# Embedding Ultra slow-roll inflaton dynamics in Warm Inflation

Sandip Biswas<EMAIL_ADDRESS>Department of Physics, Indian Institute of
Technology, Kanpur: 208016, Uttar Pradesh, India  Kaushik Bhattacharya
<EMAIL_ADDRESS>Department of Physics, Indian Institute of Technology,
Kanpur: 208016, Uttar Pradesh, India  Suratna Das<EMAIL_ADDRESS>Department of Physics, Ashoka University, Rajiv Gandhi Education City, Rai,
Sonipat: 131029, Haryana, India

###### Abstract

Slow-roll of the inflaton field defines the standard dynamics of the
inflationary epoch. However, the inflationary system deviates from slow-roll
when it encounters an extremely flat region of the inflaton potential, and
enters a phase dubbed Ultra slow roll. In this article, we explore the
possibility of realizing an Ultra slow-roll phase in a particularly
interesting inflationary scenario, called Warm Inflation. In the Warm
inflationary scenario a thermalized, sub-dominant radiation bath coexists with
the inflaton energy density as an effect of dissipative dynamics. We show in
this article that though the background dynamics indicate Ultra slow-roll when
the potential becomes extremely flat, in Warm Inflation models, where the
dissipation coefficient is a sole function of the temperature of the radiation
bath, the system fails to maintain the thermal equilibrium as soon as it
enters the Ultra slow-roll phase. As thermal equilibrium is a key feature of
Warm Inflation, and as it is not yet known how to deal with Warm Inflation
without thermal equilibrium, we could not analyze such systems any further in
this article. However, we demonstrate that brief periods of Ultra slow-roll
phase, which smoothly ends into standard slow-roll, can be accommodated in WI
models where the dissipation coefficient is not only a function of the
temperature of the radiation bath but also depends on the amplitude of the
inflaton field. We theoretically determine the criteria of successfully
embedding Ultra slow-roll in WI while the system remain in thermal
equilibrium, and also demonstrate numerically that such short Ultra slow-roll
phases can indeed be embedded in specific Warm Inflation models which comply
with the theoretically determined criteria.

##  I Introduction

Cosmic Inflation Kazanas:1980tx ; Guth:1980zm ; Sato:1981ds ; Sato:1980yn ;
Linde:1981mu ; Albrecht:1982wi , a brief period of near-exponential expansion
of the early universe, has now become an integrated part of the standard big
bang cosmology that not only solves the fine-tuning problems of the hot big
bang model but also helps generating seeds for the large-scale structures we
see today. In the standard scenario, cosmic inflation is driven by the
potential energy of a slow-rolling scalar field, dubbed inflaton. To maintain
the slow-rolling of the field during inflation, the potential of the inflaton
field needs to be sufficiently flat. However, there are regimes when the slow-
rolling conditions cannot remain valid. One such situation occurs when the
potential becomes extremely flat, such as around inflection points. The system
then deviates from the standard slow-rolling and enters a phase of so-called
“Ultra slow-roll” Dimopoulos:2017ged . The term ‘Ultra slow-roll’ was first
coined in Kinney:2005vj , and the dynamics of the system during such a phase
were studied. However, a similar situation was first investigated in
Inoue:2001zt . The attractive behaviour and stability of an Ultra slow-roll
phase were investigated in Pattison:2018bct . Such a departure from slow-roll
often turns out to be useful as such a phase allows the cosmological
perturbations to grow sufficiently enough to generate Primordial Black Holes
Motohashi:2017kbs , a viable candidate for Dark Matter Carr:2016drx .111An
issue regarding whether or not the large-scale cosmological perturbations,
probed by the Cosmic Microwave Background, receive large non-perturbative
corrections from the enhanced perturbations on small scales due to Ultra slow-
roll is being debated in the literature Kristiano:2022maq ; Riotto:2023hoz ;
Kristiano:2023scm ; Riotto:2023gpm ; Franciolini:2023lgy ; Choudhury:2023vuj .
If they do, then such mechanisms of generating Primordial Black Holes will be
ruled out. The issue is yet to be resolved.

In the standard inflationary scenario, the couplings of the inflaton field
with other particles are considered to be negligible. Thus, all other energy
densities present before inflation are diluted away exponentially by the time
inflation ends, and, therefore, to onset the standard hot big bang evolution
post-inflation, a separate phase of reheating Kofman:1994rk  is called for.
This standard inflationary scenario will be referred to as ‘Cold Inflation’
(CI) in this article. However, there is an alternate inflationary scenario,
dubbed ‘Warm Inflation’ (WI) Berera:1995ie  (for recent reviews on WI, see,
e.g., Kamali:2023lzq ; Berera:2023liv ), where those couplings of the inflaton
field to other particles play a significant role during the inflationary
evolution in dissipating inflaton’s energy density into a subdominant, yet
non-negligible, radiation bath. As a constant radiation bath is maintained
throughout WI, this helps WI to transit smoothly into a radiation dominated
universe once WI ends. Thus, WI, unlike CI, does not call for a reheating
phase post inflation. Moreover, despite being sub-dominant, the produced
radiation energy density, $\rho_{r}$, satisfies the condition
$\rho_{r}^{1/4}>H$ (where $H$ is the Hubble parameter during inflation), which
upon assuming thermalization of the radiation bath yields the condition $T>H$
($T$ being the temperature of the thermalized radiation bath). This
thermalization condition is maintained throughout the evolution of WI, and
plays a major role in determining the cosmological perturbations produced
during WI Kamali:2023lzq ; Berera:2023liv . Graceful exit of WI, which is a
much more complex process than in CI, has been extensively studied in
Das:2020lut .

