# Issues in Palatini ${\cal{R}}^{2}$ inflation : Bounds on the Reheating
Temperature

A. B. Lahanasa<EMAIL_ADDRESS>a National and Kapodistrian University of
Athens, Department of Physics, Nuclear and Particle Physics Section, GR–157 71
Athens, Greece

###### Abstract

We consider ${\cal{R}}^{2}$-inflation in Palatini gravity, in the presence of
scalar fields coupled to gravity. These theories, in the Einstein frame, and
for one scalar field $h$, share common features with $K$ \- inflation models.
We apply this formalism for the study of single-field inflationary models,
whose potentials are monomials, $V\sim h^{n}$, with $n$ a positive even
integer. We also study the Higgs model non-minimally coupled to gravity. With
${\cal{R}}^{2}$-terms coupled to gravity as $\sim\alpha{\cal{R}}^{2}$, with
$\alpha$ constant, the instantaneous reheating temperature $T_{ins}$, is
bounded by $T_{ins}\leq{0.290\,m_{Planck}}/{\,\alpha^{1/4}}$, with the upper
bound being saturated for large $\alpha$. For such large $\alpha$ need go
beyond slow-roll to calculate reliably the cosmological parameters, among
these the end of inflation through which $T_{ins}$ is determined. In fact, as
inflaton rolls towards the end of inflation point, the quartic in the velocity
terms, unavoidable in Palatini gravity, play a significant role and can not be
ignored. The values of $\alpha$, and other parameters, are constrained by
cosmological data, setting bounds on the inflationary scale $M_{s}\sim
1/\sqrt{\alpha}$ and the reheating temperature of the Universe.

Keywords: Modified Theories of Gravity, Inflationary Universe

PACS: 04.50.Kd, 98.80.Cq

##  I Introduction

The Palatini formulation of General Relativity (GR), or first-order formalism,
is an alternative to the well-known metric formulation, or second-order
formalism. In the latter the space time connection is determined by the metric
while in the Palatini approach the connection $\Gamma^{\mu}_{\lambda\sigma}$
is treated as an independent variable Sotiriou:2006hs ; Sotiriou:2006qn ;
Sotiriou:2008rp ; Borunda:2008kf ; DeFelice:2010aj ; Olmo:2011uz ;
Capozziello:2011et ; Clifton:2011jh ; Nojiri:2017ncd . It is through the
equations of motion that $\Gamma^{\mu}_{\lambda\sigma}$ receive the well known
form of the Christoffel symbols, describing thus a metric connection. Within
the context of GR the two formulations are equivalent. However in the presence
of fields that are coupled in a non-minimal manner to gravity this no longer
holds Sotiriou:2006hs ; Sotiriou:2006qn ; Sotiriou:2008rp , and the two
formulations describe different physical theories.

Encompassing the popular inflation models into Palatini Gravity, in an effort
to describe the cosmological evolution of the Universe, leads to different
cosmological predictions, from the metric formulation, due to the fact that
the dynamics of the two approaches differ. A notable example is the
Starobinsky model, for instance, where except the graviton there exists an
additional propagating scalar degree of freedom, the scalaron, whose mass is
related to the coupling of the ${\cal{R}}^{2}$ term. In the Einstein frame
this emerges as a dynamical scalar field, the inflaton, moving under the
influence of the celebrated Starobinsky potential, Starobinsky:1980te ;
Mukhanov:1981xt ; Starobinsky:1983zz . Within the framework of the Palatini
Gravity, in any $f({\cal{R}})$ theory Sotiriou:2008rp , there are no extra
propagating degrees of freedom, that can play the role of the inflaton, and
hence the inflaton has to be put in by hand as an additional scalar degree of
freedom.

