# Thermal Perturbations from Cosmological Constant Relaxation

Lingyuan Ji Department of Physics and Astronomy, Johns Hopkins University,
3400 N. Charles St., Baltimore, Maryland 21218, USA  David E. Kaplan
Department of Physics and Astronomy, Johns Hopkins University, 3400 N. Charles
St., Baltimore, Maryland 21218, USA  Surjeet Rajendran Department of Physics
and Astronomy, Johns Hopkins University, 3400 N. Charles St., Baltimore,
Maryland 21218, USA  Erwin H. Tanin Department of Physics and Astronomy, Johns
Hopkins University, 3400 N. Charles St., Baltimore, Maryland 21218, USA

###### Abstract

We probe the cosmological consequences of a recently proposed class of
solutions to the cosmological constant problem. In these models, the universe
undergoes a long period of inflation followed by a contraction and a bounce
that sets the stage for the hot big bang era. A requirement of any successful
early universe model is that it must reproduce the observed scale-invariant
density perturbations at CMB scales. While these class of models involve a
long period of inflation, the inflationary Hubble scale during their
observationally relevant stages is at or below the current Hubble scale,
rendering the de Sitter fluctuations too weak to seed the CMB anisotropies. We
show that sufficiently strong perturbations can still be sourced thermally if
the relaxion field serving as the inflaton interacts with a thermal bath,
which can be generated and maintained by the same interaction. We present a
simple model where the relaxion field is derivatively (i.e. technically
naturally) coupled to a non-abelian gauge sector, which gets excited
tachyonically and subsequently thermalizes due to its nonlinear self-
interactions. This model explains both the smallness of the cosmological
constant and the amplitude of CMB anisotropies.

##  I Introduction

Cosmological observations have shown that the universe is presently undergoing
accelerated expansion due to a form of energy density dubbed “dark energy”.
The inferred measured value of the dark energy density is $\sim 10\text{
meV}^{4}$. This measurement constrains the value of the cosmological constant
(CC) to be no bigger than $\sim 10\text{ meV}^{4}$, an energy density that is
at least 60 orders of magnitude smaller than known perturbative contributions
to it from the Standard Model. The extreme fine tuning necessary to cancel
these known contributions is known as the cosmological constant problem (see
Weinberg:1988cp  for a review). A compelling possibility is to accept the
expected large CC value, with all its radiative corrections included, as a
starting point and have it compensated by a scalar field whose dynamics lowers
its potential energy over time, gradually relaxing the effective CC in the
process Abbott:1984qf . Given an appropriate interaction structure and
cosmology, the relaxation can be made to naturally stop when the CC is
sufficiently fine-tuned, thereby naturally generating a tiny CC.

Refs. Graham:2017hfr ; Graham:2019bfu  put forward a simple, proof-of-
principle model for this class of solution that not only accomplishes CC fine-
tuning from a naturally large initial value, but also explains how this story
could fit into the hot-big-bang epoch. In this scenario, the universe
undergoes an expansion, a contraction, and an expansion again. During the
first expansion, the universe is in an extremely long period of inflation.
This period of inflation is produced by the rolling of a scalar field that has
a small, but technically natural, slope. As the field rolls down its
potential, the effective CC steadily decreases and eventually becomes slightly
negative. At this time, the large contributions to the CC have been
dynamically cancelled. But, we are left with a universe that is cold and empty
due to the long period of inflation. However, the negative value of the CC
halts the expansion of the universe and makes it contract. During this period
of contraction, various small energy densities in the universe (such as the
kinetic energy of the scalar itself) blueshift, producing a contracting hot
universe with a small negative CC. The energy density produced in the
contraction could conceivably excite UV degrees of freedom that could trigger
a bounce, causing the universe to re-expand as a hot universe but with a
small, dynamically relaxed CC. In this scenario, our current hot big bang
epoch exists during this stage of re-expansion.

The principal accomplishment of this class of models is that it allows for a
factorized approach to solve the CC problem. Specifically, it permits UV
physics at very high energies to play a role in solving the CC problem. In
these models, the tuning necessary to cancel the CC is accomplished in the
infrared by a slowly rolling scalar field. The UV plays a role at high
energies/temperatures where this dynamics can trigger a bounce causing the re-
expansion of a contracting universe.

One important issue that has not been addressed in the cosmology of the
minimal CC-relaxation model described above is the origin of the approximately
scale-invariant spectrum of scalar perturbations observed in the cosmic
microwave background (CMB). The minimal model has an inflationary period, and
therefore scale invariance, built into it. However, in order to solve the CC
problem the Hubble rate during its observationally-relevant last tens of
$e$-folds must at most be as low as its value today, $H_{0}\sim
10^{-42}\mathinner{\mathrm{GeV}}$. If the CMB anisotropies were to be seeded
by the vacuum fluctuations during such an extremely low energy inflation, the
resulting scalar power spectrum would be far too small. One may hope that the
superhorizon evolution of the curvature perturbations in the contraction phase
after inflation would lead to their growth, however under general assumptions
superhorizon curvature perturbations are conserved in the absence of non-
adiabatic pressure perturbations Wands:2000dp ; Cardoso:2008gz 111In some
gauges, e.g longitudinal gauge, there exist a curvature perturbation mode that
is growing when the universe is contracting, however it has been shown, again
under general assumptions, that this mode does not contribute to the
observationally-relevant post-bounce constant mode Creminelli:2004jg ;
Bozza:2005wn ; Bozza:2005qg ; Cardoso:2008gz .. The latter is essentially
because the homogeneous background $\rho$ and the large-scale perturbations
$\delta\rho$ of the total energy density follow the same evolution equation,
which leads to their ratio $\delta\rho/\rho$ being fixed. This suggests the
need for a non-adiabatic pressure perturbation contribution or a different
source of curvature perturbations.

In this paper, we improve on the the minimal CC-relaxation model, while
keeping the broad-brush cosmological history described above unchanged, by
coupling a field $\phi$ that relaxes the CC to a Yang-Mills sector axially.
Such a derivative coupling generates a thermal bath of radiation which, in
turn, damps the motion of $\phi$ without giving it a thermal mass
Berghaus:2019whh . The thermal fluctuations in the radiation fields cause
fluctuations in $\phi$ that are potentially much stronger than those of
quantum origin Bastero-Gil:2011rva . With appropriate choices of parameters,
this model simultaneously solves the cosmological constant problem and
explains the observed CMB anisotropies. Moreover, the thermal damping on the
rolling of $\phi$ improves considerably the degree of CC fine-tuning that can
be achieved through relaxation.

The rest of the paper is organized as follows. We present in Section II the
particle physics content of our cosmological constant relaxation model as well
as a brief summary of its cosmology. We then describe the background evolution
of the universe through the key cosmological epochs in our model and the
validity of the effective field theory under consideration in Section III.
Next, we show in Section IV that our model can explain the density
perturbations seen in the CMB, before concluding with Section V.