WI has certain attractive features than its more conventional counterpart, CI.
First of all, as mentioned above, WI is not in need of an extra reheating
phase — physics of which is not yet fully understood. Secondly, WI yields a
more enhanced scalar curvature power spectrum compared to CI Hall:2003zp ;
Ramos:2013nsa , which lowers the tensor-to-scalar ratio significantly. This
allows WI to accommodate potentials, such as quartic self-coupling potentials
($\lambda\phi^{4}$), which are otherwise ruled out in CI for generating way
too much tensor-to-scalar ratios Bartrum:2013fia . It has been shown in
Bastero-Gil:2011clw  that the observed baryon asymmetry in the nature can be
explained by the dissipative effects of the WI alone, which is absent in CI.
This also leads to observable baryon isocurvature perturbations which can help
in checking the consistency of the WI models Bastero-Gil:2014oga . Certain WI
models can also generate Primordial Black Holes without invoking any departure
from slow-roll Arya:2019wck ; Bastero-Gil:2021fac ; Correa:2022ngq ;
Arya:2023pod . Moreover, it has been shown recently that, while CI fails to
comply Kinney:2018nny  with the de Sitter Swampland Conjecture in String
Theory proposed in Ooguri:2018wrx ; Garg:2018reu , one can easily overcome the
obstacles in WI due to its very construction and can successfully accommodate
the criteria of the conjecture within the framework of WI Das:2018hqy ;
Motaharfar:2018zyb ; Das:2018rpg ; Das:2019hto ; Das:2019acf ; Das:2020xmh .
Therefore, WI is preferred over CI as an inflationary paradigm in low energy
effective field theories which decent from ultraviolet complete theories of
gravity, such as String Theory.

In this article, we investigate what happens to the dynamics of WI when it
encounters an extremely flat region of a potential, like an inflection point.
We discussed before that, in CI, the system significantly deviates from slow-
rolling and enters a phase of Ultra slow-roll in a similar situation. Thus,
can we expect a similar behaviour in WI as well? We found that though the
background dynamics does show signatures of Ultra slow-roll, the system
deviates from its thermal equilibrium exponentially fast in models of WI where
the dissipative coefficient is a sole function of the temperature of the
radiation bath. As thermal equilibrium of the system is a key feature of WI,
it is not clear what happens to the dynamics of WI if the thermal equilibrium
is lost during any phase of its evolution. Therefore, we could not analyze
such systems any further in this article. However, we showed that WI models
with dissipative coefficients depending on the temperature of the thermal bath
as well as on the amplitude of the inflaton field can successfully realize
brief periods of Ultra slow-roll while maintaining the overall thermalization
of the system.

We have organized the rest of the article as follows. In Sec. (II), we analyze
the Ultra slow-roll dynamics in standard CI and define the criteria which
distinguish Ultra slow-roll from slow-roll in terms of Hubble slow-roll
parameters. In Sec. (III), we briefly discuss the WI dynamics under slow-roll
and show how the thermalization of the system is maintained during the slow-
roll phase. In Sec. (IV) we determine the theoretical criteria which can lead
to an Ultra slow-roll phase in WI while maintaining thermal equilibrium of the
system. This follows by the Sec. (V), where we analyze numerically a specific
WI model (which can accommodate an Ultra slow-roll phase according to the
theoretical criteria developed in Sec. (IV)) with two potentials with
extremely flat regions, and show that brief periods of Ultra slow-roll phases
can indeed be realized while maintaining the thermalization of the system. We
also show that the system smoothly enters a standard slow-roll phase after
these brief periods of Ultra slow-roll. In Sec. (VI), we discuess the main
results obtained in this article and then conclude.

##  II Ultra slow-roll in Cold Inflation

To figure out how WI behaves when the potential becomes extremely flat, we
need to first understand how the dynamics deviates from slow-roll (and enters
an Ultra slow-roll phase) in CI in a similar situation. In this section we
will closely follow the arguments given in Dimopoulos:2017ged , and, in the
following section, we will generalise the arguments presented here for the
case of WI.

In canonical CI models, a single scalar inflaton field, $\phi$, evolves
according to the Klein-Gordon equation given as

$\displaystyle\ddot{\phi}+3H\dot{\phi}+V,_{\phi}=0,$ (1)

where the overdot denotes derivative with respect to the cosmic time $t$ and
$V,_{\phi}=dV/d\phi$. Following Dimopoulos:2017ged , we will call the three
terms in the above equation as the acceleration term, the friction term, and
the slope term, respectively. The Friedmann-Lemaître-Robertson-Walker scale
factor, $a(t)$, evolves according to the Friedmann equations:

$\displaystyle 3M_{\rm Pl}^{2}H^{2}$ $\displaystyle=$
$\displaystyle\frac{\dot{\phi}^{2}}{2}+V(\phi),$ (2) $\displaystyle 2M_{\rm
Pl}^{2}\dot{H}$ $\displaystyle=$ $\displaystyle-\dot{\phi}^{2},$ (3)

where $H\equiv\dot{a}/a$ is the Hubble parameter and $M_{\rm Pl}$ is the
reduced Planck mass. The background evolution is characterized in terms of the
Hubble slow-roll parameters defined by Martin:2012pe :