The differences between metric and Palatini formulation in the cosmological
predictions, as far as inflation is concerned, arise from the non-minimal
couplings of the scalars, that take-up the role of the inflaton. These
couplings are different in the two approaches. This has been first pointed out
in Bauer:2008zj  and has attracted the interest of many authors since,
Koivisto:2005yc ; Tamanini:2010uq ; Bauer:2010jg ; Enqvist:2011qm ;
Borowiec:2011wd ; Stachowski:2016zio ; Fu:2017iqg ; Rasanen:2017ivk ;
Tenkanen:2017jih ; Racioppi:2017spw ; Markkanen:2017tun ; Jarv:2017azx ;
Rasanen:2018ihz ; Racioppi:2018zoy ; Carrilho:2018ffi ; Enckell:2018kkc ;
Bombacigno:2018tyw ; Enckell:2018hmo ; Antoniadis:2018ywb ; Antoniadis:2018yfq
; Rasanen:2018fom ; Almeida:2018oid ; Takahashi:2018brt ; Kannike:2018zwn ;
Tenkanen:2019jiq ; Shimada:2018lnm ; Wu:2018idg ; Kozak:2018vlp ;
Jinno:2018jei ; Edery:2019txq ; Rubio:2019ypq ; Jinno:2019und ;
Giovannini:2019mgk ; Tenkanen:2019xzn ; Bostan:2019uvv ; Bostan:2019wsd ;
Tenkanen:2020xbb ; Gialamas:2019nly ; Racioppi:2019jsp , with still continuing
activity, Tenkanen:2020dge ; Lloyd-Stubbs:2020pvx ; Tenkanen:2020cvw ;
Das:2020kff ; McDonald:2020lpz ; Shaposhnikov:2020fdv ; Bekov:2020dww ;
Enckell:2020lvn ; Lykkas:2021vax ; Jarv:2020qqm ; Karam:2021wzz ;
Gialamas:2020snr ; Karam:2020rpa ; Gialamas:2020vto ; Karam:2021sno ;
Gialamas:2021enw ; Gialamas:2021rpr ; Annala:2021zdt ; Racioppi:2021ynx ;
Giovannini:2021due ; Cheong:2021kyc ; Mikura:2021clt ; Dioguardi:2021fmr ;
Ito:2021ssc ; Racioppi:2021jai ; Rigouzzo:2022yan ; Dimopoulos:2020pas ;
Dimopoulos:2022tvn ; Dux:2022kuk ; Dimopoulos:2022rdp ; He:2022xef .

Measurements of the cosmological parameters, by various collaborations, has
tighten the allowed window of these observables which in turn constrain, or
even exclude, particular inflationary models, Planck:2018jri ; Planck:2018vyg
; Ade:2018gkx ; BICEP:2021xfz . In particular, the spectral index $n_{s}$ and
the bounds on the tensor-to-scalar ratio $r$ impose severe restrictions and
not all models can be compatible with the observational data 111 In this work,
standard assumptions are made for neutrino masses and their effective number.
Relaxing these it induces substantial shifts in $n_{s}$ Gerbino:2016sgw . .
The precise measurements of the primordial scalar perturbations, and of the
associated power spectrum amplitude $A_{s}$, imply constrains for the scale of
inflation in models encompassed in the framework of the metric or Palatini
formulation, which are more stringent in the case of Palatini Gravity as has
been shown in Gialamas:2019nly .

In this work we shall consider ${\cal{R}}^{2}$ theories, in the framework of
the Palatini Gravity, and study the cosmological predictions of some popular
models existing in literature, with emphasis on the maximal reheating
temperature, or instantaneous reheating temperature. We will show that there
are strict theoretical bounds on it which are saturated when the couplings
$\alpha$, associated with the ${\cal{R}}^{2}$-term, is large. To this goal
need go beyond slow-roll approximation, to extract reliable predictions, since
quartic in the velocity terms of the inflaton play a crucial role. Assuming
instantaneous reheating the cosmological data impose upper bounds on $\alpha$,
or same, lower bound on the inflationary scale, which also hold for lower
reheating temperatures.

This paper is organized as follows :

In section II, we present the salient features and give the general setup of
$f({\cal{R}})$ \- Palatini Gravity 222 Throughout this paper the Ricci scalar
will be denoted by ${\cal{R}}$.  , in the presence of an arbitrary number of
scalar fields, coupled to Palatini Gravity in a non-minimal manner, in
general. Although this is not new, as this effort has been undertaken by other
authors, as well, we think that the general, and model-independent,
expressions we arrive at, are worth being discussed. We focus on
${\cal{R}}^{2}$ theories for which the passage to the Einstein frame is
analytically implemented. These theories have a gravity sector, specified by
two arbitrary functions, sourcing, in general, non-minimal couplings of the
scalars involved in Palatini Gravity, and a third function which is the scalar
potential. In the Einstein frame, and when a single field is present, these
models have much in common with the $K$ \- inflation models
ArmendarizPicon:1999rj .