##  II Model

Our starting Lagrangian contains an axion-like field $\phi$ with a linear
potential which couples axially to a dark, pure $SU(N_{c})$ Yang-Mills sector

$\mathcal{L}=\frac{1}{2}\left(\partial\phi\right)^{2}+g^{3}\phi-\frac{1}{4}G^{a}_{\mu\nu}G^{a\mu\nu}-\frac{\alpha}{8\pi}\frac{\phi}{f_{G}}G^{a}_{\mu\nu}\tilde{G}^{a\mu\nu}$
(1)

Here, $\alpha\equiv e_{G}^{2}/4\pi$ is the fine-structure constant of the
gauge sector and $e_{G}$ (not $g$) denotes the associated coupling strength.
The potential $V=-g^{3}\phi$ serves as the effective CC to be relaxed and we
will sometimes refer to it as the CC. Any $\phi$-independent contribution to
the CC are accounted for by appropriately shifting $\phi$ by a constant value.

To start with, the $\phi$ field is large and negative, with its large and
positive potential $V=-g^{3}\phi$ driving a period of cold inflation, where
the yet unoccupied gauge sector plays no role. The $\phi$ field proceeds to
roll down its potential and, once it gains enough kinetic energy, the gauge
fields become tachyonic, grow rapidly, and subsequently thermalize. Shortly
after, the universe enters a period of warm inflation where the gauge-field
thermal bath affects the background evolution through the thermal friction it
exerts on $\phi$ and sources the dominant density perturbations from its
thermal fluctuations. We assume the modes corresponding to the fluctuations
seen in the CMB become superhorizon during this period. Warm inflation ends
when $\phi$ has relaxed to a small negative value, which is followed by $\phi$
going through zero to positive values, making the potential $-g^{3}\phi$
negative and causing the universe to crunch. The now blue-shifting gauge-field
radiation soon dominates the universe. Once it hits a high enough temperature,
we assume that some UV dynamics are triggered causing the universe to bounce
and re-expand, entering the standard hot big bang epoch. An example of such UV
dynamics was discussed in Graham:2017hfr , but many other such models may be
possible. A sketch of the full cosmological history is shown in Fig. 1.

Figure 1: The scale factor v.s. comoving Hubble scale sketch (not-to-scale) of
our model. While it is shown here that the scale factor today is greater than
that at the end of warm inflation, the reverse is also possible.

Our simple model achieves not only the tuning of the effective CC from a
naturally large positive value to a small negative one, but also the right
level of CMB anisotropies. The phenomenology of our model is largely
determined by the slope $-g^{3}$ of the $\phi$ potential and the thermal-
dissipation coefficient $\Upsilon$ describing the damping of $\phi$ during
warm inflation in our model Berghaus:2019whh

$\Upsilon=\frac{T^{3}}{f^{2}}$ (2)

Here $f$ is related to the Lagrangian parameters $\alpha$ and $f_{G}$ through
the relation $f\propto f_{G}/\alpha^{5/2}$ with an $O(0.1)$ proportionality
constant that depends mildly on $\alpha$ and the details of the Yang-Mills
sector. We will treat the more directly relevant quantity, $f$, together with
$g$ as our free model parameters. The viable parameter space is shown in Fig.
2, as we will show.

Figure 2: Viable parameter space: the colored regions are ruled out and the
black line (c.f. (31)) yields the right amplitude of the scalar power spectrum
$P_{\zeta}\sim 10^{-9}$. The gray region (c.f. (26)) is ruled out as the CC
would vary too quickly today. The orange region (c.f. (27)) requires the CC
relaxation to continue beyond the regime of validity of the higher dimensional
operator that couples $\phi$ to the non-abelian gauge sector. The cyan region
(c.f. (21)) does not go through the strong regime of warm inflation.

##  III Background

###  III.1 Cold Inflation

We suppose that the universe starts in a cold inflationary phase with a Hubble
scale $H_{i}$, driven by the potential $V_{\rm i}$ of the slow-rolling $\phi$
field. In order to avoid eternal inflation, we require the classical rolling
to dominate over quantum fluctuation, $\dot{\phi}_{i}H_{i}^{-1}\sim
g^{3}/H_{i}^{2}\gtrsim H_{i}$, which can be rewritten as

$V_{\rm i}\lesssim M_{\rm P}^{2}g^{2}$ (3)

where $M_{\rm P}$ is the reduced Planck mass. Hence, the highest possible
initial CC, $V_{\rm i}$, that is amenable to our proposed CC-relaxation
mechanism is set by the highest viable values of the slope parameter $g$ from
the considerations below. As we will see, the combined requirements of solving
the CC problem and having the dynamics of the CC relaxation be within the
regime of validity of our effective field theory (EFT) puts an upper bound of
$g\lesssim 10^{-18}\mathinner{\mathrm{GeV}}$, corresponding to $V_{\rm
i}\lesssim\mathinner{\mathrm{GeV}}^{4}$. If we add to this the requirement of
explaining the CMB anisotropies, $g$ is further limited to be $g\lesssim
10^{-20}\mathinner{\mathrm{GeV}}$, which corresponds to $V_{\rm
i}\lesssim\left(100\mathinner{\mathrm{MeV}}\right)^{4}$.

###  III.2 Thermalization of the Gauge Sector

Meanwhile, the non-abelian gauge fields play no role as the universe inflates,
that is, until they are considerably excited. The equations of motion for the
gauge fields $A^{a\mu}$, after a spatial-Fourier and a helicity decomposition,
reads

$\ddot{A}^{a\pm}_{k}+H\dot{A}^{a\pm}_{k}+\left(\omega_{k}^{\pm}\right)^{2}A^{a\pm}_{k}+(\text{non-
abelian terms})=0$ (4)

where $k\propto a^{-1}$ denotes a physical wavenumber and

$\left(\omega_{k}^{\pm}\right)^{2}=k\left(k\mp 2\xi
H\right),\quad\xi\equiv\frac{\alpha\dot{\phi}}{4\pi f_{G}H}$ (5)

For a given $k$, the non-abelian terms in (4) are negligible if $k\gtrsim
e_{G}|\left<A^{a\mu}(t,x)\right>_{L\sim k^{-1}}|$, where
$\left<A^{a\mu}(t,x)\right>_{L\sim k^{-1}}$ is the gauge-field amplitude in
the configuration space averaged over an $L\sim k^{-1}$ sized domain.
Initially, the gauge fields have only horizon-size fluctuations with the
typical amplitude $\left|\left<A^{a\mu}(t,x)\right>\right|\sim H$, and so the
evolution of modes with $k\gg e_{G}H$ is well described by the terms shown in
(4), according to which $+$ helicity modes with $k<2\xi H$ are tachyonic. This
provides a way to excite the gauge sector even if it begins in a Bunch-Davies
vacuum.

