paper we present a simple modification to the RS model where $N$ can
be made parametrically large without running into this cosmological bound. We
construct a scenario where the confinement scale grows with temperature, and
hence the universe can remain in the confined phase at all times in early
cosmology. For this reason, we call our mechanism avoided deconfinement. In
order to achieve this we consider the RS I model with the IR brane stabilised
by a Goldberger-Wise (GW) field $\Phi$ Goldberger1999 . By introducing new
scalars to the IR brane, we can generate a potential which stabilises the IR
brane at high temperatures. The mechanism we use to achieve this is
reminiscent of non-restoration of electroweak symmetry at high temperatures,
as considered in refs Weinberg1974 ; Meade2018 ; Baldes:2018nel ;
Glioti:2018roy ; Matsedonskyi2020 . Similar mechanisms have also been proposed
to avoid monopoles Langacker1980 ; Salomonson1985 ; Dvali1995  or domain walls
Dvali1995a  in Grand Unified theories, as models of CP violation at high
temperature Mohapatra1979 ; Mohapatra1979a , and in the $O(N)\times O(N)$
models of Orloff1996 ; Chai2020 .

|
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Figure 1: Left hand diagram shows the high temperature behaviour of the RS
model, while the right hand diagram shows the high temperature behaviour in
the AD model. Beyond a certain temperature in the RS model the IR brane is
stabilised behind the location of the horizon in the AdS-S phase, indicating
that the model is unstable against black hole formation. In the AD model, this
instability is lifted by introducing a temperature dependence to the
stabilisation mechanism in such a way that the IR brane is stabilised outside
the would-be horizon at high temperatures.

The modification we make to the RS model can lead to dramatic departures from
its standard cosmological history. Above a critical temperature $T_{c}$, the
confinement scale varies almost linearly with temperature $T$ leading to
$T$-dependent mass scales on the IR brane,

$\displaystyle M_{{\rm
ir}}(T)\propto\mu(T)=\mu(0)\left(\frac{T}{cT_{c}}\right)^{1/(1+\epsilon)}$ (3)

with $\epsilon\ll 1,c\sim\mathcal{O}(1)$. For a mass scale $M_{{\rm
ir}}>T_{c}$, this can imply that the ratio $T/M_{{\rm ir}}(T)$ reaches 1 at
very high temperatures, or potentially not at all (similar to low reheating
temperature models). Taking the standard model (SM) to be localised on the IR
brane, the $T$-dependence of the electroweak and QCD scales is as in equation
(3). If $v_{\rm ew}>T_{c}$, the electroweak phase transition occurs at
temperatures far above the TeV scale or is completely avoided. Furthermore, at
high temperature, fields localised in the UV of the RS model may have had
significant overlap with fields localised towards the IR of the theory, a
feature which may have applications to models of baryogenesis and dark matter
production.

The initial condition for our mechanism to work is that the universe exits
inflation in the RS phase with a stabilized IR brane. A relatively simple way
to achieve this is to have inflation with Hubble rate below the confinement
scale of the gauge group, or with an additional stabilization of the IR brane
during inflation. It will be an interesting future direction to study the
interplay of AD with inflationary models. After inflation the universe reheats
and the AD mechanism prevents the brane from falling behind the would-be AdS-S
horizon (see figure 1). Note that at high enough temperatures the AdS-S phase
will still be the preferred thermodynamic phase of the theory, but in the
avoided deconfinement model the RS phase is classically stable. The
probability of tunnelling from the RS to the AdS-S phase is exponentially
suppressed by $N^{2}$ factors, and can be made vanishingly small in the
large-$N$ limit.

The rest of this paper is organised as follows. In section 2 we describe the
early universe cosmology and summarise the details of the confinement phase
transition in various generalizations of the RS model that have been
considered in the literature. We go on to describe the avoided deconfinement
(AD) model in section 3 and show how the model leads to a stabilised IR brane
at high temperatures. In section 4 we present the low energy effective
Lagrangian and discuss some of the experimental signatures of the model. In
section 5 we then discuss the unique early universe cosmology of the model and
how this relates to other non-standard cosmological histories in the
literature. We also discuss potential applications of the model to
baryogenesis, dark matter production and potential gravitational wave
signatures in this section, before concluding and summarising in section 6.

##  2 The Supercooled Randall-Sundrum Model

In this section we review the standard cosmology of the RS type I model and
Goldberger-Wise field, and its dual gauge theory description via the gauge-
gravity duality. In the standard treatment of gauge-gravity duality at finite
temperature, the gauge theory partition function is defined on a manifold
$\mathcal{M}=S_{1}\times R^{3}$ with the temporal direction compactified on a
circle of radius $\beta=\pi/T$. The corresponding gravitational theory is
defined on a 5-dimensional manifold with $\mathcal{M}$ as the boundary. In
computing the gravitational partition function, all possible geometries
$\Sigma$ which satisfy the boundary condition $\partial\Sigma=\mathcal{M}$
must be integrated over Witten1998 . The partition function will however be
dominated by classical gravity solutions. Each semi-classical gravitational
solution $\Sigma_{i}$ which satisfies the boundary condition is interpreted as
a different phase of the CFT. At a given temperature, the geometry which
minimises the Euclidean action will give the dominant contribution to the
partition function, and therefore correspond to the preferred phase of the
CFT.

In the RS model, the UV brane cuts off the AdS space, and hence plays the role
of the boundary $\partial\Sigma$. The dual gauge theory is interpreted as a
field theory coupled to 4D gravity, defined on the manifold $\mathcal{M}$. One
of the possible classical solutions is,

$\displaystyle ds^{2}_{\rm RS}$
$\displaystyle=k^{2}\rho^{2}dt^{2}-\frac{d\rho^{2}}{k^{2}\rho^{2}}-\rho^{2}k^{2}dx_{i}^{2},\,$
(4)

with the space in the $\rho$ direction cut off at the position of the IR and
UV branes so that $\rho_{\rm ir}<\rho<\rho_{\rm uv}$. Here and throughout this
paper, we work in a frame where $\rho_{\rm uv}$ is fixed to the value
$\rho_{\rm uv}=k^{-1}$, where $k$ is the AdS curvature. A convenient
definition of the temperature of the 5D theory is the local temperature at the
UV brane. We will simply refer to this temperature as $T$. Thermal effects
tend to push the IR brane towards the horizon, rendering the RS solution
unstable Creminelli2002  at arbitrarily small temperatures in the absence of
stabilization. This instability can be lifted using the GW mechanism
Goldberger1999 .

The quasi-conformal theory dual to the RS model is a strongly coupled gauge
theory Arkani-Hamed2001 ; Rattazzi2001  with $\mathcal{O}(N^{2})$ degrees of
freedom, where $N$ can be determined by matching the entropy of the black hole
with the entropy of the high temperature phase of the gauge theory Gubser2001
,

$\displaystyle\frac{N^{2}}{16\pi^{2}}\simeq
12\left(\frac{M_{5}}{k}\right)^{3}.$ (5)

This relation can be modified by $\mathcal{O}(1)$ factors depending on strong
coupling effects in different gauge theory models. We see that the large-$N$
aspect of the 4D gauge theory is a crucial feature of these models, since it
corresponds to the hierarchy between the curvature scale $k$ and the 5D Planck
scale $M_{5}$ in the 5D gravitational theory. The $\rho$ direction can be
thought of as the RG scale of the conformal theory, with small $\rho$
corresponding to the IR of the theory. The UV and IR branes of the RS model
correspond to UV and IR cut-offs in the gauge theory. The cutoff at the IR
brane represents a spontaneous breaking of conformality in the IR due to
confinement in the gauge theory, while the UV brane represents explicit
breaking by the cutoff at the Planck scale Gubser2001 . The RS model with the
IR brane therefore corresponds to the confined phase of the conformal theory.
The GW mechanism corresponds to introducing a nearly marginal operator to the
CFT which explicitly breaks the conformal symmetry of the theory. The coupling
of this operator is dual to a scalar field in the RS model with a small bulk
mass. Introducing the GW scalar generates an effective potential for the
radion (identified with $\mu=k^{2}\rho_{{\rm ir}}$ in co-ordinates where the
location of the UV brane is fixed Charmousis2000 ), with a minimum at small
$\mu$ – the IR brane will then be stabilised at the minimum of this potential.

The RS solution with the IR brane becomes classically unstable at high
temperatures. There is another classical solution that contributes to the
finite temperature partition function given by the AdS-Schwarzschild (AdS-S)
geometry,

$\displaystyle ds^{2}_{\rm AdS-S}$
$\displaystyle=f(\rho)dt^{2}-\frac{d\rho^{2}}{f(\rho)}-\rho^{2}k^{2}dx_{i}^{2},\quad
f(\rho)=k^{2}\left(\rho^{2}-\frac{\rho_{h}^{4}}{\rho^{2}}\right)\,.$ (6)

The position of the horizon $\rho_{h}$ is set by the temperature $\rho_{h}=\pi
T/k^{2}$. The solution is cut off at $\rho=\rho_{{\rm uv}}$ by the UV brane as
before. The AdS-S solution is dual to the deconfined phase of the gauge
theory, with the Hawking temperature and entropy of the AdS black hole equal
to the corresponding quantities in the gauge theory. The AdS-S solution is
classically stable for any non-zero temperature, and is the thermodynamically
preferred phase of the theory at high temperatures.

As the universe cools below a critical temperature, the RS phase with the IR
brane becomes preferred and there is a first order phase transition between
the two phases which proceeds through a tunnelling process connecting the two
solutions. This tunnelling process is strongly suppressed, however, due to the
large change in free-energy in the two phases. The requirement that this phase
transition completes places bounds on $N$ for the model to be cosmologically
viable. These bounds typically require $N\sim\mathcal{O}(1)$, which is in
tension with the assumption of working in the large-$N$ limit.

###  2.1 (De)confinement Phase Transition in the RS Model

We consider the RS model with the GW stabilization mechanism. The bulk
Lagrangian contains gravity and the GW field ($\Phi$),

$\displaystyle S_{\rm bulk,RS}[G_{AB},\Phi]$ $\displaystyle=\int
d^{4}x\,d\rho\,\sqrt{G}\left[-2M_{5}^{3}R+24M_{5}^{3}k^{2}+\frac{1}{2}G^{AB}\partial_{A}\Phi\partial_{B}\Phi-\frac{1}{2}m_{\Phi}^{2}\Phi^{2}\right]\,.$
(7)

We also include brane localized terms,

$\displaystyle S_{{\rm uv},\rm RS}$ $\displaystyle=\int d^{4}x\sqrt{-g_{{\rm
uv}}}\left[-\lambda_{{\rm uv}}(\Phi^{2}-v_{{\rm
uv}}^{2})^{2}-24M_{5}^{3}k+\delta\Lambda_{{\rm uv}}\right]$ (8) $\displaystyle
S_{{\rm ir},\rm RS}$ $\displaystyle=\int d^{4}x\sqrt{-g_{\rm
ir}}\left[-\lambda_{{\rm ir}}(\Phi^{2}-v_{{\rm
ir}}^{2})^{2}+24M_{5}^{3}k+\delta\Lambda_{{\rm ir}}\right]\,.$ (9)

where $g_{{\rm uv},{\rm ir}}$ are the induced metrics on the UV and IR branes.
In the presence of the GW stabilization mechanism, only one combination of the
brane tension detuning parameters $\delta\Lambda_{i}$ needs to be tuned,
corresponding to the tuning of the 4D cosmological constant. Depending on the
sign of $m_{\Phi}^{2}$, these parameters may be required to lie in a certain
range for there to be a local minimum for the radion away from $\mu=0$. For
simplicity, here we set each of the detuning parameters to 0 and assume
$m_{\Phi}^{2}>0$. We also assume that the stabilization occurs in the limit of
small backreaction. These assumptions are not crucial and do not affect the
qualitative results. With these assumptions, the metrics in equations (4) and
(6) continue to be approximate classical solutions.