In section III, we discuss the arising background equations of motions and
discuss the slow-roll mechanism, paying special attention to end of inflation
and its validity within the slow-roll scheme. We find that in some cases need
go beyond slow-roll to determine the end of inflation which controls the
instantaneous reheating temperature and the cosmological parameters.

In section IV we discuss various aspects of the inflationary evolution of
these models, in the general case, and extract useful conclusions, which hold
even when the evolution of inflaton, as it approaches the minimum of the
scalar potential, deviates significantly from slow-roll.

Section V deals with the instantaneous reheating temperature and its bounds
set on it which are dictated by the pertinent backrground equations. Strict
upper bounds are derived which are saturated when the parameter $\alpha$,
defining the coupling of the ${\cal{R}}^{2}$-terms to gravity, is large. These
could not have been predicted within the slow-roll scheme. Moreover, assuming
that reheating is instantaneous, we explore the bounds set by the cosmological
observables, on the parameters of particular inflation models, namely the
class of models in which the scalar field $h$, is characterized by monomial
potentials $\sim h^{n}$, with $n$ a positive even integer, and the Higgs
model. The power spectrum amplitude $A_{s}$ results to fine tuning of the
parameters of the potential, while the spectral index $n_{s}$ and the tensor
to scalar ratio $r$, set bounds on $\alpha$, and therefore bounds on the
inflation scale and the instantaneous reheating temperature, $T_{ins}$. The
latter can be as large as $\sim 10^{15}\,GeV$, the larger values attained for
the smaller allowed value of the parameter $\alpha$.

In sections VI, we end up with our conclusions.

##  II The model

In this section we shall outline the general setup, and follow the methodology
and notation used in Gialamas:2019nly . More details, if needed, can be found
in this reference. The starting point is an action involving scalar fields
$h^{J}$ which are coupled to Palatini gravity in the following manner,

$\displaystyle
S\,=\,\int\,d^{4}x\,\sqrt{-g}\,\left(\,f({\cal{R}},h)+\dfrac{1}{2}G_{IJ}(h)\,\partial
h^{I}\partial h^{J}-V(h)\,\right)\,.$ (1)

In it ${\cal{R}}$ is the scalar curvature, in the Palatini formalism, and
$f({\cal{R}},h)$ an arbitrary function of the scalars $h^{J}$ and ${\cal{R}}$.
Following standard procedure we write this action in the following manner,
introducing an auxiliary field $\Phi$,

$\displaystyle
S\,=\,\int\,d^{4}x\,\sqrt{-g}\,\left(\,f(\Phi,h)+f^{\prime}(\Phi,h)\,({\cal{R}}-\Phi)+\dfrac{1}{2}G_{IJ}(h)\,\partial
h^{I}\partial h^{J}-V(h)\,\right)\,.$ (2)

In this $f^{\prime}(\Phi,h)$ denotes the derivative with respect $\Phi$. This
action can be written as follows, in Jordan frame,

$\displaystyle
S\,=\,\int\,d^{4}x\,\sqrt{-g}\,\left(\,\;\psi\,{\cal{R}}+\dfrac{1}{2}G_{IJ}(h)\,\partial
h^{I}\partial h^{J}-\psi\Phi+f(\Phi,h)-V(h)\,\right)\,,$ (3)

where $\psi$ in defined by,

$\displaystyle\psi=\dfrac{\partial f(\Phi,h)}{\partial\Phi},\quad\text{with \,
inverse}\quad\Phi=\Phi(\psi,h)\,.$ (4)

One can go to the Einstein frame by performing a Weyl transformation of the
metric

$\displaystyle g_{\mu\nu}=\,{{\bar{g}}_{\mu\nu}}\,/\,{2\psi}$ (5)

and that done the theory receives the following form,

$\displaystyle
S\,=\,\int\,d^{4}x\,\sqrt{-\bar{g}}\,\,\left(\;\,\dfrac{\overline{\cal{R}}}{2}+\dfrac{1}{4\psi}\,G_{IJ}(h)\,\partial
h^{I}\partial
h^{J}-\dfrac{1}{4\psi^{2}}\,(\psi\Phi-\,f(\Phi,h)+V(h))\,\right)\,.$ (6)

We can further eliminate the field $\psi$, using its equation of motion,

$\displaystyle\psi\,(\partial h)^{2}\,=\,\psi\,\ \Phi-2\,f(\Phi,h)+2V(h)\,,$
(7)

where, in order to speed up notation, we have denoted $G_{IJ}(h)\,\partial
h^{I}\partial h^{J}=(\partial h)^{2}$. Note that (7) is not solvable, in
general, however in ${\cal{R}}^{2}$-theories this is feasible.