The value of $\xi\propto H^{-2}$ increases as $\phi$ rolls down the potential
with a Hubble-friction supported terminal velocity $\dot{\phi}=g^{3}/3H$. When
$\xi$ goes considerably above unity, comoving gauge-field modes become
tachyonic well before they cross the horizon and get amplified exponentially
until the growth is stopped by Hubble friction slightly outside the horizon.
The total energy density of the gauge fields soon reaches a steady state where
the gain from tachyonic instability is balanced by Hubble dilution, at a value
Anber:2009ua

$-\frac{1}{4}\left<G^{a}_{\mu\nu}G^{a\mu\nu}\right>_{H}\sim
N_{c}^{2}10^{-4}\frac{e^{2\pi\xi}}{\xi^{3}}H^{4}$ (6)

In the many Hubble times that follow, the steady-state gauge field energy
density grows according to (6) as $\xi$ increases, until it reaches the value
DeRocco:2021rzv

$-\frac{1}{4}\left<G^{a}_{\mu\nu}G^{a\mu\nu}\right>_{\rm NL}\sim\frac{(\xi
H)^{4}}{4\pi\alpha}$ (7)

whereupon non-abelian terms in (4) become comparable to the abelian ones for
$k\sim\xi H$ modes. By equating (6) and (7), one can calculate iteratively the
$\xi$ when this occurs to be

$\xi_{\rm
NL}\approx\left[\frac{6.7-\ln(N_{c}^{2}\alpha)}{2\pi}\right]+\frac{7}{2\pi}\ln\left[\frac{6.7-\ln(N_{c}^{2}\alpha)}{2\pi}\right]+\ldots$
(8)

At this point, some of the modes that would otherwise be growing tachyonically
acquire positive non-abelian contributions to their squared frequency,
terminating said growth. Non-linear interactions among these modes then lead
to a net transfer of energy from the amplified modes to other less excited
modes (including those with $-$ helicity) Kurkela:2011ti . The entropy is
maximized when the gauge sector is in thermal equilibrium with a peak energy
$\sim T_{\rm th}$ that is given by energy conservation, i.e. by equating
$\sim(N_{c}^{2}\pi^{2}/30)T_{\rm th}^{4}$ with the
$-\left<G^{a}_{\mu\nu}G^{a\mu\nu}\right>/4$ at that time. However, such a
thermal configuration can only be achieved if the near-equilibrium
thermalization rate $\sim 10N_{c}^{2}\alpha^{2}T_{\rm th}$ is greater than $H$
Laine:2021ego  in the first place, which is equivalent to

$-\frac{1}{4}\left<G^{a}_{\mu\nu}G^{a\mu\nu}\right>\gtrsim 3\times
10^{-5}\frac{N_{c}^{2}}{\left(N_{c}\alpha\right)^{8}}H^{4}$ (9)

In what follows, we call the critical $\xi$ when the above is satisfied
$\xi_{\rm scat}$.

We will focus on the case in which at the point the gauge fields hit the non-
linear regime they have enough energy to immediately thermalize. This requires
$\xi_{\rm NL}\gtrsim\xi_{\rm scat}$, which for $N_{c}={\cal O}(1)$ translates
to

$N_{c}\alpha\gtrsim 0.1$ (10)

which comes from plugging (7) and (8) into (9), and solving for $N_{c}\alpha$
numerically. The temperature $T_{\rm th}$ at which the gauge sector first
thermalizes will be ${\cal O}(1)$ higher than the Hubble rate $H_{\rm th}$ at
that time, and is roughly given by

$T_{\rm th}^{2}\sim H_{\rm th}^{2}\sim\frac{g^{3}}{f}$ (11)

where we used $\dot{\phi}\sim g^{3}/3H$, (5), (8), and (10). The opposite
case, $\xi_{\rm NL}\lesssim\xi_{\rm scat}$, is also interesting. We choose to
avoid it simply because we have less control of the evolution of the universe
in that case, as it requires some knowledge of the non-linear dynamics of the
non-abelian fields when $\xi\gtrsim\xi_{\rm NL}$ but before (9) is satisfied.
We speculate on what would happen in that regime in Appendix A.

In the above and in what follows, we assume that all the relevant scales of
fluctuations (the higher of the Hubble rate $H$ or the temperature $T$ at a
given time) during the CC relaxation are above the confinement scale of the
gauge sector, i.e. the gauge sector is always weakly coupled. The lowest
fluctuation energy scale occurs when the gauge sector begins to thermalize,
when $H\sim H_{\rm th}$ (the Hubble rate $H$ was higher before and once the
gauge sector thermalize its temperature is always higher than $H$ and will
only increase). So, as long as $\alpha$ is perturbative at that point, it will
also be perturbative at all other epochs.

###  III.3 Warm Inflation

Once the non-abelian gauge fields are thermalized, they behave as a thermal
bath of dark radiation with energy density

$\rho_{\rm DR}=\frac{\pi^{2}}{30}g_{*}T^{4}$ (12)

This dark radiation then damps the motion of $\phi$ with a dissipative
coefficient $\Upsilon=T^{3}/f^{2}$ as a form of backreaction. The number of
degrees of freedom $g_{*}$ will depend on the gauge group of the Yang-Mills
fields under consideration. For simplicity, we will from now on specialize to
the $SU(2)$ case, for which $(\pi^{2}/30)g_{*}={\cal O}(1)$.

The background equations in this epoch take the form

$\displaystyle\ddot{\phi}+(3H+\Upsilon)\dot{\phi}-g^{3}$ $\displaystyle=0$
(13) $\displaystyle\dot{\rho}_{\rm DR}$
$\displaystyle=\Upsilon\dot{\phi}^{2}-4H\rho_{\rm DR}$ (14) $\displaystyle
3M_{\rm P}^{2}H^{2}$ $\displaystyle=\rho_{\rm
DR}+\frac{1}{2}\dot{\phi}^{2}-g^{3}\phi$ (15) $\displaystyle M_{\rm
P}^{2}\dot{H}$ $\displaystyle=-\frac{1}{2}\dot{\phi}^{2}-\frac{2}{3}\rho_{\rm
DR}$ (16)

These admit attractor solutions, dubbed warm inflation, where the universe is
inflating ($V\gg\rho_{\rm DR}$ and $\dot{H}\ll H^{2}$) and thermal
fluctuations dominate the fluctuations of $\phi$. During warm inflation, the
rolling of $\phi$ is overdamped with negligible $\ddot{\phi}$, the thermal
bath is kept in a quasi-equilibrium with negligible $\dot{\rho}_{\rm DR}$, and
these imply

$\dot{\phi}\approx\frac{g^{3}}{\Upsilon+3H},\quad\rho_{\rm
DR}\approx\frac{\Upsilon}{4H}\dot{\phi}^{2}$ (17)

In the parameter space regimes of our interest, the universe lands in the weak
regime ($\Upsilon\lesssim H$) when the gauge sector first thermalizes at the
temperature $T_{\rm th}$ found in (11). Incidentally, this initial temperature
is comparable to the the steady state value $T\sim g^{6}/f^{2}H_{\rm th}^{3}$,
derived from (17), at that time. The radiation temperature then proceeds to
track its steady-state value and scale as $T\propto H^{-3}$ until it is high
enough to enter the strong regime ($\Upsilon\gtrsim H$). This happens when

$H_{\rm strong}\sim g\left(\frac{g}{f}\right)^{4/5}$ (18)