We can obtain the potential for the radion by integrating over the classical
solution for the GW field. In the limit where the brane localized terms fix
$\Phi(\rho_{{\rm uv},{\rm ir}})=v_{{\rm uv},{\rm ir}}$, the 4D effective
potential for the radion, $\mu\equiv k^{2}\rho_{{\rm ir}}$ is Creminelli2002 ,

$\displaystyle V(\mu)$ $\displaystyle=\epsilon v_{{\rm
uv}}^{2}k^{4}+\left[(4+2\epsilon)\mu^{4}(v_{{\rm ir}}-v_{{\rm
uv}}(\mu/k)^{\epsilon})^{2}-\epsilon v_{{\rm
ir}}^{2}\mu^{4}\right]+\mathcal{O}(\mu^{8}/k^{4})\,,$ (10)

with $\epsilon=\sqrt{4+m_{\Phi}^{2}/k^{2}}-2$. The minimum is obtained for:

$\displaystyle\mu_{{\rm TeV}}$ $\displaystyle=f(\epsilon)k\left(\frac{v_{\rm
ir}}{v_{\rm uv}}\right)^{1/\epsilon}\,,$ (11) $\displaystyle f(\epsilon)$
$\displaystyle=\left[\frac{4+\epsilon+\sqrt{\epsilon(4+\epsilon)}}{4+2\epsilon}\right]^{1/\epsilon}\sim\mathcal{O}(1)\,.$
(12)

A relatively modest hierarchy in $v_{{\rm uv},{\rm ir}}$ and $\epsilon\sim
1/10$ can generate an exponential hierarchy between $k$ and $\mu_{{\rm TeV}}$.
At energies $\lesssim\mu_{\rm TeV}$ the effective theory is a 4D theory with a
tower of Kaluza-Klein (KK) states with masses $\sim\mu_{{\rm TeV}}$. In the 4D
theory, dimensionful parameters involving fields localized on the IR brane,
such as the Higgs mass parameter, scale with $\mu_{{\rm TeV}}$, thus
explaining the electroweak hierarchy elegantly.

At low-temperatures, $T<\mu_{\rm TeV}$, both classical solutions – the
stabilized RS solution with UV and IR branes, and the AdS-S solution with a UV
brane and a black hole horizon – are (meta)stable. However, at high-
temperatures $T\gg\mu_{\rm TeV}$, the minimum of the radion potential is
behind the AdS-S horizon $\mu_{\rm TeV}<\rho_{h}k^{2}$, indicating that the
AdS-S solution is the only classical solution. During the early universe the
universe is in the AdS-S phase; to get to the RS phase the universe needs to
undergo a first order phase transition.

The tunnelling rate per unit volume for the phase transition is,

$\displaystyle\Gamma$ $\displaystyle\simeq R_{c}^{-4}\exp(-S_{b})$ (13)

where $S_{b}$ is the bounce action for the tunnelling transition, and $R_{c}$
is the radius of the critical bubble Coleman1977 ; Linde1983 . The field
configuration for the transition from the AdS-S phase to the RS phase involves
moving the black hole horizon to the far IR, $\rho_{h}\to 0$, and then
nucleating the IR brane at $\rho=0$ and bringing it to larger values of
$\rho$. Therefore this field configuration probes the geometry in the region
where the local temperature is super-Planckian and stringy corrections would
be relevant. However, in the case where the transition temperature is low, and
there is an approximate conformal symmetry in the IR, the dominant
contribution to the bounce is dictated by the radion dynamics and can be
estimated while ignoring the gravitational contribution to the bounce
Agashe2020 . Even so, since the field configuration probes the geometry in the
far IR, the bounce action for this configuration can depend sensitively on the
details of the GW stabilisation, and other physics in the IR. We summarize the
results for the bounce action that have been considered in various limits in
the literature next.

###  2.2 Bounce action from the radion

In a large class of models, the phase transition is captured by the dynamics
of the radion Creminelli2002 ; Agashe2019 ; Agashe2020 . The general radion
effective field theory can be understood in terms of the dual 4D theory. The
4D theory is a near-conformal field theory coupled to gravity. The
gravitational sector breaks the conformal symmetry explicitly, but below the
gravitational cutoff an approximate conformal symmetry survives. For a
stabilised RS geometry with a light radion, the 4D effective theory below the
KK scale $\mu_{\rm TeV}$ is well-described by an effective theory of a
spontaneously broken (approximate) conformal symmetry, with the light
radion/dilaton as the pseudo-Nambu-Goldstone boson.

In this section we work in this 4d picture to study a few such generalisations
that have been studied in the literature. As we will see, in each case the
first order phase transition is highly suppressed. The effective Lagrangian
for the dilaton Creminelli2002 ; Coradeschi2013  can be written as,

$\displaystyle\mathcal{L}_{\rm eff}$
$\displaystyle=\frac{N^{2}}{16\pi^{2}}\left[(\partial\mu)^{2}-\lambda(g(\mu))\mu^{4}\right]\,,$
(14)

where the $\mu$-dependence in $g(\mu)$ denotes the explicit breaking of
conformal symmetry due to the GW deformation. We expect the dilaton to be the
lightest bound state of the gauge theory Goldberger2000 ; Pomarol2019  as it
is the pNGB of the broken dilation symmetry, so is the only relevant degree of
freedom in the IR of the theory. The $N^{2}$ factor makes explicit the fact
that the dilaton is interpreted as a glueball state in a 4D large-N gauge
theory.

The free energy in the (de)confined phase is well approximated by,

$\displaystyle F_{\rm confined}$ $\displaystyle=V(\mu_{{\rm
TeV}})=-\frac{N^{2}}{16\pi^{2}}\lambda_{\rm TeV}\mu_{\rm TeV}^{4}+V_{0}$
$\displaystyle F_{\rm deconfined}$
$\displaystyle=C-2\pi^{4}(M_{5}/k)^{3}T^{4}=C-\frac{\pi^{2}}{96}N^{2}T^{4}$
(15)

where we have defined $\lambda_{\rm TeV}\equiv|\lambda(g(\mu_{\rm TeV}))|$,
and added a constant $V_{0}$ to ensure that the vacuum energy at the minimum
is zero. Notice that $\lambda(g(\mu_{\rm TeV}))<0$ for $\mu_{\rm TeV}$ to be
the minimum of the potential. $C$ can be calculated by matching the free
energy at $\mu=T=0$. The critical temperature can be calculated by equating
the free energy in the two phases at the transition,

$\displaystyle C-\frac{\pi^{2}}{96}N^{2}T_{c}^{4}$
$\displaystyle\simeq-\frac{N^{2}}{16\pi^{2}}\lambda_{\rm TeV}\mu_{\rm
TeV}^{4}+V_{0}$ (16) $\displaystyle\Rightarrow T_{c}$
$\displaystyle\simeq\left[\frac{6\lambda_{\rm
TeV}}{\pi^{4}}\right]^{1/4}\mu_{\rm TeV}\,.$ (17)

When $\lambda_{\rm TeV}\ll 1$, the transition temperature $T_{c}\ll\mu_{\rm
TeV}$, and the approximation of radion domination is well justified.

If the phase transition is prompt, it completes for $T\sim T_{c}$. In this
case the bubble has O(3) symmetry, and the action can be estimated in the
thin-wall regime (see e.g. Kaplan2006 ),

$\displaystyle\frac{S_{3}}{T}$
$\displaystyle\sim\frac{N^{2}}{8}\left[\frac{1}{\lambda_{\rm
TeV}}\right]^{3/4}\frac{T_{c}}{T}\left(1-\left(\frac{T}{T_{c}}\right)^{4}\right)^{-2}\,.$
(18)

This explicitly shows the general enhancement of the bounce action by $N^{2}$,
and often also by the weak coupling $\lambda_{\rm TeV}$.

We can evaluate the bounce action for the GW model considered in section 2.1
above. The quartic $\lambda_{\rm TeV}$ in this case is,

$\displaystyle\lambda_{\rm TeV}$
$\displaystyle=\frac{16\pi^{2}}{N^{2}}\epsilon^{3/2}v_{\rm ir}^{2}$ (19)

which leads to the following parametric form for the bounce action
Creminelli2002 ,

$\displaystyle\frac{S_{3}}{T}$
$\displaystyle\sim\frac{N^{7/2}}{\epsilon^{9/8}v_{\rm
ir}^{3/2}}\frac{T_{c}}{T}\left(1-\left(\frac{T}{T_{c}}\right)^{4}\right)^{-2}\,.$
(20)

The action is not only enhanced by the factor of $N^{2}$, but also by the
small quartic coupling of the radion, which increases the dependence on $N$ to
$N^{7/2}$. There is an additional enhancement by $1/\epsilon$, related to the
fact that the scale symmetry violation at $\mu_{\rm TeV}$ is parametrised by
$\epsilon$. The exact power of $\epsilon$ that appears can depend on the
implementation of the GW mechanism Creminelli2002 ; Nardini2007 ; Agashe2019 ;
Agashe2020 .

More generally, we can see that the action is enhanced for small scale
symmetry violation encoded in $\beta_{\lambda}\ll 1$. The zero temperature
minimum is determined by the running quartic, $\lambda(g(\mu))$,

$\displaystyle\partial V(\mu)/\partial\mu=[4\lambda(g(\mu_{\rm
TeV}))+\beta_{\lambda}(g(\mu_{\rm TeV}))]\mu^{3}=0\,.$ (21)

Thus for a nearly scale invariant theory at $\mu_{\rm TeV}$, $\lambda_{\rm
TeV}$ will be generically small.

If the transition is not prompt, then it will then take place in the
supercooled regime, $T\ll T_{c}$. In the case where there is a barrier in the
radion potential between $\mu_{\rm TeV}$ and $\mu\sim 0$, the bounce
configuration is essentially the same as the zero temperature tunnelling, and
has an O(4) symmetric bounce action Creminelli2002 ; Nardini2007

$\displaystyle S_{4}$ $\displaystyle\sim\frac{N^{2}}{16\pi^{2}\lambda_{\rm
TeV}}.$ (22)

As before, the explicit factor $N^{2}$ appears. Again, we see that in the case
of the simplest RS+GW model above, the parametric dependences are even
stronger,

$\displaystyle S_{4}$
$\displaystyle\sim\frac{N^{4}}{(4\pi)^{4}\epsilon^{3/2}v_{\rm ir}^{2}}.$ (23)

If there is no barrier between $\mu\sim 0$ and $\mu=\mu_{\rm TeV}$, then the
“release point” for the radion field can be very small $\mu\sim T$ even for
supercooled transition, so the smallest bounce action is still obtained by an
O(3) symmetric bounce. For example, in the case where conformal symmetry is
restored in the IR ($\epsilon>0$ for the GW field) Agashe2019 , the radion
potential near the origin is $V\sim\lambda(0)\mu^{4}$. The bounce action is
then the $T\ll T_{c}$ limit of equation (18),

$\displaystyle\frac{S_{3}}{T}$
$\displaystyle\sim\frac{N^{2}}{8[\lambda(0)]^{3/4}}$ (24)

which is no longer suppressed by the small parameter $\epsilon$. In the case
where $\epsilon\lambda_{\rm TeV}$ is not parametrically small, radion dynamics
no longer suffice to estimate the bounce. However, it may still be possible to
estimate the bounce in the 5D effective theory Agashe2020  and is found to be
$\mathcal{O}(N^{2})$.

When $\epsilon<0$, the GW field profile grows towards the IR. Consequently,
the higher order terms in the GW potential might become important and the
approximate conformality $\partial_{\log\mu}g(\mu)\sim\epsilon$ might be
broken as we approach $\mu\lesssim\mu_{\rm TeV}$. In such cases the
enhancement of the bounce action by $1/\epsilon$ will be absent, even though
the EW/Planck hierarchy is set by small epsilon. This can be explicitly seen
in explicit holographic constructions Bigazzi2020 ; Hassanain2007 , or in RS
models with more general stabilisation mechanisms Konstandin2010 ; Nardini2007
; Konstandin2011 ; Dillon2017 ; VonHarling2017 ; Bruggisser2018 ;
Bruggisser2018a ; Megias2018a ; Baratella2019 ; Agashe2019 ; Agashe2020 ;
Fujikura2019 ; Megias2020 ; Bunk2018 .

We see from the examples above that while the details of the bounce action
depend on the actual theory, it takes the form $S_{b}\simeq N^{2}/\lambda$ in
each case, with $\lambda\lesssim 1$. This has far-reaching consequences for
early universe cosmology. Either the universe is required to be reheated to
temperatures lower than the confinement scale, or there is a strong constraint
on the maximal $N$ allowed.