In the following we shall focus on such theories, with a single field $h$
present, with $f(h,{\cal{R}})$ quadratic in the curvature, having therefore
the form

$\displaystyle
f({\cal{R}},h)\,=\,\dfrac{g(h)}{2}\,{\cal{R}}\,+\,\dfrac{{\cal{R}}^{2}}{12M^{2}(h)}\,.$
(8)

Since a single scalar field is assumed its kinetic term can be always brought
to the form $\,(\partial h)^{2}/2$, that is in the action (1) the field can be
taken canonically normalized. Therefore in this theory there are three
arbitrary functions, namely $g(h),M^{2}(h),V(h)$, and any choice of them
specifies a particular model. We have set the reduced Planck mass
$m_{Planck}\equiv m_{P}=(8\pi G_{N})^{-1/2}$ dimensionless and equal to unity
and thus all quantities in (8) are dimensionless. When we reinstate dimensions
the functions $g,V$ have dimensions $mass^{2},mass^{4}$, respectively, while
$M^{2}$ is dimensionless. Note that a non-trivial field dependence of the
functions $g(h)$, and/or $M^{2}(h)$, is a manifestation of non-minimal
coupling of the scalar $h$ to Palatini Gravity. We recall that in Palatini
formalism there is no a scalaron field, associated with an additional
propagating degree of freedom, which in the Einstein frame of the metric
formulation plays the role of the inflaton.

With the function $f({\cal{R}},h)$, as given by (8), we get from Eq. (4),

$\displaystyle\psi=\dfrac{g(h)}{2}\,+\,\dfrac{\Phi}{6M^{2}(h)}\,,$ (9)

and (7) is solved for $\psi$ in a trivial manner, yielding

$\displaystyle\psi=\dfrac{4\,V+3M^{2}g^{2}}{2(\partial h)^{2}+6M^{2}g}\,.$
(10)

In this way $\psi$, an hence $\Phi$, from (9), are expressed in terms of
$h,(\partial h)^{2}$. Plugging $\psi,\Phi$ into (6) we get, in a
straightforward manner

$\displaystyle
S\,=\,\int\,d^{4}x\,\sqrt{-{g}}\,\,\left(\;\,\dfrac{{\cal{R}}}{2}+\dfrac{K(h)}{2}\,(\partial
h)^{2}\,+\,\dfrac{L(h)}{4}\,(\partial h)^{4}\,-\,U_{eff}(h)\,\right)\,.$ (11)

In this action we have suppressed the bar in the scalar curvature and also in
$\sqrt{-g}$, and in order to simplify notation we have denoted
$\partial_{\mu}h\partial^{\mu}h$ by $(\partial h)^{2}$ and
$(\partial_{\mu}h\partial^{\mu}h)^{2}$ by $(\partial h)^{4}$. Note the
appearance of quartic terms $(\partial h)^{4}$ in the action. As for the
functions $K,L,U_{eff}$, appearing in (11), they are analytically given by

$\displaystyle
L(h)\,=\,(3M^{2}g^{2}+4V)^{-1}\,,\,K(h)\,=\,3M^{2}gL\,,\,U_{eff}=3M^{2}VL\,.$
(12)

Observe that since terms up to ${\cal{R}}^{2}$ have been considered, in
$f({\cal{R}},h)$ , higher than $(\partial h)^{4}$ terms do not appear in the
action (11).