Once the universe is in the strong regime, the steady-state temperature scales
more slowly with the Hubble rate, $T\propto H^{-1/7}$. It can be seen from the
background equations that the conditions for slow-roll inflation,
$V\gg\rho_{\rm DR}$ and $\dot{H}\ll H^{2}$, break at about the same time, when
the Hubble rate and gauge-sector temperature are of the order of

$H_{\rm end}=\left(\frac{f^{4}g^{12}}{M_{\rm P}^{7}}\right)^{1/9},\quad T_{\rm
end}=\left(M_{\rm P}f^{2}g^{6}\right)^{1/9}$ (19)

marking the end of warm inflation. The field value at that point is around

$\phi_{\rm end}=-\left(\frac{f^{8}}{g^{3}M_{\rm P}^{5}}\right)^{1/9}M_{\rm P}$
(20)

In order to have an extended period of strong warm inflation, where density
perturbations are enhanced, we require $H_{\rm strong}$ to be well above
$H_{\rm end}$, which amounts to

$f\ll\left(M_{\rm P}^{5}g^{3}\right)^{1/8}$ (21)

###  III.4 Crunching

Though inflation has ended, $\phi$ continues to roll down at its thermal-
friction-supported terminal velocity $\dot{\phi}_{\rm end}\approx
g^{3}/\Upsilon_{\rm end}$. It soon crosses zero, overshoots to positive
values, and passes a point where the now negative $V=-g^{3}\phi$ cancels out
the $\rho_{\rm DR}\gg\dot{\phi}^{2}/2$ term in Eq. (15) completely. At that
point the universe comes to a halt ($H=0$) and begins to crunch ($H<0$). This
Hubble sign flip causes the energy density $\rho_{\rm DR}$ of the dark
radiation to start blueshifting as $a^{-4}$. Soon, it takes over as the
dominant component and remains to be so for most of the crunching phase. Due
to the increasing thermal friction $\Upsilon\propto T^{3}$ and Hubble rate
$H\propto T^{2}$ during the crunching, the distances $\phi$ travels in the
subsequent Hubble times continue to diminish. Hence, most of the $\phi$
displacement after the end of warm inflation is already covered in the first
Hubble time or so, and $\phi$ eventually parks at a value $\sim\dot{\phi}_{\rm
end}H_{\rm end}^{-1}\sim\left|\phi_{\rm end}\right|$.

We performed a numerical simulation to confirm the background evolution around
the end of warm inflation and the start of crunching discussed above (see Fig.
3). We set $f=100\,{\rm keV}$, $g=10^{-23}\,{\rm GeV}$, and evolve the
equations of motion numerically using the absolute number of $e$-folds $N_{\rm
abs}\equiv\int\left|H\right|dt$ relative to the $H=0$ point when the universe
starts crunching as a proxy for time. The results suggest that the Hubble rate
does not change significantly in the last tens of $e$-folds of the warm
inflation. The same plot also shows that the stalling value of $\phi$ is
comparable to its absolute value $\left|\phi_{\rm end}\right|$ at the end of
warm inflation (a few $e$-folds before the $H=0$ point).

Figure 3: Evolution of $\phi$, $T$, and $H$ from the slow-roll regime to where
$\phi$ stalls in the crunching stage before the bounce. Dashed lines indicate
negative values. Quantities are normalized to their initial slow-roll value,
and are plotted against the absolute number of $e$-folds. The inset plot is a
zoom-in around the $H=0$ point. Here, we take $f=100\,{\rm keV}$ and
$g=10^{-23}\,{\rm GeV}$.

###  III.5 Bouncing and Expanding to the Present Epoch

At this point, the universe is still crunching, the CC is tuned to a tiny but
negative value, and the Standard Model sector is empty. The details of what
happens next will depend on the full high energy theory that takes over
outside the realm of validity of the EFT shown in Eq. (1). That said, we know
that several things need to happen in order to produce the observed features
of the universe: (1) the universe must bounce (stop crunching and start
expanding), (2) the Standard Model sector must be excited, and (3) the CC must
be made positive and at the right level. The bounce can be achieved either by
violating NEC Alberte:2016izw  or by inducing vorticity Graham:2017hfr . It is
conceivable that the Standard Model sector as well as the degrees of freedom
responsible for the bounce, whatever the mechanism is, are generated as the
radiation is blue-shifted to sufficiently high temperatures. While we expect
the $\phi$ contribution to the CC, $-g^{3}\phi$, to remain negative, the CC
also receives contributions from other sectors that are decoupled from the
relaxation dynamics of $\phi$. Some of these sectors may have multiple vacua
and possibly transitioned between them as the universe was heated before the
bounce and cooled after the bounce222A concrete realization of this scenario
can be found in Appendix A of Graham:2019bfu .. Such a vacuum switching adds a
$\phi$-independent contribution to the CC which would show up in the EFT under
consideration as a constant shift $\Delta V$ in the potential $V$ of $\phi$

$V=-g^{3}\phi\quad\longrightarrow\quad V=-g^{3}\phi+\Delta V$ (22)

This shift must be positive in order to explain the observed CC value today.

In the interest of solving the CC problem, we require our relaxation process
to tune the CC to a negative value whose magnitude $\sim g^{3}\left|\phi_{\rm
end}\right|$ is comparable to or less than the shift $\Delta V$ which, in
turn, must be comparable or close to the value observed today
$\sim(2\mathinner{\mathrm{meV}})^{4}$,

$g^{3}\left|\phi_{\rm end}\right|\lesssim\Delta
V\sim(2\mathinner{\mathrm{meV}})^{4}$ (23)

This theoretical constraint allows the process that shifts the relaxed tiny
negative CC to the positive value observed today to be natural, i.e. not fine-
tuned. Apart from that, we also want to avoid the possibility that the now
correct CC, $V\sim(2\mathinner{\mathrm{meV}})^{4}$, would un-tune itself due
to the subsequent rolling of $\phi$. Whether, or to which degree, this occurs
has some dependence on the UV completion of our EFT (Eq. (1)). We assume here
that the thermal friction on $\phi$ increases with temperature, in which case
most of the $\phi$ rolling would occur at low energies where our EFT is valid,
and leave the discussion on the opposite possibility (that the thermal
friction diminishes at high temperatures) to the next subsection.