If the rate of tunnelling is smaller than Hubble, the universe will get stuck
in the false vacuum Guth1983 . Since the true vacuum at zero temperature is
assumed to have a (nearly) vanishing cosmological constant, the deconfined
vacuum has a large positive cosmological constant $C\sim N^{2}T_{c}^{4}$ at
low temperatures and starts to inflate with $H\simeq NT_{c}^{2}/M_{\rm pl}$.
In a Hubble volume, the probability of completing the phase transition within
a Hubble time is,

$\displaystyle P$ $\displaystyle=\Gamma/H^{4}$ (25)

If $P\ll 1$, then the universe eternally inflates. This gives us a bound on
$N$,

$\displaystyle N$ $\displaystyle\lesssim 2\sqrt{\lambda\log\frac{M_{\rm
pl}}{T_{c}}}$ (26)

We have replaced the inverse critical radius by $T_{c}$; unless the bubble
size is exponentially smaller, this is a reasonable approximation. In many
models considered above, $\lambda$ is parametrically small. For the RS+GW
model above, $S_{b}$ is enhanced both by $N$ as well as $1/\epsilon$, making
it impossible to satisfy the constraint above. Even without these
enhancements, the calculations above assume dilaton dominance, which requires
$\lambda_{\rm TeV}\ll 1$ Agashe2019 . Therefore, it is hard to get
$S_{b}\lesssim N^{2}$ in a controlled approximation. This translates into a
bound $N\lesssim 12$ for $T_{c}\sim 1\,{\rm TeV}$. From equation 5, we see
that the hierarchy between the 5D Planck scale and the AdS curvature is
$(M_{5}/k)\lesssim 1$. This lack of hierarchy makes the 5D effective
gravitational theory very delicate.

One avenue to evade this cosmological bound is to avoid reheating the universe
above the TeV scale. This may require a more intricate inflationary mechanism,
as well as solutions to baryogenesis at the electroweak scale or below
Baldes:2018nel ; Bruggisser2018 ; Bruggisser2018a . In the next section we
outline the avoided deconfinement mechanism, where the GW stabilisation of the
radion is temperature dependent and the IR brane is stabilised at arbitrarily
high temperatures. This allows for parametrically large hierarchies between
$M_{5}$ and $k$, and an early cosmology without a stringent restriction on the
reheat temperature.

##  3 5D Model for Avoided Deconfinement

In this section we modify the RS model with GW field ($\Phi$) by including
extra scalars localised to the IR brane***Localised fields on the IR brane may
be required to arise from corresponding bulk modes with masses below the 5D
cutoff Fichet:2019owx . These bulk modes will then have an associated tower of
KK states, but this detail will not affect our discussion.. Given a suitable
set of parameters, the effect of this will be to realise a model where the new
scalars provide a metastable minimum for the radion at high temperature,
avoiding the formation of the AdS-S black hole.

We make a simple modification to the RS model described in equation (9) by
adding scalar field(s) $S$ to the IR brane. The action is:

$\displaystyle S$ $\displaystyle=S_{\rm bulk,RS}+S_{{\rm uv},\rm RS}+S_{{\rm
ir},\rm RS}+S_{{\rm ir},\rm AD}\,.$ (27)

where $S_{\rm bulk,RS}$ and $S_{{\rm uv}/{\rm ir},\rm RS}$ are the RS model
bulk and brane actions which are unchanged from their definition in equations
(7), (8),and (9). We continue to choose the detuning parameter
$\delta\Lambda_{\rm ir}=0$. As usual, this simplifying assumption can be
relaxed. The modified IR brane action, $S_{{\rm ir},\rm AD}$ includes $N_{s}$
real scalars $S$ localised to the brane. The additional terms in the IR brane
action is

$\displaystyle S_{{\rm ir},\rm AD}$ $\displaystyle=k^{4}\int_{\rho=\rho_{\rm
ir}}d^{4}x\sqrt{-g_{\rm ir}}\sum_{i=1}^{N_{s}}\left[\frac{1}{2k^{2}}g_{{\rm
ir}}^{\mu\nu}\partial_{\mu}S_{i}\,\partial_{\nu}S_{i}-\frac{\lambda_{s}}{4}(S_{i}^{2}-v_{s}^{2})^{2}-\frac{\gamma}{6}S_{i}^{3}\right],$
(28)

where we have explicitly included factors of $k$ so that the parameters
$\lambda_{s},\gamma,v_{s}$ as well as the field $S$ are dimensionless. We will
suppress the index $i$ on $S$ for notational simplicity. In order for the
potential to remain bounded from below, the coupling $\lambda_{s}$ must be
positive. The value of the masses and quartic couplings of each field $S$ do
not have to be equal, but for simplicity we will take them to be the same. For
unequal couplings our results below can be reinterpreted using statistical
averages over the $S$ ensemble. Each $S$ has an approximate $Z_{2}$ symmetry
that is spontaneously broken at zero temperature. The coupling $\gamma$ that
weakly breaks the $Z_{2}$ symmetry for each $S$ is introduced to avoid domain
wall problems, and can be very small in a technically natural way.

Before presenting the consequences of adding these extra scalars, we summarize
the choice of parameters for which our approximations are under theoretical
control. The primary goal of the AD setup is to generate a classically stable
minimum for the radion at high temperatures for arbitrary $N$, therefore
avoiding a confinement phase transition entirely and putting the large-$N$
approximation on a firmer footing. It is then worth highlighting the validity
of the large-$N$ expansion, especially in light of adding extra matter on the
IR brane. Requiring the gravitational loop counting parameter to be small,
$N_{\rm species}/N^{2}\lesssim 1$, restricts the number of scalars we can add.
As we show below, the AD mechanism does require $N_{s}>1$ to operate. However,
in order to obtain a classically stabilized radion at high temperatures,
$N_{s}$ can be parametrically smaller than $N^{2}$. In this case, the black
hole phase does have a lower free energy, but the tunneling rate from the AD
phase to the black hole phase is exponentially suppressed by a tunnelling
exponent of order $\sim N^{3}$. We present an estimate of the tunneling rate
in appendix A. Thus, the parameter $N$ can be taken arbitrarily large while
keeping other parameters in our model fixed, ensuring that the $1/N$ expansion
is well under control.

In order to understand the other parametric scalings in the Lagrangian, it is
useful to characterize the cosmological history of the AD model by the
following three scales:

  1. 1.

the temperature $T_{s}$ at which the scalars $S$ undergo a (crossover) phase
transition;

  2. 2.

the temperature $T_{c}$ at which the AD construction begins to take effect and
the position of the IR brane begins to vary with temperature; and

  3. 3.

the zero-temperature radion vev, $\mu_{\rm TeV}$.

The AD model requires the hierarchy $T_{s}<T_{c}$, due to the fact that the
addition of the new scalars $S$ only generates the desired finite-temperature
effects in the symmetric phase. At temperatures $T>T_{c}$ the position of the
IR brane is moved from the GW minimum due to thermal effects. As mentioned
above, in this temperature range the confined phase is metastable, in contrast
to the usual RS model where the confined phase becomes classically unstable at
high temperature. Another condition on the parameter space comes from the
requirement that the IR brane is stabilized at a radius where the local
temperature is small enough such that the backreaction on the bulk geometry is
small. At the confinement scale, this condition implies

$\displaystyle T_{c}<\frac{\mu_{\rm TeV}}{\pi},$ (29)

This condition becomes stronger logarithmically in temperature, and can fail
at very high temperatures, which sets a maximum temperature $T_{\rm max}$ for
the AD mechanism to operate. As we explain in further detail in section 3.2,
this leads to the following condition on the parameters of the model:

$\displaystyle 1>\frac{\pi T_{c}}{\mu_{\rm
TeV}}>\sqrt{\frac{6}{N_{s}}}\frac{m_{\varphi}}{m_{s}}>v_{s}$ (30)

where $m_{\varphi},m_{s}$ are the masses of the radion and scalar ($s$)
fluctuations at zero temperature. Since the mass of the radion $m_{\varphi}$
cannot be too small phenomenologically, this leads us to require a moderately
large number of scalar fields $N_{s}$ for the AD model to work.

###  3.1 Finite temperature effective potential for the radion

We work in a regime where the local 5D temperature remains below
$(M_{5}^{3}k^{2})^{1/5}$ everywhere in the 5th dimension, so that the finite
temperature effects have a negligible backreaction effect on the bulk
geometry. At any temperature $T$ and the position of the IR brane $\rho_{\rm
ir}(T)$, we can solve the equations of motion for the GW field on the
background RS metric with the same boundary conditions, $\Phi(\rho_{{\rm
ir}/{\rm uv}}(T))=k^{3/2}v_{{\rm ir}/{\rm uv}}$. The bulk solution is,

$\displaystyle\Phi(\rho)=A\rho^{-4-\epsilon}+B\rho^{\epsilon}$ (31)

where $A,B$ are fixed by the boundary conditions. As above, $\epsilon\approx
m_{\Phi}^{2}/(4k^{2})$, which we take to be positive. We expand in
fluctuations around this classical solution with the size of the extra
dimension equal to $\rho_{\rm ir}(T)$. We decompose the bulk fluctuations into
Kaluza-Klein modes and integrate over the 4D modes to derive the finite
temperature effective potential.

The temperature-dependent effective potential can be broken up into the tree-
level potential and the one-loop potential Delaunay2008 ; Curtin2018 :

$\displaystyle V_{\rm eff}(T)$ $\displaystyle=V_{\rm tree}+\Delta V^{\rm
CW}_{1}+\Delta V^{T}_{1}(T)$ (32)

where we have separated the 1-loop contribution into a piece that goes to 0 at
zero temperature $\Delta V^{T}_{1}(0)=0$. The zero temperature Coleman-
Weinberg potential, $\Delta V^{\rm CW}_{1}$ includes the usual UV-divergences
one would encounter in these calculations. The tree-level 4D action is
obtained as above by integrating the classical solution over the extra
dimension. The potential is given by,

$\displaystyle V_{\rm tree}(\mu,S_{i})$
$\displaystyle=\mu^{4}\left[(4+2\epsilon)(v_{\rm ir}-v_{\rm
uv}(\mu/k)^{\epsilon})^{2}-\epsilon v_{\rm
ir}^{2}+\sum_{i=1}^{N_{s}}\left(\frac{\lambda_{s}}{4}(S^{2}_{i}-v_{s}^{2})^{2}+\frac{\gamma}{6}S_{i}^{3}\right)\right]$
(33)

where we have used $\mu=k^{2}\rho_{\rm ir}(T)$. We have also suppressed the
$T$-dependence in the notation.

The one-loop contribution is obtained by integrating over the fluctuations.
The finite temperature contribution to the potential depends on the effective
mass of the fluctuations around the classical solution. The relevant particles
in our case are the radion, the new scalars and the SM fields. Including the
kinetic term for the radion Coradeschi2013 ; Chacko:2014pqa  and the scalars,
we find the following action for the fluctuations:

$\displaystyle S$ $\displaystyle=\int d^{4}x\left[\mathcal{L}_{\rm
SM}+\frac{1}{2}(\partial\varphi)^{2}+\frac{1}{2}(\partial s_{i})^{2}-V_{\rm
tree}\left(\mu\left(1+\frac{\varphi}{F_{\varphi}}\right),S_{i}+\frac{s_{i}}{\mu}\right)\right]$
(34)

We have introduced the canonically normalized fluctuations for the radion
$\varphi$ and the scalars $s$. The radion decay constant

$\displaystyle F_{\varphi}=\frac{N}{2\sqrt{2}\pi}\mu.$ (35)

The indices in the kinetic term are now contracted with the 4d Minkowski
metric.