$\displaystyle\epsilon_{i+1}=\frac{d\ln\epsilon_{i}}{dN},$ (4)

where $N\equiv\ln a$ denotes the number of e-folds. Starting with
$\epsilon_{0}\propto 1/H$, the consecutive first slow-roll parameter
$\epsilon_{1}$ and the second slow-roll parameter $\epsilon_{2}$ can be
expressed as

$\displaystyle\epsilon_{1}$ $\displaystyle\equiv$
$\displaystyle-\frac{\dot{H}}{H^{2}},$ (5) $\displaystyle\epsilon_{2}$
$\displaystyle\equiv$
$\displaystyle\frac{\dot{\epsilon}_{1}}{\epsilon_{1}H}=2\epsilon_{1}+\frac{\ddot{H}}{H\dot{H}}.$
(6)

Inflation requires $\epsilon_{1}<1$, and the validity of slow-roll
approximation is ensured by the general conditions $|\epsilon_{i}|\ll 1$.
During inflation the potential term dominates over the kinetic term. Thus,
from the Friedmann equations given in Eq. (2) and Eq. (3, we see that
$\epsilon_{1}\sim 3\dot{\phi}^{2}/(2V(\phi))$, ensuring $\epsilon_{1}\ll 1$
during inflation. And during slow-roll, the acceleration term ($\ddot{\phi}$)
is negligible compared to the friction ($H\dot{\phi}$) and the slope
($V,_{\phi}$) terms in the Klein-Gordan equation given in Eq. (1), which then
can be approximated as

$\displaystyle 3H\dot{\phi}+V,_{\phi}\simeq 0.$ (7)

This suggests that

$\displaystyle\frac{\ddot{H}}{H\dot{H}}=\frac{2\ddot{\phi}}{H\dot{\phi}}\simeq
2\epsilon_{1}-2\eta_{V},$ (8)

where $\eta_{V}$ is one of the two potential slow-roll parameters:

$\displaystyle\epsilon_{V}\equiv\frac{M_{\rm
Pl}^{2}}{2}\left(\frac{V,_{\phi}}{V}\right)^{2},\quad\quad\eta_{V}\equiv
M_{\rm Pl}^{2}\frac{V,_{\phi\phi}}{V}.$ (9)

Both these potential slow-roll parameters quantify the flatness of the
inflaton potential during inflation. As the potential requires to be nearly
flat for the slow-rolling of the field, the potential slow-roll parameters
require to be much smaller than unity during slow-roll inflation. Therefore,
from Eq. (6) we see that $|\epsilon_{2}|\sim|4\epsilon_{1}-2\eta_{V}|\ll 1$.
Thus, both $\epsilon_{1}$ and $|\epsilon_{2}|$, being much smaller than unity,
ensure the slow-roll dynamics of the inflaton field.

However, when the potential becomes extremely flat, the slope term in the
Klein-Gordan equation becomes negligible, and it becomes

$\displaystyle\ddot{\phi}+3H\dot{\phi}\simeq 0,$ (10)

yielding $\epsilon_{2}\sim-6+2\epsilon_{1}$. We note that $\epsilon_{1}$
remains much smaller than unity even when the potential becomes extremely
flat, as the potential term dominates over the kinetic term. Therefore,
inflation does not stop when the potential becomes extremely flat. However,
$\epsilon_{2}$ becomes of the order of unity ($\epsilon_{2}\sim-6$). This
clearly indicates that the scalar field dynamics deviates from slow-roll when
the potential becomes extremely flat, and enters a new phase of evolution,
dubbed the Ultra slow-roll.

##  III Warm Inflation: The slow-roll regime

During WI, the inflaton field, $\phi$, dissipates its energy to a radiation
bath, maintaining a non-negligible radiation energy density, $\rho_{r}$,
throughout. This feature distinguishes WI from the standard CI scenario.
Therefore, the equation of motion of the inflaton field also differs from that
of in the CI scenario. The equations governing the dynamics of the inflaton
field, $\phi$, and the radiation bath, $\rho_{r}$, in WI can be written as

$\displaystyle\ddot{\phi}+3H\dot{\phi}+V,_{\phi}=-\Upsilon(\phi,T)\dot{\phi},$
(11) $\displaystyle\dot{\rho}_{r}+4H\rho_{r}=\Upsilon(\phi,T)\dot{\phi}^{2}.$
(12)

Here, $\Upsilon$ is the dissipative term which can depend on the amplitude of
the inflaton field, $\phi$, as well as the temperature of the radiation bath,
$T$. It is assumed that the radiation bath, generated by the dissipation of
the inflaton field, is in near thermal equilibrium throughout WI, and thus a
temperature $T$ can be defined. Many microphysical models of Warm Inflation
have been studied over the years and the decay rates (under the assumption of
thermal equilibrium) have been calculated. For such studies one can look into
Bastero-Gil:2010dgy ; Berera:2008ar  and Kamali:2023lzq  and for a more recent
review. Form these studies, one can see that a general form of the dissipative
coefficient in Warm Inflation can be written in the form

$\displaystyle\Upsilon(\phi,T)=C_{\Upsilon}T^{p}\phi^{c}M^{1-p-c},$ (13)

where $C_{\Upsilon}$ is a dimensionless constant carrying the signatures of
the microscopic model used to derive the dissipative coefficient (such as the
different coupling constants), and $M$ is some appropriate mass-scale, so that
the dimensionality of the dissipative coefficient is preserved,
$[\Upsilon]=$[mass] in our system of units where $\hbar=c=1$. The numerical
powers of $T$ and $\phi$, which are $p$ and $c$ respectively, can take
positive or negative values, however, we will restrict ourselves to $|p|<4$
due to the stability of the WI models Moss:2008yb ; Bastero-Gil:2012vuu .