The most relevant constraint on the dark energy today comes from its measured
energy density $\sim(2\mathinner{\mathrm{meV}})^{4}$ and the constraints on
its equation of state $w_{\rm DE}$, which in our model is given by

$w_{\rm
DE}\approx\frac{-(2\mathinner{\mathrm{meV}})^{4}+\dot{\phi}_{0}^{2}/2+\rho_{\rm
DR,0}/3}{(2\mathinner{\mathrm{meV}})^{4}+\dot{\phi}_{0}^{2}/2+\rho_{\rm
DR,0}}$ (24)

The Planck TT, TE, EE+lowE+lensing+Sne+BAO constraints the equation of state
today to be $-1<w_{\rm DE}<-0.95$ Planck:2018vyg . Note that in cases of our
interest $\rho_{\rm DR}\gg(\Upsilon/H)\dot{\phi}^{2}\gg\dot{\phi}^{2}$, and so
the latter constraint is essentially a constraint on the amount of dark
radiation today:

$\rho_{\rm DR,0}\lesssim 0.6\mathinner{\mathrm{meV}}^{4}$ (25)

In the likely scenario where $\dot{\phi}$ and $\rho_{\rm DR}$ are tracking the
steady state solution (17) today333Assuming that the $\rho_{\rm DR}$ at the
time of BBN is consistent with the $\Delta N_{\rm eff}$ bound, $\dot{\phi}$
and $\rho_{\rm DR}$ would naturally evolve to reach the steady state given by
(17) before today., we can use (17) to express $\rho_{\rm DR}$ in terms of the
current Hubble rate $H_{0}\approx 10^{-42}\mathinner{\mathrm{GeV}}$ and the
model parameters $f$ and $g$

$\rho_{\rm DR,0}\sim\left(\frac{f^{2}g^{6}}{4H_{0}}\right)^{4/7}\lesssim
0.6\mathinner{\mathrm{meV}}^{4}$ (26)

It turns out that this is more stringent than the theoretical constraint (23),
and so can be regarded as our model requirement for solving the cosmological
constant problem.

###  III.6 EFT Validity

The EFT under consideration, Eq. (1), is trustworthy throughout the CC-
relaxation process if at any given time the cutoff444We adopt $f_{G}/\alpha$
as the cutoff of our EFT here, but the actual cutoff scale will depend on how
the EFT is UV completed. For example, in some UV completions with $\phi$
coupled to heavy fermions Kim:1979if ; Shifman:1979if , the scale at which the
EFT breaks down is closer to $f_{G}$ than $f_{G}/\alpha$. $f_{G}/\alpha\sim
10\alpha^{3/2}f$ of the higher dimensional operator that couples $\phi$ to the
Yang-Mills sector is above the highest relevant fluctuation scale, i.e. the
higher of the Hubble rate and the gauge sector temperature. The Hubble rate is
the only relevant scale during cold inflation and its highest value $H_{i}$
occurs at the very beginning, while the temperature $T$ dominates over $H$
throughout warm inflation and is highest at the end of it. In cases of our
interest, we always have $T_{\rm end}\gg g\gtrsim H_{i}$. Thus, it suffices to
require

$\alpha\gtrsim\left(\frac{T_{\rm end}}{f}\right)^{2/3}$ (27)

Since Section III.2 we have specialized to the regime where
$0.1\lesssim\alpha\lesssim 1$ (for $N_{c}={\cal O}(1)$) at the time of the
gauge sector thermalization. The running of the coupling constant for a pure
Yang-Mills theory tells us that by the end of warm inflation, we would have
$\alpha\sim\left[1/0.1+(11/6\pi)N_{c}\ln\left(T_{\rm end}/H_{\rm
th}\right)\right]^{-1}\sim 0.01$. In light of this, (27) turns into a
constraint on $f$ and $g$.

Let us return to the issue of the CC un-tuning itself due to the rolling of
$\phi$. We have discussed previously the case where the thermal friction on
$\phi$ becomes stronger with increasing temperature in the UV regime, in which
case most of the $\phi$ rolling occurs at low energies. We now consider the
extreme scenario where the thermal friction disappears completely outside of
the realm of validity of our EFT, assuming that the UV regime is radiation
dominated throughout. In that case, when the temperature goes above the cutoff
$f_{G}/\alpha$ during the contraction, at the Hubble scale $H_{\rm cutoff}\sim
10^{2}\alpha^{3}f^{2}/M_{\rm P}$, the initially small $\dot{\phi}$ (due to
thermal friction) immediately accelerates to $g^{3}/H_{\rm cutoff}$ in a
Hubble time due to the slope $-g^{3}$ of the $\phi$ potential. Following that,
$\dot{\phi}$ scales up as $a^{-3}$ due to Hubble anti-friction up to the
bounce point and proceeds to scale down as $\dot{\phi}\sim a^{-3}$ (all the
while with $\dot{\phi}\gtrsim g^{3}/H$) until $\dot{\phi}$ is back at
$g^{3}/H_{\rm cutoff}$ at $H\sim H_{\rm cutoff}$, where the thermal friction
turns back on and slows $\dot{\phi}$ down significantly. Given this scaling
and that $H^{-1}\propto a^{2}$ during radiation domination, the field
excursion per Hubble time $\dot{\phi}H^{-1}$ would be maximum at the bounce
point, where $\dot{\phi}H^{-1}\sim\left(g^{3}/H_{\rm
cutoff}^{2}\right)\left(T_{\rm bounce}/10\alpha^{3/2}f\right)$. Requiring the
CC to not change by more than $\sim(2\mathinner{\mathrm{meV}})^{4}$ at that
time amounts to

$\frac{T_{\rm
bounce}}{10^{16}\mathinner{\mathrm{GeV}}}\lesssim\left(\frac{\alpha}{0.01}\right)^{15/2}\left(\frac{f}{10\mathinner{\mathrm{MeV}}}\right)^{5}\left(\frac{g}{10^{-20}\mathinner{\mathrm{GeV}}}\right)^{-6}$
(28)

To reiterate, this is only when the thermal friction disappears as soon as the
temperature is above the cutoff. Thus, this is a model-dependent constraint.
Since this is not a forbidding constraint, the success of the CC relaxation
per se does not hinge on the UV physics, even if the only viable UV model
requires the friction to disappear.

##  IV Perturbations

CMB observations by the Planck satellite found at 95% confidence level the
following amplitude and spectral index of the scalar power spectrum
Akrami:2018odb

$P_{\zeta}(k_{*})\approx 2.1\times 10^{-9},\quad n_{s}(k_{*})=0.9649\pm
0.0084$ (29)

at around the pivot scale whose wavenumber today is $k_{*}=0.05\text{
Mpc}^{-1}$. These observed perturbations are often attributed to the vacuum
fluctuations during an inflationary period taking place at some energy scale
well above MeV, the big bang nucleosynthesis (BBN) temperature, in a universe
that has a beginning and has been expanding since.

Our CC-relaxation model does include a period of inflation, albeit one that is
separated by a crunch and a bounce from the hot big bang epoch. When it comes
to explaining CMB anisotropies, this fact by itself is not an issue as the CMB
scale perturbations are frozen as soon as they turn superhorizon during the
pre-bounce inflation and therefore insensitive to the subsequent cosmological
history until around the epoch of recombination Creminelli:2004jg . A robust
feature of our CC-relaxation model is that the last stages of the pre-bounce
inflation must occur at $\lesssim\mathinner{\mathrm{meV}}$ energy scales in
order to solve the cosmological constant problem. Given the fact that the
vacuum contribution to the scalar power spectrum $P_{\zeta}$ is related to the
inflationary Hubble rate $H$ as $P_{\zeta}\propto H^{2}$, its low energy scale
poses a significant challenge in getting the right level of curvature
perturbations.