The field-dependent masses of these particles are defined as the second
derivative of the potential w.r.t. the corresponding field. The SM particle
masses and the radion and $s$ masses all scale with $\mu$. For a large number
of scalars $N_{s}\gg 1$, the thermal potential is dominated by loops of
$s_{i}$†††Note that since the SM contribution to the $\mu$ potential is
proportional the mass of the SM field, only the contributions from $t,W,Z,h$
are sizeable.. The field-dependent masses of $s_{i}$ are,

$\displaystyle m_{s,i}^{2}$
$\displaystyle=\mu^{2}\left[-\lambda_{s}v_{s}^{2}+3\lambda_{s}S_{i}^{2}+\gamma
S_{i}\right]$ (36)

The other modes in the spectrum are the KK modes of the graviton, the GW field
and other fields in the bulk. The mass of the $n$th KK mode is approximated by
Gherghetta2000 :

$\displaystyle m_{n}$
$\displaystyle\simeq\left(n+\frac{2+\epsilon}{2}-\frac{3}{4}\right)\pi\mu.$
(37)

The $\rho$-coordinate of the would-be AdS-S horizon is $\rho_{h}=\pi T/k^{2}$.
Therefore, the condition that the IR brane is stabilised outside the AdS-S
horizon, $\rho_{\rm ir}(T)>\rho_{h}$ implies that the higher KK modes are not
excited at any $T$, and can be safely neglected in the thermal potential.

The Coleman-Weinberg potential is given by,

$\displaystyle V_{1}^{\rm CW}(\mu,S_{i})$
$\displaystyle=\sum_{i=1}^{N_{s}}\frac{1}{64\pi^{2}}m_{s,i}^{4}\left(\log\left[\frac{m_{s,i}^{2}}{\mu_{R}^{2}}\right]-\frac{3}{2}\right)$
(38)

where $\mu_{R}$ is a renormalisation scale. A convenient choice of the
renormalisation scale for the dynamics on the IR brane is $\mu$ itself
Sundrum:2003yt . With this choice, we do not generate large hierarchies of
scale on the IR brane and the one-loop corrections at zero-temperature stay
small. We have included all terms allowed by the scale symmetry of the radion
and the $Z_{2}$ symmetries in the $s$-sector, as well as leading terms
violating these symmetries parameterized by the small parameters
$\epsilon,\gamma$. Thus, the higher order terms can be safely neglected and we
will simply absorb the UV divergent pieces into a redefinition of couplings
and masses as renormalized quantities.

The finite-temperature one-loop contributions from $s_{i}$ are

$\displaystyle\Delta V_{1}^{T}(\mu(T),S_{i}(T),T)$
$\displaystyle=\sum_{i=1}^{N_{s}}\frac{T^{4}}{2\pi^{2}}\int
dk\,k^{2}\log\left[1-\exp\left(-\sqrt{k^{2}+\frac{m_{s,i}^{2}(\mu,S_{i})}{T^{2}}}\right)\right]$
(39)
$\displaystyle\equiv\sum_{i=1}^{N_{s}}\frac{T^{4}}{2\pi^{2}}J_{b}\left(\frac{m_{s,i}^{2}(\mu,S)}{T^{2}}\right)$
(40)

We approximate the thermal function $J_{b}$ by assuming that $m_{s}(T)\ll T$.
At high temperature the thermal function can be approximated as,

$\displaystyle J_{b}(y^{2})$
$\displaystyle\approx-\frac{\pi^{4}}{45}+\frac{\pi^{2}}{12}y^{2}-\frac{\pi}{6}y^{3}-\frac{1}{32}y^{4}\left(\log\frac{y^{2}}{\pi^{2}}+2\gamma_{E}-\frac{3}{2}\right)\,.$
(41)

where $\gamma_{E}$ is the Euler-Mascheroni constant. The field-dependent
$\log$ pieces cancel between the Coleman-Weinberg terms and the thermal
corrections. The renormalisation scale $\mu_{R}\simeq\mu$ scales with the
temperature (as we show below), and hence we do not get any enhanced large-log
pieces, and we can safely ignore the terms of $\mathcal{O}(y^{4})$. Then,

$\displaystyle\Delta V_{1}^{T}(\mu,S,T)$
$\displaystyle\simeq\frac{N_{s}}{24}T^{2}\mu^{2}\left(-\lambda_{s}v_{s}^{2}+3\lambda_{s}{S^{2}}+\gamma{S}\right)$
$\displaystyle\qquad-\frac{N_{s}}{12\pi}T\mu^{3}\left(-\lambda_{s}v_{s}^{2}+3\lambda_{s}{S^{2}}+\gamma{S}+\lambda_{s}\frac{T^{2}}{4\mu^{2}}\right)^{3/2}$
(42)

where we have used the fact that each of the scalar vevs $S_{i}=S$ when
$\\{\lambda_{s},v_{s},\gamma\\}$ are taken to be the same for each $s_{i}$.
The extra term involving $T^{2}$ in the second term above is a result of
performing the leading daisy resummation, where we replace the field-dependent
mass $m_{s,i}^{2}$ by $m_{s,i}^{2}+\Pi_{i}$ in equation (40), with $\Pi_{i}$
the leading temperature contribution to the one-loop thermal mass.

Thus, we see that at high temperatures $T>T_{s}\simeq\frac{1}{2}v_{s}\mu_{\rm
TeV}$, thermal effects drive to restore the (approximate) $Z_{2}$ symmetry in
$S$, so that $\langle S\rangle\ll 1$. This generates a tachyonic direction for
$\mu$, providing a finite temperature stabilization. As the universe cools,
the $S$ symmetry gets broken, and the radion settles down close to its zero
temperature minimum dictated by the GW part of the potential. The thermal
potential can be minimized numerically, and for a range of parameters the
radion remains stabilized outside the would-be AdS-S horizon at high
temperatures.

###  3.2 High temperature radion stabilization

The minimum of the radion potential can be simply approximated in two distinct
regimes:

$\displaystyle\mu(T)$ $\displaystyle=\left\\{\begin{array}[]{ll}\mu_{{\rm
TeV}}&T<T_{c}\\\ \mu_{{\rm
TeV}}\left(\frac{T}{cT_{c}}\right)^{\frac{1}{1+\epsilon}}&T\gg
T_{c}\end{array}\right.$ (45)

where the constant $c$ is given by:

$\displaystyle c^{2}=\frac{4v_{\rm uv}^{2}}{\epsilon^{3/2}v_{\rm
ir}^{2}}\left(\frac{\mu_{\rm TeV}}{k}\right)^{2\epsilon}$ (46)

The zero temperature value of the radion minimum $\mu_{\rm TeV}$ is well
approximated by equation 11, up to an $\mathcal{O}(\gamma v_{s}^{3})$
correction to the $\mu^{4}$ co-efficient which is a result of the potential
for $S$ not vanishing at the zero-temperature minimum. The transition
temperature $T_{c}$ is is the temperature at which the radion starts to move
and is given by,

$\displaystyle T_{c}^{2}$
$\displaystyle\simeq\frac{6}{N_{s}}\frac{m_{\varphi}^{2}}{m_{s}^{2}}\mu_{\rm
TeV}^{2}=\frac{24\mu_{\rm
TeV}^{2}}{N_{s}(\lambda_{s}v_{s}^{2})}\epsilon^{3/2}v_{\rm ir}^{2}$ (47)

where $m_{\varphi},m_{s}$ here are the zero-temperature masses for
$\varphi,s$. Since $\mu(T)/T$ is slowly growing, there is a maximum
temperature $T_{\rm max}$ beyond which the IR brane would fall behind the
horizon,

$\displaystyle T_{\rm max}$ $\displaystyle\sim\mu(T_{\rm max})/\pi\Rightarrow
T_{\rm max}=\frac{\mu_{\rm TeV}}{\pi}\left(\frac{\mu_{\rm TeV}}{\pi
cT_{c}}\right)^{1/\epsilon}$ (48)

This sets a (mild) bound on the reheat temperature of the universe. This is at
an exponentially high scale for $\delta\equiv\pi T_{c}/\mu_{\rm TeV}\ll 1/c$.
The AD transition temperature should also be higher than the $Z_{2}$ symmetry
restoration temperature, $T_{c}>T_{s}$. These two requirements give us the
following inequalities on our parameter space,

$\displaystyle\delta>\sqrt{\frac{6}{N_{s}}}\frac{m_{\varphi}}{m_{s}}>v_{s}$
(49)

For illustration, we choose the following benchmark values‡‡‡The value of
$\epsilon$ was chosen such that $\mu_{\rm TeV}\simeq 100\ {\rm TeV}$. A more
generic choice works equally well.

$\displaystyle\left\\{k=6\times 10^{16}\ {\rm GeV},\epsilon=4.13\times
10^{-2},v_{\rm uv}=10^{-3},v_{\rm ir}=3\times 10^{-4},\right.$
$\displaystyle\left.N_{s}=100,\lambda_{s}=1,v_{s}=2\times
10^{-3},\gamma=-10^{-8}\right\\},$ (50)

and find the following parameters

$\displaystyle\mu_{\rm TeV}$ $\displaystyle\simeq 1.8\times 10^{-12}k\simeq
100\ {\rm TeV}$ (51) $\displaystyle T_{c}$ $\displaystyle\simeq 700\ \ {\rm
GeV}\,.$ (52)

The maximum temperature $T_{\rm max}$ for this case is around $5\times
10^{11}\ {\rm GeV}$. The temperature evolution of various scales in this
benchmark is illustrated in figure 2.

|
---|---
Figure 2:  Plots showing the dependence of mass scales in the theory with
temperature, for parameters in equation (50). The left plot shows the radion
expectation value ($\mu$), the scalar expectation value ($S\mu$) and the QCD
scale, while the second plot shows the same quantities and the KK scale with
the dependence on the radion factored out. The vertical line shows the
critical temperature $T_{c}\sim 700\ \ {\rm GeV}$. The dashed black line in
the first plot shows the horizon location $k^{2}\rho_{h}$ as a function of
temperature.

Comparing the AD model to the usual RS model, the high temperature behaviour
is vastly different. In the RS model, as we move to high temperature, thermal
effects drive the IR brane towards the AdS boundary, where it eventually
collapses to form an AdS-S black hole Hebecker2001 . The model then remains in
the black hole phase until tunnelling to the RS phase through a first order
phase transition which is highly suppressed in calculable models, as was shown
in section 2. In contrast, the model presented here describes a situation
where the IR brane is stabilised closer to the UV end of the warped direction
at high temperatures, then falls approximately linearly with temperature into
the IR before stabilising at a constant value deep in the IR as in the RS
case. It should be noted that at very high temperatures, the black hole phase
can still have a lower free energy than the AD phase, but the IR brane remains
meta-stabilised. The only way the system can transition to the black hole
phase is through a first order phase transition which is exponentially
suppressed.

The AD model also introduces a novel temperature dependence of the mass scales
on the IR brane, which scale linearly with the radion. Assuming that the SM is
confined to the IR brane, the left panel of figure 2 shows the dependence of
the QCD scale with temperature in the AD model. The other dimensional
parameters of the SM, such as the electroweak vacuum value, exhibit a similar
scaling above the critical temperature, $T_{c}$, of the theory. Above $T_{c}$
the temperature only increases marginally relative to the other scales of the
theory. This leads to the KK modes being frozen out to arbitrarily high
temperature, as shown in the right panel of figure 2. A similar behaviour
would occur for any other scales of the theory which are above $T_{c}$ – they
are frozen out to much higher temperatures than in the usual RS model. In
section 5 we discuss this behaviour and its potential implications for BSM
phenomenology in more detail.

##  4 Low Temperature Phenomenology

In this section we show we study the constraints on the avoided deconfinement
model from collider results and ALP searches. For simplicity we have assumed
that the SM is confined to the IR brane. More realistic models typically have
some or all SM fields propagating in the bulk, which can offer an explanation
of the hierarchical Yukawa couplings in the SM Gherghetta2000 ; Agashe2003 ;
Casagrande2008  but also lead to new constraints from flavour-violating
processes Agashe2005 ; Bauer2009 . The constraints on the RS model have been
well-studied. Here we will focus on the new features that are required for the
AD mechanism to work.

The qualitative features that the mechanism requires can be estimated using
the inequality in equation 30. For $\delta\ll 1$, we need,

$\displaystyle\frac{m_{s}}{\mu_{\rm TeV}}$ $\displaystyle\simeq v_{s}<\delta$
$\displaystyle\frac{m_{\varphi}}{\mu_{\rm TeV}}$
$\displaystyle<\delta^{2}\sqrt{N_{s}}\,.$ (53)

Experimental constraints on a very light radion will lead us to require a
large number of scalars, $N_{s}\gg 1$, with masses below the confinement
scale.