The above Lagrangean may feature, under conditions, K - inflation models
ArmendarizPicon:1999rj , which involve a single field, described by an action
whose general form is

$\displaystyle
S\,=\,\int\sqrt{-g}\;\left(\,\dfrac{{\cal{R}}}{2}+p(h,X)\,\right)\,d^{4}x\,.$
(13)

where $X\equiv(1/2)\partial_{\mu}h\partial^{\mu}h$. The cosmological
perturbations of such models were considered in Garriga:1999vw  and the
importance of a time-dependent speed of sound $c_{s}$ in K - inflation models
was emphasized in Lorenz:2008je  and cosmological constraints were derived,
where improved expressions for the density perturbations power spectra were
used. Specific models were also considered in Li:2012vta . See also
Nojiri:2019dqc ; Odintsov:2019ahz ; Mikura:2021ldx ; Pareek:2021lxz ;
Odintsov:2021lum ; Oikonomou:2021edm , for more recent works on these models,
in various contexts.

In a flat Robertson-Walker metric, where the background field $h$ is only time
dependent, the energy density and pressure are given by

$\displaystyle\rho(h,X)\,=\,{K(h)}X+3\,L(h)X^{2}+U_{eff}(h)\quad,\quad
p(h,X)\,=\,{K(h)}X+\,L(h)X^{2}-U_{eff}(h)\,,$ (14)

with $X$ being, in this case, half of the velocity squared,
$X={\dot{h}}^{2}/2$.

We shall assume that the function $L(h)$ is always positive to avoid phantoms,
which may lead to an equation of state with $w<-1$. This may occur when $L<0$
and $X$ becomes sufficiently large. However, there is no restriction on the
sign of $K(h)$ which may be negative in some regions of the field space,
signaling that the kinetic term has the wrong sign in those regions. Obviously
the sign of $K(h)$ should be positive at the minimum of the potential. Options
where $K$ is negative in some regions, although interesting, will not be
pursued in this work. Besides, we shall assume that the potential is positive
$U_{eff}(h)\geq 0$ and bears a Minkowski vacuum. This ensures that the energy
density is positive definite even when the velocity is vanishing. The location
of the Minkowski vacuum can be taken to be at $h=0$, without loss of
generality, by merely shifting appropriately the field $h$. Then having a
positive definite potential which vanishes at $h=0$ entails $U_{eff}(0)=0$ and
also $U_{eff}^{\prime}(0)=0$. When inflation models are considered, the
inflaton will roll down towards this minimum signaling the end of inflation
and beginning of Universe thermalization.

Concerning the potential $U_{eff}$, appearing in the Lagrangian (11) in the
Einstein frame, using Eq. (12) it is trivially shown that it can be cast in
the following form 333 The quantity $R$ should not be confused with the Ricci
scalar ${\cal{R}}$.  ,

$\displaystyle
U_{eff}=\dfrac{1}{4}\,\left({3\,M^{2}}\,-\dfrac{1}{R}\right)=\dfrac{3M^{2}}{4}\,\left(1\,-gK\right)\quad\text{where}\quad
R=\dfrac{L}{K^{2}}\quad.$ (15)

The two forms of the potential above are equivalent, if the relation
$K=3M^{2}gL$ of Eq. (12) is used. The quantity $R$ appearing in this equation
may play an important role, as we shall see, in inflationary evolution. From
(15) we see that positivity of $U_{eff}\geq 0$ entails to having
$R^{-1}\leq{3M^{2}}$. In terms of the potential $V(h)$ appearing in the action
(1) this simply reads $V\geq 0$, as can be seen from the last of Eqs. (12) .
Dealing with positive definite potentials, an upper bound is then established,

$\displaystyle U_{eff}\leq\dfrac{3M^{2}}{4}\,.$ (16)

as is evident from (15). Although not necessary, this upper bound can be
easily saturated, for large $h$, by choosing appropriately the functions
involved. Actually the asymptotic behavior of these functions, for large $h$,
control the behavior of the potential in this regime. Choosing for instance
the function $R$ to increase, as $h$ becomes large, then saturation of the
above bound is easily obtained If, moreover, we opt that the function $M^{2}$
approaches or even be a constant, for large $h$-values, while the function $R$
increases in this regime, then the potential reaches a plateau which may yield
enough inflation. This is a rather plausible scenario, which may drive
successful inflation, and is obtainable under rather mild assumptions.
However, other less obvious choices may be available.

The models studied in this work can be, in general, classified in three main
categories :

  * •

Models with $g,\,M^{2}=constants$, named M1 for short for future reference.