We define the dimensionless parameter $Q$ as

$\displaystyle Q\equiv\frac{\Upsilon}{3H},$ (14)

which is the ratio of the two friction terms appearing in the equation of
motion of the inflation field, one due to dissipation ($\Upsilon\dot{\phi}$)
and the other due to the Hubble expansion ($3H\dot{\phi}$). If the Hubble
friction dominates over the friction due to dissipation, i.e. $Q<1$, we call
it a weak dissipative regime of WI. On the other hand, when the friction due
to dissipation dominates the equation of motion of the inflaton field ($Q>1$
in such cases), we call it a strong dissipative regime of WI.

Apart from the two potential slow-roll parameters defined in Eq. (9), there
are two additional slow-roll parameters in WI Hall:2003zp ; Berera:2008ar :

$\displaystyle\beta\equiv M_{P}^{2}\frac{\Upsilon_{,\phi}V_{,\phi}}{\Upsilon
V},\quad\quad\delta\equiv\frac{TV_{,\phi T}}{V_{,\phi}},$ (15)

which are required to ensure the slow-roll dynamics. During slow-roll
$\epsilon_{V}$, $\eta_{V}$, $\beta$ and $\delta$ are all smaller than $1+Q$ in
both weak and strong dissipative regime.

In the slow-roll regime, i.e. when the acceleration term in Eq. (11) is
negligible with respect to all the other terms, we can approximate this
equation as

$\displaystyle 3H(1+Q)\dot{\phi}+V,_{\phi}\approx 0.$ (16)

However, as the inflaton potential energy density always dominates over the
kinetic term and the radiation energy density during WI, the Friedmann
equation given in Eq. (2) can be written as

$\displaystyle 3M_{\rm Pl}^{2}H^{2}\approx V(\phi).$ (17)

As during WI a constant radiation bath is maintained by the dissipation of
energy of the inflaton field into the radiation bath, we can assume
$\dot{\rho}_{r}\approx 0$, and can approximate Eq. (12) as

$\displaystyle 4H\rho_{r}\approx\Upsilon\dot{\phi}^{2}.$ (18)

Also, as the radiation bath is near equilibrium, we can define a temperature
$T$ of the radiation bath as

$\displaystyle\rho_{r}=\frac{\pi^{2}}{30}g_{*}T^{4},$ (19)

where $g_{*}$ is the relativistic degrees of freedom of the radiation bath. We
can then determine the evolution of $T$ with respect to the e-foldings $N$
during slow-roll as

$\displaystyle\frac{d\ln
T}{dN}=\frac{\dot{T}}{HT}\simeq\frac{1}{4-p}\left(\frac{\epsilon_{V}}{1+Q}-\frac{\beta}{1+Q}\right)\ll
1.$ (20)

Here we have used the relation $dN=Hdt$. The above equation ensures that
during slow-roll, the temperature of the radiation bath evolves very slowly,
maintaining the near equllibrium condition.

With all these conditions of slow-roll WI, we find that during a slow-roll
phase in WI

$\displaystyle\epsilon_{1}$ $\displaystyle=$
$\displaystyle\frac{\epsilon_{V}}{1+Q},$ (21) $\displaystyle\epsilon_{2}$
$\displaystyle=$
$\displaystyle-2\frac{\eta_{V}}{1+Q}+\left(\frac{4+3Q}{1+Q}-\frac{p}{4-p}\frac{Q}{1+Q}\right)\frac{\epsilon_{V}}{1+Q}$
(22) $\displaystyle+\left(\frac{4}{4-p}\frac{Q}{1+Q}\right)\frac{\beta}{1+Q}.$

In deriving these relations we have used the general form of the dissipative
coefficient given in Eq. (13). As $\epsilon_{V}$, $\eta_{V}$ and $\beta$ are
all smaller than $1+Q$ during slow-roll we see that $\epsilon_{1}$ and
$|\epsilon_{2}|$ are both smaller than unity during slow-roll. In the next
section we will see that this situation will change when the potential becomes
extremely flat, as it happens in CI.

##  IV Realising Ultra slow-roll within Warm Inflation

We note in Sec. (II) that in CI, the evolution enters an Ultra slow-roll phase
when the potential becomes extremely flat, and we quantify it by showing that
the magnitude of the second Hubble slow-roll parameter, $\epsilon_{2}$,
becomes of the order of unity, while the first Hubble slow-roll parameter,
$\epsilon_{1}$, remains much smaller than 1. Does a similar situation arise in
WI when the potential becomes extremely flat? While answering this question,
one also needs to keep in mind that WI maintains a radiation bath in near
equilibrium throughout the slow-roll evolution. Can WI enter an Ultra slow-
roll phase while maintaining the thermal equilibrium of the radiation bath
present during the evolution?