In section 4.1 we write down the effective Lagrangian – the relevant degrees
of freedom being the SM fields, the radion $\varphi$, and the AD scalar(s)
$s$. We ignore higher dimensional operators which arise from integrating out
KK modes and $1/N$ suppressed stringy corrections, taking these to be
negligible. In section 4.2 we then describe the dominant experimental
constraints in different regions of parameter space.

###  4.1 Effective Lagrangian at zero temperature

The tree-level interactions of the radion $\varphi$ with the SM fields can be
written compactly by the replacing the mass terms in the SM by a $\varphi$
dependent mass,

$\displaystyle\mathcal{L}^{(\rm tree)}[m_{i}]\to\mathcal{L}^{(\rm tree)}_{\rm
int}\left[m_{i}\left(1+\frac{\varphi}{F_{\varphi}}\right)\right]$ (54)

where we the decay constant $F_{\varphi}$ was defined in equation 35. This
produces Yukawa-like interactions of $\varphi$ with the fermions, as well as
trilinear and quartic couplings with the Higgs and the gauge bosons. This form
of the radion potential is dictated by the AdS isometries (and hence is valid
in the limit of $\epsilon\ll 1$). The self-interaction terms for the radion
are generated by the GW mechanism,

$\displaystyle\mathcal{L}_{\rm radion}$
$\displaystyle=\frac{1}{2}m_{\varphi}^{2}\varphi^{2}\left(1+\frac{5}{3}\frac{\varphi}{F_{\varphi}}+\frac{11}{12}\frac{\varphi^{2}}{F_{\varphi}^{2}}\right)$
(55)

where we have only kept terms to leading order in $\epsilon$. Finally, the
scalars $s$ interact with the SM through the radion portal.

$\displaystyle\mathcal{L}^{(\rm tree)}_{\rm s}$
$\displaystyle=\lambda_{s}v_{s}^{2}\mu_{\rm
TeV}^{2}s^{2}\left(1+\frac{\varphi}{F_{\varphi}}\right)^{2}+\lambda_{s}v_{s}\mu_{\rm
TeV}s^{3}\left(1+\frac{\varphi}{F_{\varphi}}\right)+\frac{1}{4}\lambda_{s}s^{4}+\mathcal{O}(\gamma)$
(56)

The terms suppressed by the explicit $Z_{2}$ violating coupling $\gamma$ are
assumed to be very small, and do not contribute significantly to the zero-
temperature phenomenology. Notice that we do not generate terms of the form of
$\varphi$-s mixing, or $s\varphi^{3}$. The classical solution sets the linear
term in $s$ to zero, and the $\varphi$ field coupling as $(1+\varphi)$ then
does not have a linear coupling to $s$ around the vacuum. A small Higgs portal
coupling of the form $\kappa sH^{\dagger}H$ can be added in order for $s$ to
be able to decay safely before BBN.

At loop level, there are also induced couplings between the EM and QCD field
strengths proportional to their $\beta$-functions:

$\displaystyle\mathcal{L}^{(1-\rm loop)}_{\rm int}\supset\frac{\alpha_{\rm
EM}}{8\pi F_{\varphi}}b_{\rm
EM}\varphi\,F_{\alpha\beta}F^{\alpha\beta}+\frac{\alpha_{s}}{8\pi
F_{\varphi}}b_{\rm G}\varphi\,G^{a}_{\alpha\beta}G^{a\,\alpha\beta}\,,$ (57)

with the dominant contributions to these terms coming from quark and $W$-boson
loops. In the case where the SM is confined to the IR brane, $b_{\rm
EM}=11/3$, $b_{\rm G}=-\frac{11}{3}N_{c}+2n/3$, where $n$ is the number of
quarks lighter than the radion Blum2014 .

###  4.2 Experimental constraints

The low energy phenomenology of the model is largely determined by the
physical masses of the radion and $s$ fields, as well as the KK scale. These
are related to the fundamental parameters of the model by:

$\displaystyle m_{{\rm KK}}$ $\displaystyle\simeq\frac{5\pi}{4}\mu_{\rm
TeV}\,,$ (58) $\displaystyle m_{\varphi}$
$\displaystyle=2\sqrt{2}v\epsilon^{3/4}\mu_{\rm TeV}\,,$ (59) $\displaystyle
m_{s}$ $\displaystyle=\sqrt{2\lambda_{s}}v_{s}\mu_{\rm TeV}\,.$ (60)

Collider searches limit the KK scale in RS models to be above $m_{{\rm
KK}}\gtrsim 4.25$ TeV Sirunyan2018 ; Sirunyan2019 , requiring the KK
resonances to be out of the kinematic reach of current colliders. Due to the
approximate shift symmetry of the GW field (broken only by the small parameter
$\epsilon$), the radion is parametrically lighter than the KK scale. The AD
scalar masses are similarly suppressed, with $m_{s}$ proportional to the
combination $(\lambda_{s}v_{s}^{2})^{1/2}$, which is chosen to be small for
the $s$ phase transition to happen well before the deconfinement transition.
Therefore, the radion and the AD scalars can be kinematically accessible at
colliders Giudice2018 , however their couplings to the SM are suppressed by
$\mu_{\rm TeV}$. Collider constraints translate into a bound $\mu_{\rm
TeV}\gtrsim 2\,{\rm TeV}$ Blum2014 , which is weaker than direct bounds on the
KK scale.

If the radion mass is below the GeV scale, bounds on the
${\varphi\gamma\gamma}$ coupling from supernova cooling§§§Whether supernova
bounds on the radion coupling apply depends on the radion coupling to nucleons
Abu-Ajamieh2017 . In the case where the SM quarks and gluons are on the IR
brane, this coupling is too large for the radion to contribute significantly
to supernova cooling., cosmology and beam dump experiments can give the
strongest bounds on the model. These limits have been derived for axion-like
particles Masso1995 ; Jaeckel2015 ; Dobrich2015 , which translate into a bound

$\displaystyle F_{\varphi}\gtrsim 4.25\times 10^{7}\,{\rm TeV}\,.$ (61)

For a heavier radion $m_{\varphi}>1\ {\rm GeV}$, these constraints are no
longer applicable. The radion mass will be above a GeV for parameters,

$\displaystyle v\epsilon^{3/4}$ $\displaystyle>7\times 10^{-5}\
\left(\frac{\mu_{{\rm TeV}}}{5\ {\rm TeV}}\right)^{-1}\,.$ (62)

The AD scalars can couple to the SM fields through the radion portal. For
light AD scalars, the couplings to photons/gluons generated at higher loop
order might still provide significant constraints. If the AD scalars are above
the 1 GeV scale, these constraints are also absent.

##  5 Cosmology

The mechanism of avoided deconfinement has dramatic implications for early
universe cosmology. The main departure from a standard cosmology is due to the
scaling of the radion expectation value with temperature. This leads to the
interesting consequence that while the universe may be reheated to a very high
temperature (exciting heavy fields on the UV brane, for instance), from the IR
brane point-of-view, the cosmology resembles a low-reheat cosmology. We aim to
highlight some of the applications of the AD model to cosmology, but leave a
detailed study of these implications for future work.

Figure 2 shows the characteristic dependence on temperature of dimensionful
parameters on the IR brane. In particular, $m_{KK}>T$ to arbitrarily high
temperature, as required by the condition that the IR brane be stabilised
outside the horizon at a given temperature. Therefore, KK modes play no role
in early universe cosmology from the point of view of the IR brane. In the
high temperature regime $T>T_{c}$, the radion expectation value scales with
temperature as:

$\displaystyle\mu(T)\propto T^{\frac{1}{1+\epsilon}}\,.$ (63)

This introduces a scaling of the other dimensionful quantities of the theory
with $T$. The KK scale, the Higgs mass parameter and the QCD scale and are all
proportional to $\mu(T)$¶¶¶More generally, $\Lambda_{\rm
QCD}\propto(\mu(T))^{n}$, where $n=1$ is true for the case where the SM is
confined to the IR brane, $n<1$ if some of the SM quark fields are bulk
fields., which means that the ratio of the temperature to these mass scales
(denoted $\Lambda$) varies with $T$ as:

$\displaystyle\frac{T}{\Lambda}\propto T^{\frac{\epsilon}{1+\epsilon}}\,.$
(64)

The consequence of this scaling is that the ratio $T/\Lambda$ may reach unity
at significantly higher temperatures than is the case in standard RS
cosmology. For example, as we show below, if the critical temperature $T_{c}$
is below the electroweak symmetry breaking scale, then the electroweak
symmetry restoration phase transition may occur at much higher temperatures
than in the usual case, or never occur at all.

Figure 3: Higgs expectation value, $v_{\rm ew}$, as a function of temperature
for the parameters of equation (50) (blue) and equation (65) (red). The dashed
line is the critical temperature, $T_{c}=18\ {\rm GeV}$ for the choice of
parameters in equation (65).

###  5.1 Electroweak Phase Transition

In this section we show that the electroweak phase transition can occur in the
avoided deconfinement phase at temperatures much higher than the weak scale.
To illustrate some of these effects, we use a new set of parameters, with:

$\displaystyle v_{\rm ir}=1.5\times 10^{-4},\,v_{\rm uv}=7.5\times
10^{-4},\,\epsilon=0.05\,$ (65)

and all other parameters as in equation (50), which leads to a radion
stabilised at $\mu_{\rm TeV}=4.73$ TeV at zero temperature. We note that with
these parameters, the model in its simplest for does not satisfy the bound on
the radion mass (62). We expect that a more complete model with additional
breaking of scaling invariance can lead to an unsuppressed radion mass and a
less severe bound than equation (62). This could happen, for example, through
additional terms in the GW action Chacko:2014pqa , allowing more fields to
propagate in the bulk, or by considering a more general geometry for the fifth
dimension Hassanain2007 . We will leave the detailed model building for future
work.

The Higgs potential at finite temperature gets thermal corrections from the
top Yukawa, gauge couplings and its quartic coupling. In addition, the Higgs
mass parameter scales with $\mu(T)$. The Higgs thermal mass in the low- and
high-temperature limits is given by:

$\displaystyle\mu_{h}^{2}(T>T_{c})$
$\displaystyle=T^{2}\left(-\lambda\frac{v_{{\rm
ew}}^{2}}{c^{2}T_{c}^{2}}\left(\frac{cT_{c}}{T}\right)^{2\epsilon}+\frac{\lambda_{t}^{2}}{4}+\frac{3g^{2}}{16}+\frac{g^{\prime
2}}{16}+\frac{\lambda}{2}\right),$ (66) $\displaystyle\mu_{h}^{2}(T<T_{c})$
$\displaystyle=-\lambda v_{{\rm
ew}}^{2}+T^{2}\left(\frac{\lambda_{t}^{2}}{4}+\frac{3g^{2}}{16}+\frac{g^{\prime
2}}{16}+\frac{\lambda}{2}\right).$ (67)

where $v_{\rm ew}\simeq 246\ {\rm GeV}$. The electroweak phase transition
(EWPT) happens at the temperature where the Higgs mass parameter become
positive. This happens for $T<T_{c}$ if $T_{c}$ is above the electroweak
scale, in which case there is no modification to the phase transition in
comparison to the SM. However, if $T_{c}$ is below the electroweak scale, the
EWPT will occur at a temperature:

$\displaystyle T_{*}=cT_{c}\left(\frac{T_{{\rm
ew}}}{cT_{c}}\right)^{\frac{1}{\epsilon}}$ (68)

where $T_{{\rm ew}}$ is the temperature of the EWPT in the SM. For small
$\epsilon$, even a modest ratio $T_{{\rm ew}}/T_{c}$ can lead to the EWPT
occurring at a temperature which is orders of magnitude above the scale
predicted by the SM. Figure 3 shows $-\mu_{h}^{2}$ as a function of
temperature the two sets of parameters defined in equations (50) & (65). For
the second set of parameters the EWPT doesn’t occur until the universe reaches
a temperature of order $\sim 5\times 10^{3}$ GeV.