We show in this section that thermal perturbations during warm inflation can
provide enough enhancements of the curvature perturbations relative to the
vacuum one to match the amplitude seen in CMB. However, due to the lack of
curvature in the linear potential $-g^{3}\phi$ in our model, the resulting
scalar power spectrum is always blue-tilted, in contrast to the red-tilted
spectrum suggested by CMB observations. This is, nevertheless, a mild problem
that can be addressed in isolation, while respecting the required shift
symmetry for the CC relaxation.

Moreover, our pre-bounce inflationary period automatically solves all the
other problems that the standard, post-big-bang inflation solves. The horizon
and flatness problems are solved by the sheer amount of expansion during the
inflationary period as in traditional inflationary scenarios. While the
thermal fluctuations during warm inflation give rise to highly amplified
scalar perturbations, they do not source extra tensor perturbations. The
tensor perturbations remain to be sourced by quantum fluctuations, which in
our case is suppressed due to the smallness of the Hubble rate near the end of
warm inflation. Hence, our model predicts a vanishingly small tensor to scalar
ratio $r$. Warm inflation also yields unique non-gaussianity signatures that
are within the current observational bounds. In the strong dissipative regime
that is of interest to us, $\Upsilon\gg 3H$, warm inflation predicts $f_{\rm
NL}={\cal O}(10)$ Moss:2011qc ; Berghaus:2019whh .

###  IV.1 Scalar Amplitude

The curvature power spectrum for modes exiting the horizon during the strong
warm inflation (with $\Upsilon\gg H$) is given by Bastero-Gil:2011rva

$P_{\zeta}\approx 5\times
10^{-10}\left(\frac{\Upsilon}{3H}\right)^{8}\left(\frac{T}{H}\right)\left(\frac{H^{2}}{\dot{\phi}}\right)^{2}$
(30)

The exact amplitude $P_{\zeta}(k_{*})$ evaluated at the CMB pivot scale down
to ${\cal O}(1)$ factors will depend not only on the spectral index but also
on the number of $e$-folds separating the horizon exit of the pivot scale and
the end of warm inflation. These are more model-dependent and will be
discussed in the next subsection. Our focus here is on getting the amplitude
at the right order of magnitude. An extrapolation of the small spectral tilt
observed in the CMB suggests that $P_{\zeta}(k)$ would not vary by orders of
magnitudes in the range of scales that become superhorizon in the last tens of
$e$-folds of warm inflation. Hence, a good estimate for $P_{\zeta}(k_{*})$ is
given by Eq. (30) for the mode that exited the horizon at end of warm
inflation

$P_{\zeta}\sim 10^{-13}\left(\frac{g^{12}M_{\rm P}^{11}}{f^{23}}\right)^{2/3}$
(31)

where we have used (2), (17), (19), and taken $T\sim T_{\rm end}$, etc. Thus,
$P_{\zeta}(k_{*})\sim 10^{-9}$ can be achieved with

$f\sim g^{12/23}M_{\rm P}^{11/23}$ (32)

Given the strong dependence of the amplitude Eq. (31) on $f$ and $g$, we
expect this rough parameteric requirement to be robust against model
variations. To explain both the CMB anisotropies and the CC problem at once,
we need to satisfy the above plus the dark energy constraint Eq. (26). This
amounts to

$g\lesssim 10^{-20}\mathinner{\mathrm{GeV}},\quad f\lesssim
10\mathinner{\mathrm{MeV}}$ (33)

###  IV.2 Scalar Spectral Index

The spectral index evaluated at the pivot scale $k_{*}$ is given by

$\displaystyle n_{s}-1=\left.\frac{d\ln
P_{\zeta}}{dN}\right|_{k=k_{*}}=\frac{6}{7}\left(11\epsilon_{V}-8\eta_{V}\right)$
(34)

where $dN$ is differential number of $e$-folds and

$\displaystyle\epsilon_{V}$ $\displaystyle\equiv$ $\displaystyle\frac{M_{\rm
P}^{2}}{2(1+\Upsilon_{*}/3H_{*})}\left(\frac{V_{*}^{\prime}}{V_{*}}\right)^{2}$
$\displaystyle\eta_{V}$ $\displaystyle\equiv$ $\displaystyle\frac{M_{\rm
P}^{2}}{1+\Upsilon_{*}/3H_{*}}\left(\frac{V_{*}^{\prime\prime}}{V_{*}}\right)$
(35)

are two of the warm-inflation slow-roll parameters evaluated at the horizon
crossing of the pivot scale. In the minimal model where the potential is
perfectly linear, $V(\phi)=-g^{3}\phi$, and $\phi$ interacts only with the
weakly-coupled non-abelian gauge sector that provides the dominant friction
during warm inflation, we have $\eta_{V}=0$ and consequently the scalar power
spectrum is always blue-tilted $n_{s}-1>0$.

To get a red-tilted spectrum, we need a non-zero and sufficiently positive
$\eta_{V}$ to counter the $\epsilon_{V}$ term in Eq. (34). This can be
achieved, for example, by coupling $\phi$ to a cold, confined non-abelian
sector with a decay constant $\tilde{f}$ and a confinement scale
$\tilde{\Lambda}$. Doing so adds a modulation to the linear potential
$-g^{3}\phi$

$V(\phi)=-g^{3}\phi+\tilde{\Lambda}^{4}\left[\cos\frac{\phi_{0}}{\tilde{f}}-\cos\left(\frac{\phi-\phi_{0}}{\tilde{f}}\right)\right]$
(36)

The phase $\phi_{0}$ accounts for the separation between the zero of $V(\phi)$
that is relevant to our scenario and the nearest extremum of the potential
modulation. The constant shift $\tilde{\Lambda}^{4}\cos(\phi_{0}/\tilde{f})$
ensures that the potential still crosses zero when $\phi$ does. We require
$\tilde{\Lambda}^{4}/\tilde{f}\leq g^{3}$, so that the resulting potential
alternates between steep cliffs and flat plateaus with no minima. Otherwise
$\phi$ could be trapped prematurely in one of these minima. This no-trapping
condition and the definition of $\eta_{V}$ amount to an upper bound on
$\tilde{f}$, which can be written as

$\tilde{f}<\frac{\cos\left[(\phi_{*}-\phi_{0})/\tilde{f}\right]}{\eta_{V}}\left(\frac{g^{3}}{V^{\prime}_{*}}\right)\dot{\phi}_{*}H_{*}^{-1}$
(37)

assuming that the CMB scales became superhorizon before slow-roll breaks. It
is likely that the CMB scales $k\sim 10^{-4}-10^{-1}\text{ Mpc}^{-1}$,
including the pivot scale $k_{*}$, all exited the horizon on one of the
plateaus since the rolling of $\phi$ is significantly slower there, and it
must be on a part of the plateau where $V^{\prime\prime}>0$ so that the
resulting spectral index is red. This would require the field excursion over
the $\Delta N_{\rm CMB}$ $e$-folds during which the CMB scales exited the
horizon to be much less than the oscillation period $2\pi\tilde{f}$ of the
potential, which sets a lower bound on $\tilde{f}$