To answer these questions, we first determine the evolution of the temperature
$T$ for a phase when the potential becomes extremely flat. During such an
evolution, as $V,_{\phi}\approx 0$, we can approximate Eq.(11) as

$\displaystyle\ddot{\phi}+3H(1+Q)\dot{\phi}\approx 0.$ (23)

It is to note that an Ultra slow-roll phase during an inflationary epoch, both
in CI and WI, must be a transient phase, i.e. the Ultra slow-roll phase must
be preceded by a slow-roll phase and ends into another slow-rolling phase.
Thus the initial conditions of the Ultra slow-roll phase is set by the
preceding slow-roll phase. In WI, a thermal equilibrium is assumed in the
slow-rolling phase preceding the Ultra slow-roll phase. Therefore, the
thermalization condition and the form of the dissipative coefficient during
Ultra slow-roll phase are set by the preceding slow-roll phase. With these
initial conditions, we can then let the system evolve during an Ultra slow-
roll phase during WI. Hence, we will assume that at the onset of such an
evolution a constant radiation bath is maintained, and that radiation bath
will remain near thermal equilibrium so that a temperature $T$ can be defined.
In such a case, both Eq. (18) and Eq. (19) would remain valid. From Eq.(23),
Eq. (18) and Eq. (19) it is straightforward to show that the temparature $T$
evolves as

$\displaystyle\frac{d\ln
T}{dN}=\frac{1}{4-p}\left[c\frac{\dot{\phi}}{H\phi}+\epsilon_{1}-6(1+Q)\right]\,,$
(24)

when the potential becomes extremely flat. We can see here that the first and
the last terms on the right hand side of the above eqution are not
proportional to any slow-roll parameters and can take large values. Thus,
comparing this equation with Eq. (20), which depicts the evolution of $T$
during slow-roll in WI, we see that one cannot conclude right away that the
temperature will evolve slowly during the evolution through an extremely flat
region of the potential. Can we make the temperature evolve slowly in such a
case so that the system can evolve being near equilibrium? We will address
this question by the end of this section.

We will now see what happens to the first Hubble slow-roll parameter,
$\epsilon_{1}$, when the potential becomes extremely flat. We note that, in
WI, $-2M_{\rm
Pl}^{2}\dot{H}=\rho_{\phi}+P_{\phi}+\rho_{r}+P_{r}=\dot{\phi}^{2}+(3/4)\rho_{r},$
where $\rho_{\phi}$ and $P_{\phi}$ are the energy density and the pressure of
the inflaton field, and similarly, $\rho_{r}$ and $P_{r}$ are the radiation
energy density and radiation pressure, respectively. However, using Eq. (18),
we can write

$\displaystyle-2M_{\rm Pl}^{2}\dot{H}=(1+Q)\dot{\phi}^{2}.$ (25)

Thus, using the above equation and Eq. (17), the first Hubble slow-roll
parameter can be expressed as

$\displaystyle\epsilon_{1}\sim\frac{3}{2}\frac{(1+Q)\dot{\phi}^{2}}{V(\phi)}.$
(26)

Now, as during inflation the potential energy density dominates over kinetic
energy density, we can see that in the weak dissipative regime ($Q<1$)
$\epsilon_{1}$ can remain much smaller than unity ensuring inflation to
continue even when the potential becomes extremely flat. However, in strong
dissipation regime, there are models where the $Q$ values can be as large as
$\mathcal{O}(10^{3})$ Berghaus:2019whh ; Das:2019acf ; Das:2020xmh . In such
models, it might happen that when the potential becomes extremely flat,
$Q\dot{\phi}^{2}\sim V(\phi)$, yielding $\epsilon_{1}\sim 1$, which is an
indication of end of inflation. But, there are WI models realised in strong
dissipation Bastero-Gil:2019gao  where $Q$ is of the order of 10 or 100. In
such models, inflation continues to take place even when the potential becomes
extremely flat.

Now, to determine the second Hubble parameter, $\epsilon_{2}$, during such an
evolution, we use Eq. (23), Eq. (24) and Eq. (25) to show that

$\displaystyle\frac{\ddot{H}}{\dot{H}H}$ $\displaystyle=$
$\displaystyle-6(1+Q)+\frac{4}{4-p}\frac{\Upsilon_{,\phi}\dot{\phi}}{3H^{2}(1+Q)}$
(27)
$\displaystyle+\left[\frac{1}{4-p}\frac{\Upsilon_{,T}T}{\Upsilon}+1\right]\frac{Q}{1+Q}\epsilon_{1}-\dfrac{6Qp}{4-p}\,.$

Using this relation into Eq. (6), we see that

$\displaystyle\epsilon_{2}=-6\left(1+\frac{4Q}{4-p}\right)+\frac{4}{4-p}\left[c\frac{\dot{\phi}}{H\phi}+\epsilon_{1}\right]\frac{Q}{1+Q}+2\epsilon_{1},$
(28)

where we have again used the general form of the dissipative coefficient given
in Eq. (13). Let us appraise the situation in two different regimes, strong
dissipation and weak dissipation, separately.

In strong dissipative regime ($Q\gg 1$), Eq. (28) can be approximated as

$\displaystyle\epsilon_{2}\sim\frac{4}{4-p}\left[-6Q+c\frac{\dot{\phi}}{H\phi}+\epsilon_{1}\right]+2\epsilon_{1}\,.$
(29)