A high temperature EWPT has been considered in refs Meade2018 ; Baldes:2018nel
; Glioti:2018roy ; Matsedonskyi2020  in the context of electroweak
baryogenesis. A primary motivation for these models is to avoid the bounds
which result from introducing new sources of CP violation around the weak
scale by having the EWPT occur at a temperature $T\gg v_{{\rm ew}}$. This
typically requires the introduction of a large number of fields coupled to the
Higgs sector in order to significantly increase the temperature of the phase
transition while satisfying collider bounds. In contrast, the AD model
provides a mechanism in which the electroweak phase transition can occur at
arbitrarily high temperatures due solely to the Higgs-radion interaction.
However, the Higgs potential must still be modified to make the EWPT first
order and introduce new sources of CP violation introduced in order to include
a mechanism for electroweak baryogenesis in the AD framework. Further, even
though the electroweak phase transition happens at much higher temperatures,
the scales governing local physics on the IR brane also scale with $T$. Thus,
if the CP violating operators are localized on the IR brane, their effect at
$T=T_{*}$ will be the same as that at $T=T_{c}\lesssim v_{\rm ew}$. Therefore,
they will be subject to the very constraints the models of Meade2018 ;
Baldes:2018nel ; Glioti:2018roy ; Matsedonskyi2020  were constructed to avoid.
On the other hand, if the CP violating operators are on the UV brane/bulk,
their effect does become much more important at higher temperatures. It will
be interesting to construct and study a high-temperature electroweak
baryogenesis model using avoided deconfinement in further detail.

###  5.2 High Scale Baryogenesis

As discussed in section 2, in the usual RS model the high temperature phase is
described by an AdS black hole, before a phase transition to the RS phase at a
temperature around the TeV scale or below. The period of supercooling
accompanying the phase transition significantly dilutes any pre-existing
baryon asymmetry Baratella2019 . This has motivated consideration of
baryogenesis mechanisms that combine the electroweak and RS phase transitions
Bruggisser2018 ; Bruggisser2018a ; Konstandin2011b . Baryogenesis mechanisms
which operate at temperatures significantly above the TeV scale are difficult
to realise in the RS model. This is not the case, however, for the AD model,
as the universe is never in the BH phase after inflation and does not undergo
a period of supercooling.

At high temperature the radion is stabilised closer to the UV brane. This
means that fields localised toward the IR brane may have significant overlap
with UV- localised fields at early times, with the UV and IR sectors then
decoupling at low temperature. This allows for the possibility of having
baryogenesis occur due to interactions between the IR and UV fields, which
have $\mathcal{O}(1)$ couplings in the early universe but whose interactions
are negligibly small after the radion has settled to its zero-temperature
expectation value.

Figure 4: Dark matter annihilation rate for $m_{\chi}(T=0)=10$ TeV, and
$\alpha_{\rm{DM}}\sim 10^{-2}$ in the AD model (solid blue line) and without
radion dependence (dashed blue line). The AD model parameters are as in
equation (65). The Hubble rate assuming radiation domination and
$g_{*}=106.75$ is shown in black.

###  5.3 WIMP Freeze-in

The relic abundance of a particle with weak scale mass and interactions in
standard cosmology turns out to be a good estimate for the observed dark
matter abundance. This has led to the WIMP paradigm, which is supported by the
idea that new physics at the weak scale is motivated by the electroweak
hierarchy problem. In this light, we expect a WIMP in the RS model to be
associated with a field localised to the IR brane. Thus, avoided deconfinement
may have significant implications for the WIMP freeze-out in such cases – in
fact, we show that particles with weak scale interactions can have a freeze-in
mechanism.

As noted above, the scales on the IR brane are proportional to $\mu(T)$ and if
the dark matter particle $\chi$ lives on the IR brane, the quantity
$m_{\chi}(T)/T$ changes by only an O(1) amount during the phase of avoided
deconfinement. In particular, the thermal abundance of $\chi$ can be Boltzmann
suppressed for the entire cosmic history.

If the annihilation rate of dark matter is set by a weak coupling
$\alpha_{\rm{DM}}$, then the equilibrium annihilation rate in early cosmology
can be estimated as (assuming $m_{\chi}(0)>T_{c}$),

$\displaystyle n_{eq}\langle\sigma v\rangle$
$\displaystyle\sim\frac{\pi\alpha_{\rm{DM}}^{2}}{m_{\chi}(T)^{2}}(m_{\chi}(T)T)^{3/2}\exp\left[-\frac{m_{\chi}(T)}{T}\right]\sim
T\frac{\pi\alpha_{\rm{DM}}^{2}}{m_{\chi}(0)^{2}}\exp\left[-\frac{m_{\chi}(0)}{cT_{c}}\right]+O(\epsilon\log(T/T_{c})),$
(69)

where we have ignored $\mathcal{O}$(1) factors to highlight the scaling
behavior. The form of the annihilation rate being nearly proportional to $T$
follows from the approximate conformal symmetry. Figure 4 shows the different
annihilation rates as a function of temperature in the AD model and without
radion dependence.

The annihilation rate decreases slower than the Hubble rate, which decreases
as $T^{2}$ in radiation domination, as shown in figure 4. Even though the
zero-temperature annihilation rate is weak-scale, the high-temperature
annihilation rate can be out of equilibrium because it is Boltzmann
suppressed. After the IR brane is stabilized at $T\sim T_{c}$,
$m_{\chi}(T)\sim m_{\chi}(0)$, and the equilibrium annihilation rate drops
exponentially with temperature. If the Hubble rate around $T=T_{c}$ is larger
than the equilibrium annihilation rate, this implies that:

$\displaystyle\frac{m_{\chi}(0)}{T_{c}}\gtrsim\log\frac{M_{\rm pl}}{T_{c}},$
(70)

and the annihilation is never in thermal equilibrium, so the abundance is set
by freeze in. Note that the DM-SM coupling can be sizeable, so that we can
detect $\chi$ in direct and indirect detection experiments as a WIMP. However
the usual relic abundance calculation would not apply. This can potentially
open up a large parameter space for simple WIMPs like the Wino-like
electroweak triplet, or other heavy WIMP candidates which would have a large
abundance in the standard freeze out history.

###  5.4 QCD Phase Transition and the QCD Axion

The QCD phase transition in the AD model may also be modified from the usual
picture if $T_{c}$ is below the QCD scale. In order to achieve this in our
model while satisfying the bounds on the radion and $s$ masses requires a
large number of AD scalars. However, as was the case for the electroweak phase
transition, even for $T_{c}$ slightly below the QCD scale the QCD phase
transition may occur at temperatures far above the TeV scale. This may be able
to reproduce some of the non-standard QCD dynamics discussed in refs
Ipek:2018lhm ; Croon2020 ; Berger2020 .

A cosmology where QCD confinement occurs at high temperatures, when
$\Lambda_{{\rm QCD}}(T)\gg\Lambda_{{\rm QCD}}(0)$, can also have dramatic
consequences for the abundance of the QCD axion. The axion field can have
various 5D origins; one simple possibility is that it is the fifth component
of $U(1)$ gauge field in 5D. Irrespective of its origin, the large decay
constant of the axion suggests that its wavefunction is localised near the UV
brane. Therefore, we can safely assume that $f_{a}$ is largely temperature
independent. In the confined phase of QCD, but with a temperature-dependent
confinement scale, the axion mass is given by,

$\displaystyle m_{a}(T)=\frac{f_{\pi}(T)m_{\pi}(T)}{f_{a}}$ (71)

The axion starts oscillating around the epoch of QCD confinement at a
temperature $T_{\rm osc}$ which is defined by $m_{a}(T_{\rm osc})\simeq
H(T_{\rm osc})$. The axion abundance at the onset of oscillation is

$\displaystyle\rho_{a}(T_{\rm osc})$ $\displaystyle\sim m_{a}^{2}(T_{\rm
osc})f_{a}^{2}\theta_{i}^{2}$ (72)

where $\theta_{i}$ is the initial misalignment angle. The mass of the axion
continues to decrease due to the temperature dependence of $\Lambda_{{\rm
QCD}}(T)\sim\mu(T)$. In the adiabatic approximation $\dot{m}\ll m^{2}$, the
number density of the axion scales as $a^{-3}$, and the mass redshifts as
$\sim a^{-2}$, so the axion energy density redshifts approximately as $a^{-5}$
in this epoch, whereas the background energy density is redshifting as
$a^{-4}$. This can reduce the axion abundance dramatically.

###  5.5 Gravitational Waves

The absence of a first order confinement phase transition is a necessary
feature of this mechanism that distinguishes it from the standard RS model.
The RS phase transition results in a gravitational wave signal which will be
absent in the avoided deconfinement model Megias2018a ; Randall2006 . The RS
phase transition also leads to a drop in $g_{*}$ of order $N^{2}$ as a result
of degrees of freedom confining and freezing out. If there is an observable
background of gravitational waves, such as from a cosmic string network
Cui2017 ; Cui2018  or inflation Watanabe2006 ; Jinno2012 ; Saikawa2018 , this
change in $g_{*}$ is observable as a relative decrease in the power in modes
which were below the horizon scale prior to the phase transition. The absence
of these gravitational wave signals could be used to distinguish the AD model
from RS models which do undergo a phase transition. Furthermore, the addition
of the AD scalars, which are necessarily light degrees of freedom due to the
bound (53), also leads to a potentially observable change in $g_{*}$ in the
early universe for $N_{s}$ as low as $N_{s}\sim 10$ and masses around the GeV
scale.

In addition to modifying the RS phase transition, in our set up there are
additional phase transitions associated with the $s$ fields, which can be
first order and can each happen at slightly different temperatures. This can
give us interesting forest of GW signals with a spectrum that is different
from the one expected from a single phase transition. As noted above, the
electroweak and/or the QCD phase transition may also be made first order and
can happen at very high temperatures, predicting a gravitational wave
signature from these phase transitions as well.

##  6 Discussion

In this work we have described a mechanism which addresses the cosmological
problem of eternal inflation due to suppressed confinement transitions in the
RS model. The standard RS model is described at high temperature by an AdS
black hole, with a transition to the RS phase proceeding via a first order
phase transition which is exponentially suppressed by the large number
$N^{2}$. We have shown that this situation can be avoided by introducing new
scalars localised in the IR which generate a potential that stabilises the IR
brane at high temperatures. Provided the universe exits inflation in the RS
phase, it remains there, never entering the BH phase.

There are a number of issues that would be worth exploring further. It would
be interesting to understand this phenomenon in a 4D field theory example. The
additional scalars that we have introduced on the IR brane are expected to be
emergent degrees of freedom in the 4D theory that appear after confinement. In
such an example the thermal effect of these scalars will be to drive the
confinement scale itself to higher values. This may provide a new insights on
the problem of confinement.

There are also various phenomenological applications of this mechanism which
we have merely touched upon in this paper. Avoided Deconfinement changes the
cosmological history in a unique way, where from the IR brane point of view it
is a model with effectively a low reheat temperature, but from the UV brane
point of view the temperature can get arbitrarily high. This allows us to
build realizations where the electroweak and/or QCD phase transitions happen
at very high temperatures. Since the cosmology at high temperatures is
modified, we have shown that the mechanism for generating the abundance of
various species such as WIMP dark matter, axion dark matter or baryogenesis
can be significantly modified. An interesting future direction would be to
build explicit models which realize these mechanisms, and study their
phenomenological signatures.

Gravitational waves are a powerful experimental tool for studying very early
universe cosmology, both in terms of new sources of the waves as well as
modifications of propagation of gravitational waves in the early universe
plasma. Modification of the Randall-Sundrum phase transition, or the
electroweak/QCD phase transition can change the expectations of GW signals
from these phase transitions; these phase transitions are also associated with
a change of number of degrees of freedom in the plasma, which may also be
detectable in the GW spectrum. Even if a large-$N$ confining gauge group is
part of a dark sector decoupled from the standard model, these gravitational
wave signatures can provide important information about these sectors. Thus,
the detailed phenomenological predictions of avoided deconfinement would be
important to study further even in this more general situation.

###### Acknowledgements.

We would like to thank Raman Sundrum and Soubhik Kumar for useful discussions
and comments on the manuscript. We are grateful to Anson Hook, Lisa Randall,
Matt Reece and John March-Russell for useful conversations. PA is supported by
the STFC under Grant No. ST/T000864/1. MN is funded by a joint Clarendon and
Sloane-Robinson scholarship from Oxford University and Keble college.