$\tilde{f}\gg\frac{\Delta N_{\rm CMB}}{2\pi}\dot{\phi}_{*}H_{*}^{-1}$ (38)

Given that $\Delta N_{\rm CMB}\sim 10$ and $\eta_{V}\sim 1\%$, the following
choice of parameters

$\tilde{f}\sim
10^{2}\dot{\phi}_{*}H_{*}^{-1},\quad\tilde{\Lambda}^{4}\sim\tilde{f}g^{3}$
(39)

meet the requirements discussed above (with a little tuning). For example, if
$f\sim 100\mathinner{\mathrm{keV}}$ and $g\sim
10^{-23}\mathinner{\mathrm{GeV}}$ then $\tilde{f}\sim
10^{14}\mathinner{\mathrm{GeV}}$ and $\tilde{\Lambda}\sim
10^{-14}\mathinner{\mathrm{GeV}}$ would work. Figure 4 shows that, for
appropriate choices of $\tilde{f}$ and $\tilde{\Lambda}$, there is a range of
$e$-folds $\Delta N\gtrsim 10$ with $n_{s}$ lying inside the CMB 95%
confidence level band, Eq. (29). We have also checked that the running of the
spectral index is well within the the 95% confidence level limits from Planck.
Note that modes continue to exit the horizon in the radiation-dominated
contraction following the end of warm inflation. To account for this, we have
chosen the parameters such that the pivot scale crossed the horizon at around
$N\approx 40$ $e$-folds before the end of warm inflation, instead of the
typically assumed $N\approx 60$.

Figure 4: The top panel here is the same as Fig. 3, but for the modulated
potential Eq. (36). It shows that the background evolution remains unchanged
qualitatively. The bottom panel shows the scalar spectral index $n_{s}$ as a
function of the absolute number of $e$-folds, with the gray band showing the
Planck constraints. Here, we take $f=100\,{\rm keV}$, $g=10^{-23}\,{\rm GeV}$,
$\tilde{f}=2\times 10^{14}\,{\rm GeV}$, $\phi_{0}=4.5\tilde{f}$, and
$\tilde{\Lambda}=0.97(\tilde{f}g^{3})^{1/4}$.

##  V Discussion and Conclusion

We have developed a model that dynamically relaxes the (effective)
cosmological constant to a tiny value through the rolling of a shift-symmetric
scalar field down its linear potential. This simple model not only explains
the unexpected smallness of the cosmological constant and the reheating of the
universe to the hot big bang era, but also gives rise to the right amplitude
of the primordial density perturbations at the CMB scales. It also improves
the degree of cosmological constant tuning and is even simpler compared to the
previously proposed model of the same kind Graham:2019bfu . Remarkably, this
class of models is already strongly constrained by the shift symmetry required
to ensure the radiative stability of the $\phi$ field, and yet our model can
give rise to a rich cosmology that achieves the above-mentioned feats in a way
that is consistent with all the current observations. It will be interesting
to attempt a more systematic and generalized approach to realizing this class
of solutions. Possible extensions include considering different dissipative
mechanisms and multiple rolling scalars.

A number of assumptions were made in reaching our conclusions. We assumed that
the non-linear self-interactions of the gauge fields tend to bring themselves
to a thermal equilibrium, but did not describe the thermalization process in
full. Numerical simulations will provide a fuller picture of the
thermalization process in the nonlinear regime and a dedicated study may
reveal analytically-calculable processes that lead to the thermalization of
the gauge fields, possibly before they turn nonlinear. The simple EFT shown in
(1) that achieves the CC relaxation and generates the right amplitude of
primordial density perturbations does not on its own provide: a concrete
realization of the non-singular bounce, a mechanism that turns the relaxed
tiny negative CC into the positive value $(2\mathinner{\mathrm{meV}})^{4}$
observed today, and an explanation for the slight red tilt of the scalar power
spectrum seen in the CMB. While we have provided examples for each of these,
it would be interesting to write down a complete and renormalizable model that
achieves all the key steps in this scenario, as it would provide further
consistency checks.

The rich cosmology of our scenario gives rise to a series of predictions that
can be checked against current and future cosmological surveys. First, the
range of scales spanning 60 or so $e$-folds that re-enter the horizon during
the post-bounce expansion will include not only the modes that turn
superhorizon during warm inflation, but also those during the subsequent
contraction period. The resulting scalar power spectrum will have a mixed
feature where it is nearly scale-invariant at large scales (the inflation
part) and perhaps strongly blue-tilted at small scales (the contraction part),
with a rapid transition somewhere in between. Furthermore, the non-
gaussianities in the primordial perturbations generated during the warm
inflation period in our model are small enough to be within the current
constraints, but can be tested in future observations. Our model also predicts
highly suppressed inflationary gravitational waves, however the cosmological
bounce that occurs afterward may generate potentially observable gravitational
waves with a peaked spectrum.

Unlike most early universe models, the most defining parts of our model (i.e.
those described by the EFT shown in Eq. (1)) operates at low energies, and
this gives us further opportunities to test it today. If it is indeed realized
in nature, the present-day cosmology should mirror to some extent what
happened in the pre-bounce, low-energy universe. In fact, our model says that
we are about to enter a new period of warm inflation. Most directly related to
the dynamical explanation for the cosmological constant problem is the
evolving nature of the dark energy. This can be probed through the time
variation of its equation of state today. The strong friction needed to
generate the right level of primordial density perturbations requires a
thermal bath of dark radiation coupled to the dark energy. Assuming even weak
mixing with the standard model, such a dark radiation could be probed directly
in a variety of ways Berghaus:2020ekh .

## Acknowledgements

We thank William DeRocco, Michael Fedderke, Peter Graham, and Saarik Kalia for
useful discussions. DK and SR are supported in part by the NSF under grant
PHY-1818899. SR is also supported by the DoE under a QuantISED grant for MAGIS
and the SQMS quantum center and the Simons Investigator award 827042.

##  Appendix A Gauge Sector Thermalization in the Small Coupling $\alpha$
Regime

As argued in Section III.2, one requirement for the gauge sector to thermalize
is that it must have enough energy to satisfy (9). If $\xi_{\rm
NL}\lesssim\xi_{\rm scat}$, the gauge sector does not immediately thermalize
when it becomes non-linear, and will stay in the non-linear regime without
thermalizing until it acquires enough energy from $\phi$. In this appendix, we
argue that the gauge sector would still thermalize in this case and estimate
the possible values of the Hubble $H_{\rm th}$ rate when it does. The
challenge lies mainly in determining the growth rate of the energy density of
the gauge fields in the non-linear regime, where (6) no longer applies. The
gauge field temperature right after thermalization is then given by $T_{\rm
th}\sim H_{\rm th}/10N_{c}^{2}\alpha^{2}$, which can be much higher than the
Hubble rate at that time, $H_{\rm th}$, depending on how small $N_{c}\alpha$
is.