There are couple of models where WI can be realised in strong dissipative
regime: one is presented in Bastero-Gil:2019gao , and the other is presented
in Berghaus:2019whh  (this model is dubbed Minimal Warm Inflation). In the
strong dissipative WI model presented in Bastero-Gil:2019gao , the dissipative
coefficient, despite having a complex form, varies inversely with the
temperature ($\Upsilon\propto T^{-1}$) during the evolution, and does not
depend on the inflaton amplitude. Therefore, for this kind of model $p=-1$ and
$c=0$. This yields $\epsilon_{2}\sim(-24/5)Q+(14/5)\epsilon_{1}$. We have
mentioned earlier that in this model, $Q$ can be of the order of 10 or 100
(yielding $\epsilon_{1}\ll 1$), and thus the model can enter an Ultra slow-
roll phase while inflating. However, looking at Eq. (24), we note that $d\ln
T/dN\sim-(6/5)Q$, and therefore, the temperature decreases exponentially with
the e-foldings during the Ultra slow-roll phase. It indicates that the thermal
equilibrium of the radiation bath cannot be maintained during Ultra slow-roll
in such models, and the system will deviate from the basic WI picture. In the
Minimal Warm Inflation model Berghaus:2019whh , the dissipative coefficient
varies with the cubic power of temperature ($\Upsilon\propto T^{3}$) and does
not depend on the inflaton amplitude. Therefore in this model $p=3$ and $c=0$,
yielding $\epsilon_{2}\sim-24Q+6\epsilon_{1}$. As mentioned earlier, $Q$ can
take very large values ($\mathcal{O}(10^{3})$) in such models. Despite that,
if $Q\phi^{2}\ll V(\phi)$, then $\epsilon_{1}\ll 1$ and we get
$\epsilon_{2}\sim-24Q$, indicating an Ultra slow-roll phase. Even though, like
in the previous case, we note that $d\ln T/dN\sim-6Q$, and the temperature
will exponentially fall with e-foldings taking the system away from thermal
equilibrium, and deviating from the standard WI picture.

In weak dissipative regime $(Q\ll 1)$, Eq. (28) can be approximated as

$\displaystyle\epsilon_{2}=-6+\frac{4}{4-p}\left[c\frac{\dot{\phi}}{H\phi}+\epsilon_{1}\right]Q+2\epsilon_{1}\,.$
(30)

We will deal with two kinds of models where WI is realised in weak dissipative
regime. The first one is dubbed Warm Little Inflaton Bastero-Gil:2016qru ,
where the dissipative coefficient varies linearly with the temperature
($\Upsilon\propto T$) and does not depend on the inflaton amplitude.
Therefore, in this case, $p=1$ and $c=0$, yielding
$\epsilon_{2}=-6+(10/3)\epsilon_{1}$. As we have seen before, $\epsilon_{1}\ll
1$ when the potential becomes extremely flat during weak dissipation, and
therefore, we conclude that during such a phase $\epsilon_{2}\sim-6$. Though
it indicates an onset of an Ultra slow-roll phase, we again note that, as in
the strong dissipative cases discussed above, Eq. (24) yields $d\ln
T/dN\sim-2$, indicating departure from thermal equilibrium. So far, we have
discussed WI models where the dissipative coefficient varies solely with $T$,
both in strong and weak dissipative regimes, and saw that the system veers
away from the thermal equilibrium state when an Ultra slow-roll like phase
sets in. As it is not yet known how to treat WI when the thermal equilibrium
of the radiation bath is lost, we will not further analyze the Ultra slow-roll
phase in WI models where the dissipative coefficient solely varies with the
temperature and has no dependence on the field amplitude.

We will now analyze a WI model where the dissipative coefficient is a function
of both $T$ and $\phi$. The model studied in Berera:2008ar ; Bastero-
Gil:2010dgy ; Bastero-Gil:2012akf  has a dissipative coefficient of the form
$\Upsilon\propto T^{3}/\phi^{2}$, and WI is realised in weak dissipative
regime in such a model. This model has also been verified with the Planck data
in Benetti:2016jhf ; Arya:2017zlb . Being realized in weak dissipative regime,
we can guarantee that $\epsilon_{1}$ will remain much smaller than unity even
when the potential becomes extremely flat. Our aim, now, is to keep $|d\ln
T/dN|\approx 0$ while making $|\epsilon_{2}|$ larger than unity, so that an
Ultra slow-roll phase can be realized while maintaining the thermal
equilibrium of the system. Therefore, with $\epsilon_{1}\ll 1$ and $Q\ll 1$,
Eq. (24) can be approximated as

$\displaystyle\frac{d\ln
T}{dN}\approx\frac{1}{4-p}\left(c\frac{d\ln\phi}{dN}-6\right).$ (31)

We note from this equation that, to keep $|d\ln T/dN|\approx 0$, $d\ln\phi/dN$
should be positive if $c$ is positive and vice versa. In other words,
$|c||d\ln\phi/dN|$ should always remain positive. Hence, to maintain a thermal
equilibrium we demand that during an Ultra slow-roll phase

$\displaystyle\left|\frac{d\ln\phi}{dN}\right|\sim\frac{6}{|c|}.$ (32)

If we impose this condition on $\epsilon_{2}$ given in Eq. (30) with other
conditions, like $\epsilon_{1}\ll 1$ and $Q\ll 1$, we obtain

$\displaystyle\epsilon_{2}\approx-6+\left(\frac{4}{4-p}\right)6Q.$ (33)

We can keep $|\epsilon_{2}|>1$ in two ways: keeping $\epsilon_{2}>1$ or
demanding $\epsilon_{2}<-1$. In the first case, when $\epsilon_{2}>1$, we get

$\displaystyle Q>\frac{7}{6}\left(\frac{4-p}{4}\right).$ (34)

Therefore, to ensure that WI takes place in weak dissipative regime we
constrain $Q$ as $(7/6)[(4-p)/4]<Q<1$. For the model discussed above
Berera:2008ar ; Bastero-Gil:2010dgy ; Bastero-Gil:2012akf , this condition
leads to $(7/24)<Q<1$. However, in the second case, when $\epsilon_{2}<-1$, we
get

$\displaystyle Q<\frac{5}{6}\left(\frac{4-p}{4}\right).$ (35)

This condition ensures that WI will take place in weak dissipative regime. For
the model discussed above Berera:2008ar ; Bastero-Gil:2010dgy ; Bastero-
Gil:2012akf , this condition leads to $Q<5/24$.