##  Appendix A Bounce action for deconfining phase transition

In this appendix we estimate the bounce action, $B$, which determines the
transition rate from the AD phase in the high-temperature regime to the AdS-S
or deconfined phase. At high temperatures the phase transition from the AD
phase to the black hole phase proceeds at a rate

$\Gamma\simeq T^{4}e^{-B}.$ (73)

If this is larger than $H^{4}$, where $H$ is the hubble rate, then this
indicates that the AD phase is unstable. This defines a maximum temperature
$T_{\rm max}$ above which the AD mechanism no longer works, but we will find
that tunnelling only becomes significant at temperatures equal to the
temperature which defines the classical instability of the model (defined in
equation (48)), up to $\mathcal{O}(1/N)$ corrections.

In order to determine $B$ we make the approximation that the action is
dominated by the dynamics of the radion and neglect the contribution from the
gravitational portion of the action. The justification for this is that the
gravitational action scales as $N^{2}$ with no further enhancement from small
or large parameters, while the contribution to the bounce from the radion, as
we show below, scales like $N^{2}\lambda^{-1/2}$ for a weak coupling
$\lambda$. In this approximation the relevant Euclidean action is

$\displaystyle S_{E}$
$\displaystyle=\frac{N^{2}}{4\pi}\int_{0}^{T^{-1}}dt_{E}\int
r^{2}dr\left[(\partial\mu)^{2}-\lambda_{1}T^{2}\mu^{2}+\lambda_{2}\mu^{4}\right],$
(74) $\displaystyle\lambda_{1}$
$\displaystyle=\frac{2\pi^{2}N_{s}\lambda_{s}v_{s}^{2}}{3N^{2}},$
$\displaystyle\lambda_{2}$ $\displaystyle=\frac{64\pi^{2}(v_{\rm ir}-v_{\rm
uv})^{2}}{N^{2}},$

where we have explicitly scaled out the factor of $N^{2}$. At high temperature
the minimum of the radion potential is well-approximated by equation (45)

$\mu\simeq\alpha(T)T,\qquad\alpha(T)=\frac{\mu_{\rm
TeV}}{(cT_{c}T^{\epsilon})^{\frac{1}{1+\epsilon}}}.$ (75)

After rescaling the co-ordinates and radion field as $\mu=\alpha
T\tilde{\mu}$, $x_{E}=T^{-1}\tilde{x}_{E}$ the action can be written as:

$\displaystyle S_{E}$
$\displaystyle=\frac{\alpha^{2}N^{2}}{4\pi}\int_{0}^{1}d\tilde{t}_{E}\int\tilde{r}^{2}d\tilde{r}\left[(\tilde{\partial}\tilde{\mu})^{2}-\lambda_{1}\tilde{\mu}^{2}+\lambda_{2}\alpha^{2}\tilde{\mu}^{4}\right].$
(76)

Figure 5:  Plot showing the parametric dependence of the potential in the AD
model. The right-hand side shows the radion potential at high temperature,
with depth set by $N_{s}T^{4}$ and width set by $\langle\mu\rangle=\alpha(T)T$
and the left-hand side is the potential for the black hole hawking temperature
$T_{h}$, with width set by $T$ and depth of order $N^{2}T^{4}$. The dashed
line shows the would-be horizon position on the radion side of the potential
and the break in the curve indicates the region where EFT control is lost.

The equation of motion for $\tilde{\mu}$ then implies that $\tilde{\mu}$
varies by an $\mathcal{O}(1)$ amount over a distance of order
$\Delta\tilde{r}\sim\lambda_{1}^{-1/2}\gg 1$, i.e.

$\displaystyle\left|\frac{\partial\tilde{\mu}}{\partial\tilde{r}}\right|\sim\lambda_{1}^{1/2}.$
(77)

We then make the conservative estimate that the bounce solution only requires
the radion to vary by an amount given by

$\displaystyle\Delta\mu=\alpha(T)T-k^{2}\rho_{h},$ (78)

which amounts to moving the IR brane from its stabilised location to the
position of the would-be horizon. In figure 5 this corresponds to the radion
varying from its value at the minimum of the potential to the dashed line, as
opposed to a bounce analogous to the one proposed in Creminelli2002  which
involves the radion varying to $\mu=0$ over the bounce trajectory. In terms of
$\tilde{\mu}$ this is

$\displaystyle\delta\tilde{\mu}=1-\frac{\pi}{\alpha},$ (79)

which approaches $0$ logarithmically (indicating that the IR brane is becoming
classically unstable) as $T$ approaches $T_{\rm max}$.

With this estimate the characteristic radius of the tunnelling configuration
will be $\tilde{R}_{b}\sim(\delta\tilde{\mu})\lambda_{1}^{-1/2}$. For
$\tilde{R}_{b}\gg 1$ we expect the bounce solution to be the $O(3)$ symmetric
($\tilde{t}_{E}$-independent) configuration, while for $\tilde{R}_{b}\ll 1$
the solution will obey an $O(4)$ symmetry and depend on the variable
$\tilde{\rho}=\sqrt{\tilde{t}_{E}^{2}+\tilde{r}^{2}}$. The $O(4)$ bounce
solution therefore only becomes dominant for
$\delta\tilde{\mu}\ll\lambda^{1/2}\sim 1/N$ at which point the IR brane is
close to becoming classically unstable anyway, so only the $O(3)$ symmetric
bounce is relevant for computing the lifetime of the AD phase. The integrals
determining the bounce action scale as

$\displaystyle\int\tilde{r}^{2}d\tilde{r}\left((\tilde{\partial}\tilde{\mu})^{2}-\lambda_{1}\tilde{\mu}^{2}\right)$
$\displaystyle=\lambda_{1}^{-1/2}c_{1}$ (80)
$\displaystyle\int\tilde{r}^{2}d\tilde{r}\left(\lambda_{2}\alpha^{2}\tilde{\mu}^{4}\right)$
$\displaystyle=\lambda_{1}^{-3/2}\lambda_{2}c_{2}.$

where $c_{1},c_{2}$ are $\mathcal{O}(1)$ coefficients. We can then estimate
the bounce to be:

$\displaystyle B$
$\displaystyle\simeq\frac{\alpha^{2}N^{2}(\delta\tilde{\mu})^{3}\lambda_{1}^{-3/2}}{4\pi}\left[c_{1}\lambda_{1}+c_{2}\lambda_{2}\alpha^{2}\right],$
(81)
$\displaystyle\simeq\frac{\sqrt{3}\alpha^{2}N^{3}(\delta\tilde{\mu})^{3}}{4\sqrt{2}\pi^{2}\left(N_{s}\lambda_{s}v_{s}^{2}\right)^{1/2}}\left[c_{1}+c_{2}\frac{96\alpha^{2}(v_{\rm
ir}-v_{\rm uv})^{2}}{N_{s}\lambda_{s}v_{s}^{2}}\right].$

The bounce action is therefore suppressed by the factor $N^{3}$ and
additionally by inverse powers of $v_{s}$. The lifetime of the metastable AD
vacuum is much larger than Hubble for temperatures up until
$\delta\tilde{\mu}\lesssim\mathcal{O}(1/N)$, meaning we can safely ignore the
tunneling rate from the AD to deconfined phase and consider only the classical
instability of the model.