Let us first assume that when $\xi\gtrsim\xi_{\rm NL}$, the universe settles
to a steady state where the kinetic energy of $\phi$ is approximately constant
in a Hubble time:

$\frac{d}{dt}\left(\frac{\dot{\phi}^{2}}{2}\right)=g^{3}\dot{\phi}-3H\dot{\phi}^{2}-Q\approx
0$ (40)

where

$Q=-\frac{1}{4}\left<G^{a}_{\mu\nu}G^{a\mu\nu}\right>_{\rm
NL}\Gamma\sim\frac{(\xi H)^{4}}{4\pi\alpha}\Gamma$ (41)

is the energy density transfer rate from $\phi$ to the gauge sector,
parameterized by the rate $\Gamma$ to be estimated later. In order to
thermalize, the energy density acquired by the gauge sector per Hubble time
$QH^{-1}$ must satisfy (9), which for $N_{c}={\cal O}(1)$ happens when

$\xi_{\rm th}\sim\left(\frac{H}{\Gamma}\right)^{1/4}\frac{1}{\alpha^{7/4}}$
(42)

From (40), we see that there are two possible steady states for $\dot{\phi}$
at that time

$\dot{\phi}_{\rm th}\sim\text{min}\left[\frac{g^{3}}{H_{\rm
th}},\frac{Q}{g^{3}}\right]\sim\text{min}\left[\frac{g^{3}}{H_{\rm
th}},\frac{H_{\rm th}^{5}}{\alpha^{8}g^{3}}\right]$ (43)

depending on how small $\alpha$ is. The Hubble rate at which the gauge sector
thermalizes $H_{\rm th}$ can then be found from (42), (43), and the definition
of $\xi$ (Eq. (5))

$H_{\rm th}\sim\text{max}\left[\xi_{\rm
th}^{-\frac{1}{2}}\alpha^{-\frac{3}{4}}\left(\frac{g^{3}}{f}\right)^{\frac{1}{2}},\xi_{\rm
th}^{\frac{1}{4}}\alpha^{\frac{19}{8}}(g^{3}f)^{\frac{1}{4}}\right]$ (44)

We now attempt to estimate $\Gamma$. The thermalization condition (9) requires
the energy density in the gauge sector to go well above (7). If at that time
the energy density is still concentrated around the tachyonic $k\sim\xi H$
modes, we would have $e_{G}|\left<A^{a\mu}(t,x)\right>_{L\sim(\xi H)^{-1}}|\gg
e_{G}^{1/2}\left(-\left<G^{a}_{\mu\nu}G^{a\mu\nu}\right>_{\rm
NL}/4\right)^{1/4}\sim\xi H$. It follows that $k\lesssim
e_{G}|\left<A^{a\mu}(t,x)\right>_{L\sim k^{-1}}|$ for most of these modes and
this shuts off their tachyonic instabilities. While the tendency to maximize
entropy implies that energy would continue to flow from $\phi$ to the gauge
sector even when the latter has become completely nonlinear and no longer
tachyonic, it is possible that the energy transfer rate in that regime is
highly suppressed. To be on the conservative side, we assume that tachyonic
instability is the only way for the gauge fields to acquire energy from
$\phi$. The tachyonic band would re-open if the configuration-space gauge-
field amplitude is somehow reduced. There are two ways for this to happen.
First, Hubble friction will eventually deplete it if the gauge sector stops
gaining energy from $\phi$. The second way is through the self-scatterings
among the nonlinear modes, which tend to redistribute energy from the
amplified low-wavenumber modes to the less-occupied high-wavenumber modes.
This has the effect of increasing the typical gradient of $A^{a\mu}$. The
average amplitude on the tachyonic scales
$|\left<A^{a\mu}(t,x)\right>_{L\sim(\xi H)^{-1}}|$ must therefore go down due
to cancellations (the same conclusion can be reached from energy conservation
considerations). Based on these considerations, we expect $\Gamma$ to lie in
the range $H\lesssim\Gamma\lesssim\xi H$. The lowest $\Gamma$ corresponds to
the case where the self-scattering rates are so slow that the gauge fields
need to wait a Hubble time before the tachyonic band opens up again due to
Hubble dilution, while the highest $\Gamma$ is expected when the self-
scattering rates are so fast that the tachyonic band is essentially always
open. It follows from (42) that $\alpha^{-7/4}\lesssim\xi_{\rm
th}\lesssim\alpha^{-7/5}$ and this can be plugged into (44) to give the Hubble
rate when the gauge fields thermalize.

## References

  * (1) S. Weinberg,  Rev. Mod. Phys. 61, 1 (1989).
  * (2) L. F. Abbott,  Phys. Lett. B 150, 427 (1985).
  * (3) P. W. Graham, D. E. Kaplan, and S. Rajendran,  Phys. Rev. D 97, 044003 (2018), 1709.01999.
  * (4) P. W. Graham, D. E. Kaplan, and S. Rajendran,  Phys. Rev. D 100, 015048 (2019), 1902.06793.
  * (5) D. Wands, K. A. Malik, D. H. Lyth, and A. R. Liddle,  Phys. Rev. D 62, 043527 (2000), astro-ph/0003278.
  * (6) A. Cardoso and D. Wands,  Phys. Rev. D 77, 123538 (2008), 0801.1667.
  * (7) P. Creminelli, A. Nicolis, and M. Zaldarriaga,  Phys. Rev. D 71, 063505 (2005), hep-th/0411270.
  * (8) V. Bozza and G. Veneziano,  Phys. Lett. B 625, 177 (2005), hep-th/0502047.
  * (9) V. Bozza,  JCAP 02, 009 (2006), hep-th/0512066.
  * (10) K. V. Berghaus, P. W. Graham, and D. E. Kaplan,  JCAP 03, 034 (2020), 1910.07525.
  * (11) M. Bastero-Gil, A. Berera, and R. O. Ramos,  JCAP 07, 030 (2011), 1106.0701.
  * (12) M. M. Anber and L. Sorbo,  Phys. Rev. D 81, 043534 (2010), 0908.4089.
  * (13) W. DeRocco, P. W. Graham, and S. Kalia,  (2021), 2107.07517.
  * (14) A. Kurkela and G. D. Moore,  JHEP 12, 044 (2011), 1107.5050.
  * (15) M. Laine and S. Procacci,  (2021), 2102.09913.
  * (16) L. Alberte, P. Creminelli, A. Khmelnitsky, D. Pirtskhalava, and E. Trincherini,  JHEP 12, 022 (2016), 1608.05715.
  * (17) Planck, N. Aghanim et al.,  Astron. Astrophys. 641, A6 (2020), 1807.06209.
  * (18) J. E. Kim,  Phys. Rev. Lett. 43, 103 (1979).
  * (19) M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov,  Nucl. Phys. B 166, 493 (1980).
  * (20) Planck, Y. Akrami et al.,  Astron. Astrophys. 641, A10 (2020), 1807.06211.
  * (21) I. G. Moss and T. Yeomans,  JCAP 08, 009 (2011), 1102.2833.
  * (22) K. V. Berghaus, P. W. Graham, D. E. Kaplan, G. D. Moore, and S. Rajendran,  (2020), 2012.10549.