However, we note that though the condition given in Eq. (32) allows the onset
of Ultra slow-roll in WI, while maintaining thermal equilibrium, still Eq.
(32) differs from the equation of motion of the inflaton field during Ultra
slow-roll given in Eq. (23). Therefore, as Ultra slow-roll proceeds, the
dynamics of the inflaton field will take the system away from the condition in
Eq. (32), and the temperature of the system will start to evolve indicating a
departure from thermal equilibrium. Thus, the system needs to exit from Ultra
slow-roll before the temperature evolves too much to disrupt thermal
equilibrium. We will show in the next section, by numerically evolving the
system, that such a thermally equilibrated Ultra slow-roll phase can be
realised in WI in specific cases.

##  V Numerical analysis of viable Ultra slow-roll phase in WI

In WI, the inflaton field equation of motion and the evolution of the
radiation bath (as well as the evolution of the temperature) are coupled as
the inflaton field dissipates its energy to the radiation bath throughout WI.
The system thus evolves according to the coupled equations given in Eq. (11)
and Eq. (12). We will numerically evolve these two equations in cases where WI
can enter an Ultra slow-roll phase, and then appraise its characteristics. We
will consider two different potentials, the linear potential
Dimopoulos:2017ged , and the cubic potential Pattison:2018bct , where it has
been shown previously that the system undergoes Ultra slow-roll in CI
scenarios. In both these cases, we will make use of the dissipative
coefficient $\Upsilon=C_{\Upsilon}T^{3}/\phi^{2}$ and will ensure that WI
takes place in weak dissipative regime.

###  V.1 Linear Potentail: $V(\phi)=V_{0}+M_{0}^{3}\phi$

This potential has been considered in Dimopoulos:2017ged  to analyze the
dynamics of Ultra slow-roll in CI. In this potential, when $V_{0}\gg
M_{0}^{3}\phi$, the potential becomes extremely flat, and the system can enter
an Ultra slow-roll phase. We chose the parameters $V_{0}$ and $M_{0}$
accordingly, which we have quoted in the caption of Fig. (1). We see from Fig.
(1) that initially the acceleration term ($\ddot{\phi}$) and the friction term
($3H(1+Q)\dot{\phi}$) dominate over the slope term $(V,_{\phi})$ (the graph of
$\ddot{\phi}$ and $3H(1+Q)\dot{\phi}$ overlaps for the first few e-folds),
indicating an Ultra slow-roll phase. Around 3.5 e-folds the acceleration term
becomes subdominant, and the dynamics is governed by the friction and the
slope terms, indicating an onset of the usual slow-roll phase.

Figure 1: The figure depicts the numerical evolution of the acceleration term
($\ddot{\phi}$), the friction term ($3H(1+Q)\dot{\phi}$) and the slope term
($V,_{\phi}$) present in the equation of motion of the inflaton field through
an Ultra slow-roll phase in the case of linear potentail. We have chosen the
parameters as follows: $V_{0}=(10^{-4}\,M_{\rm Pl})^{4}$, $M_{0}=2.5\times
10^{-8}\,M_{\rm Pl}$, $C_{\Upsilon}=10$ and $g_{*}=106.75$.

In Fig. (2), we note that the thermalization condition during Ultra slow-roll
given in Eq. (32) is maintained for about the first 1.5 e-folds. However, as
the condition for thermalization differs from the equation of the inflaton
field during Ultra slow-roll (Eq. (23)), the thermalization condition cannot
be maintained for a long period and the system will veer off from thermal
equilibrium, as has been pointed out in the previous section. However, the
overall thermalization condition of WI, $T>H$, will remain maintained
throughout the Ultra slow-roll phase, as has been depicted in Fig. (3).

Figure 2: Evolution of the thermalization condition given in Eq. (32) during
Ultra slow-roll in the case of linear potential. Figure 3: Evolution of the
temperature and the Hubble parameter during Ultra slow-roll in the case of
linear potential.

Fig. (4) depicts the evolution of the second Hubble parameter, $\epsilon_{2}$,
during this Ultra slow-roll evolution. We note that the Ultra slow-roll
condition, $\epsilon_{2}<-1$, is maintained till about 4 e-foldings and after
that $|\epsilon_{2}|$ becomes smaller than unity, indicating onset of slow-
roll phase. Also, Fig. (5), where the evolution of $Q$ is shown during Ultra
slow-roll, ensures that WI takes place in weak dissipative regime during this
Ultra slow-roll phase.

Hence, we can see that in such a setup an Ultra slow-roll phase can be
embedded in a weak dissipative WI model while maintaining thermal equilibrium.

Figure 4: Evolution of the second Hubble slow-roll parameter, $\epsilon_{2}$,
during Ultra slow-roll in the case of linear potential. Figure 5: Evolution of
$Q$ during Ultra slow-roll in the case of linear potentail.