## References

  * (1) G. ’t Hooft, A Planar Diagram Theory for Strong Interactions, Nucl. Phys. B72 (1974) 461.
  * (2) J. Maldacena, The large-N limit of superconformal field theories and supergravity, International Journal of Theoretical Physics 38 (nov, 1999) 1113–1133, [hep-th/9711200].
  * (3) E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253–291, [hep-th/9802150].
  * (4) S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, Gauge Theory Correlators from Non-Critical String Theory, Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics 428 (feb, 1998) 105–114, [hep-th/9802109].
  * (5) L. Randall and R. Sundrum, A Large mass hierarchy from a small extra dimension, Phys. Rev. Lett. 83 (1999) 3370–3373, [hep-ph/9905221].
  * (6) W. D. Goldberger and M. B. Wise, Modulus stabilization with bulk fields, Phys. Rev. Lett. 83 (1999) 4922–4925, [hep-ph/9907447].
  * (7) M. A. Luty and R. Sundrum, Hierarchy stabilization in warped supersymmetry, Phys. Rev. D 64 (2001) 065012, [hep-th/0012158].
  * (8) H. Verlinde, Holography and compactification, Nuclear Physics B 580 (jul, 2000) 264–274, [hep-th/9906182].
  * (9) C. S. Chan, P. L. Paul, and H. Verlinde, A note on warped string compactification, Nuclear Physics B 581 (aug, 2000) 156–164, [hep-th/0003236].
  * (10) F. Brümmer, A. Hebecker, and E. Trincherini, The throat as a Randall-Sundrum model with Goldberger-Wise stabilization, Nuclear Physics B 738 (mar, 2006) 283–305, [hep-th/0510113].
  * (11) L. Randall, The Boundaries of KKLT, Fortsch. Phys. 68 (2020), no. 3-4 1900105, [arXiv:1912.06693].
  * (12) S. Kachru, R. Kallosh, A. D. Linde, and S. P. Trivedi, De Sitter vacua in string theory, Phys. Rev. D 68 (2003) 46005, [hep-th/0301240].
  * (13) I. R. Klebanov and E. Witten, Superconformal field theory on three-branes at a Calabi-Yau singularity, Nucl. Phys. B 536 (1998) 199–218, [hep-th/9807080].
  * (14) I. R. Klebanov and A. A. Tseytlin, Gravity Duals of Supersymmetric SU(N) x SU(N+M) Gauge Theories, Nuclear Physics B 578 (feb, 2000) 123–138, [hep-th/0002159].
  * (15) I. R. Klebanov and M. J. Strassler, Supergravity and a confining gauge theory: Duality cascades and chi SB resolution of naked singularities, JHEP 08 (2000) 52, [hep-th/0007191].
  * (16) I. Bena, E. Dudas, M. Graña, and S. Lüst, Uplifting Runaways, Fortsch. Phys. 67 (2019), no. 1-2 1800100, [arXiv:1809.06861].
  * (17) S. Kachru, M. Kim, L. McAllister, and M. Zimet, de Sitter Vacua from Ten Dimensions, arXiv:1908.04788.
  * (18) S. W. Hawking and D. N. Page, Thermodynamics of black holes in anti-de Sitter space, Communications in Mathematical Physics 87 (dec, 1983) 577–588.
  * (19) E. Witten, Anti-de Sitter space, thermal phase transition, and confinement in gauge theories, Adv. Theor. Math. Phys. 2 (1998) 505–532, [hep-th/9803131].
  * (20) P. Creminelli, A. Nicolis, and R. Rattazzi, Holography and the electroweak phase transition, JHEP 03 (2002) 51, [hep-th/0107141].
  * (21) J. Kaplan, P. C. Schuster, and N. Toro, Avoiding an Empty Universe in RS I Models and Large-N Gauge Theories, hep-ph/0609012.
  * (22) S. S. Gubser, AdS / CFT and gravity, Phys. Rev. D63 (2001) 84017, [hep-th/9912001].
  * (23) L. Randall and G. Servant, Gravitational Waves from Warped Spacetime, Journal of High Energy Physics 2007 (jul, 2006) [hep-ph/0607158].
  * (24) G. Nardini, M. Quiros, and A. Wulzer, A Confining Strong First-Order Electroweak Phase Transition, JHEP 09 (2007) 077, [arXiv:0706.3388].
  * (25) T. Konstandin, G. Nardini, and M. Quiros, Gravitational Backreaction Effects on the Holographic Phase Transition, Physical Review D 82 (jul, 2010) [arXiv:1007.1468].
  * (26) T. Konstandin and G. Servant, Cosmological consequences of nearly conformal dynamics at the TeV scale, Journal of Cosmology and Astroparticle Physics (2011) [arXiv:1104.4791].
  * (27) B. M. Dillon, B. K. El-Menoufi, S. J. Huber, and J. P. Manuel, A rapid holographic phase transition with brane-localized curvature, Physical Review D 98 (aug, 2017) [arXiv:1708.02953].
  * (28) B. von Harling and G. Servant, QCD-induced Electroweak Phase Transition, Journal of High Energy Physics 2018 (nov, 2017) [arXiv:1711.11554].
  * (29) S. Bruggisser, B. von Harling, O. Matsedonskyi, and G. Servant, The Baryon Asymmetry from a Composite Higgs, Physical Review Letters 121 (mar, 2018) [arXiv:1803.08546].
  * (30) S. Bruggisser, B. von Harling, O. Matsedonskyi, and G. Servant, Electroweak Phase Transition and Baryogenesis in Composite Higgs Models, Journal of High Energy Physics 2018 (apr, 2018) 17–229, [arXiv:1804.07314].
  * (31) E. Megías, G. Nardini, and M. Quirós, Cosmological phase transitions in warped space: gravitational waves and collider signatures, Journal of High Energy Physics 2018 (sep, 2018) [arXiv:1806.04877].
  * (32) D. Bunk, J. Hubisz, and B. Jain, A perturbative RS I cosmological phase transition, European Physical Journal C 78 (2018), no. 1 [arXiv:1705.00001].
  * (33) P. Baratella, A. Pomarol, and F. Rompineve, The Supercooled Universe, JHEP 03 (2019) 100, [arXiv:1812.06996].
  * (34) K. Agashe, P. Du, M. Ekhterachian, S. Kumar, and R. Sundrum, Cosmological Phase Transition of Spontaneous Confinement, Journal of High Energy Physics 2020 (oct, 2019) [arXiv:1910.06238].
  * (35) K. Fujikura, Y. Nakai, and M. Yamada, A more attractive scheme for radion stabilization and supercooled phase transition, JHEP 02 (2020) 111, [arXiv:1910.07546].
  * (36) A. Azatov and M. Vanvlasselaer, Phase transitions in perturbative walking dynamics, JHEP 09 (2020) 085, [arXiv:2003.10265].
  * (37) E. Megías, G. Nardini, and M. Quirós, Gravitational imprints from heavy Kaluza-Klein resonances, Physical Review D 102 (2020), no. 5 [arXiv:2005.04127].
  * (38) K. Agashe, P. Du, M. Ekhterachian, S. Kumar, and R. Sundrum, Phase Transitions from the Fifth Dimension, arXiv:2010.04083.
  * (39) B. Hassanain, J. March-Russell, and M. Schvellinger, Warped Deformed Throats have Faster (Electroweak) Phase Transitions, JHEP 10 (2007) 89, [arXiv:0708.2060].
  * (40) S. Weinberg, Gauge and global symmetries at high temperature, Physical Review D 9 (jun, 1974) 3357–3378.
  * (41) P. Meade and H. Ramani, Unrestored Electroweak Symmetry, Physical Review Letters 122 (jul, 2018) [arXiv:1807.07578].
  * (42) I. Baldes and G. Servant, High scale electroweak phase transition: baryogenesis & symmetry non-restoration, JHEP 10 (2018) 53, [arXiv:1807.08770].
  * (43) A. Glioti, R. Rattazzi, and L. Vecchi, Electroweak Baryogenesis above the Electroweak Scale, JHEP 04 (2019) 27, [arXiv:1811.11740].
  * (44) O. Matsedonskyi and G. Servant, High-Temperature Electroweak Symmetry Non-Restoration from New Fermions and Implications for Baryogenesis, arXiv:2002.05174.
  * (45) P. Langacker and S. Y. Pi, Magnetic monopoles in grand unified theories, Physical Review Letters 45 (jul, 1980) 1–4.
  * (46) P. Salomonson, B. S. Skagerstam, and A. Stern, On the primordial monopole problem in grand unified theories, Physics Letters B 151 (feb, 1985) 243–246.
  * (47) G. Dvali, A. Melfo, and G. Senjanovic, Is there a monopole problem?, Physical Review Letters 75 (jul, 1995) 4559–4562, [hep-ph/9507230].
  * (48) G. Dvali and G. Senjanovic, Is there a domain wall problem?, Physical Review Letters 74 (jan, 1995) 5178–5181, [hep-ph/9501387].
  * (49) R. N. Mohapatra and G. Senjanović, Soft CP-invariance violation at high temperature, Physical Review Letters 42 (jun, 1979) 1651–1654.
  * (50) R. N. Mohapatra and G. Senjanović, Broken symmetries at high temperature, Physical Review D 20 (dec, 1979) 3390–3398.
  * (51) J. Orloff, The U.V. Price for Symmetry Non-Restoration, Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics 403 (nov, 1996) 309–315, [hep-ph/9611398].
  * (52) N. Chai, et al., Thermal Order in Conformal Theories, arXiv:2005.03676.
  * (53) N. Arkani-Hamed, M. Porrati, and L. Randall, Holography and phenomenology, JHEP 08 (2001) 17, [hep-th/0012148].
  * (54) R. Rattazzi and A. Zaffaroni, Comments on the holographic picture of the Randall-Sundrum model, JHEP 04 (2001) 21, [hep-th/0012248].
  * (55) C. Charmousis, R. Gregory, and V. A. Rubakov, Wave function of the radion in a brane world, Phys. Rev. D62 (2000) 67505, [hep-th/9912160].
  * (56) S. R. Coleman, The Fate of the False Vacuum. 1. Semiclassical Theory, Phys. Rev. D15 (1977) 2929–2936.
  * (57) A. D. Linde, Decay of the False Vacuum at Finite Temperature, Nucl. Phys. B216 (1983) 421.
  * (58) F. Coradeschi, P. Lodone, D. Pappadopulo, R. Rattazzi, and L. Vitale, A naturally light dilaton, JHEP 11 (2013) 57, [arXiv:1306.4601].
  * (59) W. D. Goldberger and M. B. Wise, Phenomenology of a stabilized modulus, Phys. Lett. B475 (2000) 275–279, [hep-ph/9911457].
  * (60) A. Pomarol, O. Pujolas, and L. Salas, Holographic conformal transition and light scalars, JHEP 10 (2019) 202, [arXiv:1905.02653].
  * (61) F. Bigazzi, A. Caddeo, A. L. Cotrone, and A. Paredes, Fate of false vacua in holographic first-order phase transitions, arXiv:2008.02579.
  * (62) A. H. Guth and E. J. Weinberg, Could the Universe Have Recovered from a Slow First Order Phase Transition?, Nucl. Phys. B212 (1983) 321–364.
  * (63) S. Fichet, Braneworld effective field theories — holography, consistency and conformal effects, JHEP 04 (2020) 016, [arXiv:1912.12316].
  * (64) C. Delaunay, C. Grojean, and J. D. Wells, Dynamics of Non-renormalizable Electroweak Symmetry Breaking, JHEP 04 (2008) 29, [arXiv:0711.2511].
  * (65) D. Curtin, P. Meade, and H. Ramani, Thermal Resummation and Phase Transitions, Eur. Phys. J. C78 (2018), no. 9 787, [arXiv:1612.00466].
  * (66) Z. Chacko, R. K. Mishra, D. Stolarski, and C. B. Verhaaren, Interactions of a Stabilized Radion and Duality, Phys. Rev. D 92 (2015), no. 5 56004, [arXiv:1411.3758].
  * (67) T. Gherghetta and A. Pomarol, Bulk fields and supersymmetry in a slice of AdS, Nuclear Physics B 586 (oct, 2000) 141–162, [hep-ph/0003129].
  * (68) R. Sundrum, Gravity’s scalar cousin, hep-th/0312212.
  * (69) A. Hebecker and J. March-Russell, Randall-Sundrum II cosmology, AdS/CFT, and the bulk black hole, Nuclear Physics B (2001) [hep-ph/0103214].
  * (70) K. Agashe, A. Delgado, M. J. May, and R. Sundrum, RS1, custodial isospin and precision tests, Journal of High Energy Physics 7 (2003), no. 8 1167–1197, [hep-ph/0308036].
  * (71) S. Casagrande, F. Goertz, U. Haisch, M. Neubert, and T. Pfoh, Flavor Physics in the Randall-Sundrum Model: I. Theoretical Setup and Electroweak Precision Tests, Journal of High Energy Physics 2008 (jul, 2008) [arXiv:0807.4937].
  * (72) K. Agashe, G. Perez, and A. Soni, Flavor structure of warped extra dimension models, Physical Review D - Particles, Fields, Gravitation and Cosmology 71 (2005), no. 1 [hep-ph/0408134].
  * (73) M. Bauer, S. Casagrande, U. Haisch, and M. Neubert, Flavor Physics in the Randall-Sundrum Model: II. Tree-Level Weak-Interaction Processes, Journal of High Energy Physics 2010 (dec, 2009) [arXiv:0912.1625].
  * (74) K. Blum, M. Cliche, C. Csaki, and S. J. Lee, WIMP Dark Matter through the Dilaton Portal, Journal of High Energy Physics 2015 (oct, 2014) [arXiv:1410.1873].
  * (75) A. M. Sirunyan, et al., Search for physics beyond the standard model in high-mass diphoton events from proton-proton collisions at s =13 TeV, Physical Review D 98 (nov, 2018) [arXiv:1809.00327].
  * (76) A. M. Sirunyan, et al., Combination of Searches for Higgs Boson Pair Production in Proton-Proton Collisions at s =13 TeV, Physical Review Letters 122 (mar, 2019) [arXiv:1811.09689].
  * (77) G. F. Giudice, Y. Kats, M. McCullough, R. Torre, and A. Urbano, Clockwork/linear dilaton: structure and phenomenology, JHEP 06 (2018) 009, [arXiv:1711.08437].
  * (78) F. Abu-Ajamieh, J. S. Lee, and J. Terning, The Light Radion Window, Journal of High Energy Physics 2018 (nov, 2017) [arXiv:1711.02697].
  * (79) E. Masso and R. Toldra, On a Light Spinless Particle Coupled to Photons, Physical Review D 52 (mar, 1995) 1755–1763, [hep-ph/9503293].
  * (80) J. Jaeckel and M. Spannowsky, Probing MeV to 90 GeV axion-like particles with LEP and LHC, Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics 753 (sep, 2015) 482–487, [arXiv:1509.00476].
  * (81) B. Döbrich, J. Jaeckel, F. Kahlhoefer, A. Ringwald, and K. Schmidt-Hoberg, ALPtraum: ALP production in proton beam dump experiments, Journal of High Energy Physics 2016 (dec, 2015) 1–27, [hep-ph/1512.03069].
  * (82) T. Konstandin and G. Servant, Natural Cold Baryogenesis from Strongly Interacting Electroweak Symmetry Breaking, JCAP 07 (2011) 024, [arXiv:1104.4793].
  * (83) S. Ipek and T. M. P. Tait, Early Cosmological Period of QCD Confinement, Phys. Rev. Lett. 122 (2019), no. 11 112001, [arXiv:1811.00559].
  * (84) D. Croon, J. N. Howard, S. Ipek, and T. M. Tait, QCD baryogenesis, Physical Review D (2020) [arXiv:1911.01432].
  * (85) D. Berger, S. Ipek, T. M. P. Tait, and M. Waterbury, Dark Matter Freeze Out during an Early Cosmological Period of QCD Confinement, arXiv:2004.06727.
  * (86) Y. Cui, M. Lewicki, D. E. Morrissey, and J. D. Wells, Cosmic Archaeology with Gravitational Waves from Cosmic Strings, Physical Review D 97 (nov, 2017) [arXiv:1711.03104].
  * (87) Y. Cui, M. Lewicki, D. E. Morrissey, and J. D. Wells, Probing the pre-BBN universe with gravitational waves from cosmic strings, Journal of High Energy Physics 2019 (aug, 2018) [arXiv:1808.08968].
  * (88) Y. Watanabe and E. Komatsu, Improved Calculation of the Primordial Gravitational Wave Spectrum in the Standard Model, Physical Review D 73 (apr, 2006) [astro-ph/0604176].
  * (89) R. Jinno, T. Moroi, and K. Nakayama, Probing dark radiation with inflationary gravitational waves, Physical Review D 86 (aug, 2012) [arXiv:1208.0184].
  * (90) K. Saikawa and S. Shirai, Primordial gravitational waves, precisely: The role of thermodynamics in the Standard Model, Journal of Cosmology and Astroparticle Physics 2018 (mar, 2018) [arXiv:1803.01038].