###  V.2 Cubic Potential:
$V(\phi)=V_{0}+\left[1+\left(\frac{\phi}{\phi_{0}}\right)^{3}\right]$

The cubic potential was considered in Pattison:2018bct  while discussing the
attractor behaviour of Ultra slow-roll in CI. This potential has an inflection
point around $\phi=0$. In WI, if we start near this inflection point region
then the system can undergo an Ultra slow-roll phase. It is to note that we
cannot analyze the system at the inflection point ($\phi=0$), as the
dissipative coefficient ($\Upsilon\propto T^{3}/\phi^{2}$) will be ill-defined
at this point. We can only analyze the system near about the inflection point.

We first note in Fig. (6), that in this case the system quickly deviates from
the thermalization condition given in Eq. (32) as soon as Ultra slow-roll
begins. This indicates that the system should not linger in an Ultra slow-roll
phase for a long time as that will lead to disruption of thermal equilibrium
of the system. In Fig. (7), we notice that the coupled equations of the system
do not let the Ultra slow-roll phase to last for long, and within nearly 1.5
e-folds the system tends to enter a slow-roll phase. We also note in Fig. (8)
that the overall thermalization condition of WI ($T>H$) is maintained
throughout the Ultra slow-roll phase.

Figure 6: Evolution of the thermalization condition given in Eq. (32) during
Ultra slow-roll in the case of cubic potential. Figure 7: The figure depicts
the numerical evolution of the acceleration term ($\ddot{\phi}$), the friction
term ($3H(1+Q)\dot{\phi}$) and the slope term ($V,_{\phi}$) present in the
equation of motion of the inflaton field through an Ultra slow-roll phase in
the case of cubic potential. We have chosen the parameters as follows:
$V_{0}=(10^{-4}\,M_{\rm Pl})^{4}$, $\phi_{0}=2.5\times 10^{-1}\,M_{\rm Pl}$,
$C_{\Upsilon}=10^{4}$ and $g_{*}=106.75$. Figure 8: Evolution of the
temperature and the Hubble parameter during Ultra slow-roll in the case of
cubic potential.

In Fig. (9) we see that the second Hubble slow-roll, $\epsilon_{2}$, will
remain larger than -1 for about 2.5 e-foldings and then we achieve
$|\epsilon_{2}|<1$ indicating an onset of an usual slow-roll phase. Also, Fig.
(10) ensures that the whole dynamics takes place in a weak dissipative regime.

Figure 9: Evolution of the second Hubble slow-roll parameter, $\epsilon_{2}$,
during Ultra slow-roll in the case of cubic potential. Figure 10: Evolution of
$Q$ during Ultra slow-roll in the case of cubic potential.

Therefore, in this system, too, an Ultra slow-roll phase, though much shorter
than the previous scenario, can be realized near an inflection point within a
weak dissipative WI model while maintaining the overall thermal equilibrium of
the system.

##  VI Discussion and Conclusion

CI undergoes an Ultra slow-roll phase when the potential becomes extremely
flat Dimopoulos:2017ged . In this manuscript we ask the question what happens
to WI, a variant inflationary scenario, in a similar situation. As has been
pointed out in the Introduction, a constant, subdominant, nearly thermally
equilibrated radiation bath coexists during WI due to the continuous
dissipation of energy of the inflaton field into this radiation bath. The
coexistence of this radiation bath along with inflaton energy densitiy is the
signature of WI which distinguishes it from the standard CI scenario.
Therefore, one naturally expects the radiation bath, along with the thermal
equilibrium of the system, to be maintained in WI even when the system passes
through an extremely flat region of the potential.

However, we found in this article that WI models with dissipative coefficients
solely dependent on the temperature ($\Upsilon\propto T^{p}$) fail to maintain
thermal equilibrium of the system when the system traverses through a very
flat region of the potential. It is not yet known what happens to the WI
dynamics when the thermal equilibrium is lost. We, therefore, could not
further analyze these systems in such circumstances.

On the other hand, we showed that the overall thermal equilibrium of the
system can be maintained throughout a phase when the WI system traverses
through an extremely flat region of the potential in cases where the
dissipative coefficients are function of both the temperature and the inflaton
amplitude. We particularly dealt with the models with dissipative coefficient
of the form $\Upsilon\propto T^{3}/\phi^{2}$. Such models have been shown to
tally well the observations when WI takes place in weak dissipative regime
Benetti:2016jhf ; Arya:2017zlb . Therefore, we treated such models in the weak
dissipative regime, and considered two potentials (linear and cubic) with
extremely flat regions to demonstrate that an Ultra slow-roll phase can indeed
be realised in WI while maintaining the overall thermal equilibrium of the
system.

Though it seems like a positive note to conclude the article, we would like to
call attention to our incapability and discomfort to deal with WI systems with
dissipative coefficients depending on the temperature alone when such systems
encounter an extremely flat region of the potential. We have observed that
these systems lose thermal equilibrium, a signature property of WI. We hope
that this article will encourage more research in this field to reveal the
true nature of the WI dynamics in such circumstances. We also leave the
analysis of the cosmological perturbations during the Ultra slow-roll phase in
WI for a future project, which is essential to determine the observational
consequences of such a departure from slow-roll during WI. Recently, a
numerical code has been developed Montefalcone:2023pvh  to study the
cosmological perturbations in standard slow-roll WI models. We aim to develop
similar codes for an Ultra slow-roll phase in WI in a future project.

###### Acknowledgements.

SD would like to thank Rudnei Ramos for many useful discussions on Warm
Inflation from time to time.

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