These are dubbed minimally coupled models. In this case the constant $g$ can
be taken equal to unity without loss of generality. This is implemented by
rescaling the metric as $g_{\mu\nu}\rightarrow g_{0}^{-1}\,g_{\mu\nu}$, where
$g_{0}=g$, accompanied by a redefinition of the field $h\rightarrow
g_{0}^{1/2}\,h$ in the action (1), with $f({\cal{R}},h)$ as given by (8) ,
before going to Einstein frame.

  * •

Models with $M^{2}=constant$ and $g$ a function of $h$. These we name M2 for
short.

In this class of models $g$ can be a function of the field $h$, $g(h)$. In
this case we can take $g(0)=1$ by rescaling the metric and the field $h$, in
the way described previously for the M1 models, with $g_{0}$ identified with
$g(0)$ . In both cases, M1 or M2, it is tacitly assumed that $g_{0}>0$.

  * •

Models with both $g,\;M^{2}$ functions of $h$. These we shall name M3.

The models $M1,M2$ cover a broad range of interesting models, studied in the
past in various contexts, and shall concern us most. Models belonging to the
class $M_{3}$ have been studied in Lykkas:2021vax .

As we discussed, we shall be interested in models with positive semi-definite
potential having a single Minkowski vacuum at a point, which without loss of
generality we can take it to be located at $h=0$. Then, besides $U_{eff}>0$,
we must have $\dfrac{dU_{eff}}{dh}>0$ for $h>0$, with the sign of the
derivative reversed when $h<0$. These imply restrictions for the functions
describing the aforementioned models. For instance, for the minimally coupled
models this entails $\dfrac{dK}{dh}<0\;(>0)$ for $h>0\;(h<0)$, using Eq. (15)
and the fact that $g>0$. The above requirements are rather mild and can be
easily satisfied. Therefore many options are available for scalar potentials
bearing the characteristics demanded for successful inflation to be possibly
implemented. This will be exemplified in specific models, to be discussed
later.

Concluding this section, we presented a general, and model independent,
framework of ${\cal{R}}^{2}$ \- theories, in the Palatini formulation of
Gravity, which may be useful for the study of inflation. In the Einstein frame
these theories may be considered as generalizations of $K$-inflation models.
This formalism will be implemented, for the study of particular inflationary
models.

##  III The equations of motion and the slow-roll

###  III.1 The inflationary equations of motion

When non-canonical kinetic terms are present the equations of motions for the
would be inflaton scalar field $h$ differ from their standard form. As a
result, the cosmological parameters describing the slow-roll evolution should
be modified appropriately. Certainly one can normalize the kinetic term of the
scalar field appropriately but this is not always very convenient. Actually
the integrations needed, in order to pass from the non-canonical to a
canonically normalized field, in most of the cases, cannot be carried out
analytically. Therefore it proves easier, in certain cases, to work directly
with the non-canonical fields and express the pertinent cosmological
observables in a manner that is appropriate for this treatment.

It is not hard to see that the field $h$ satisfies the equation of motion
given by

$\displaystyle(K+3L\,\dot{h}^{2})\ddot{h}+3H(K+L\,\dot{h}^{2})\,\dot{h}+U^{\prime}_{eff}(h)+\dfrac{1}{4}\,(2K^{\prime}+3L^{\prime}\,\dot{h}^{2})\,\dot{h}^{2}=0\,,$
(17)

where dots denote derivatives with respect time. If the field were canonical,
$K=1$, and there were no quartic in the velocity terms, that is $L=0$, the
equation above receives a much simpler form. In this, the effect of using a
non-canonical, in general, field $h$ is encoded in the function $K$. The
effect of the presence of terms $({\partial h})^{4}$ in the action is encoded
within the function $L$. The terms that depend on $L$ are multiplied by an
extra power of the velocity squared, as compared to the $K$-terms. These
cannot be neglected, as we discuss below, since they are not small in general.

We can gain more insight if we use a canonically normalized field, say $\phi$,
defined by

$\displaystyle\phi=\int_{0}^{h}{\sqrt{K(h)}}\,dh\,,$ (18)

which, however, cannot be always presented in a closed form, as we have
already remarked. The constant of integration has been chosen, without loss of
generality, so that $h=0$ corresponds to the value $\phi=0$ too. To avoid
ghosts we shall assume that $K>0$, so that the integration above makes sense.
Actually if $K$ is negative the kinetic term of the field $\phi$ would have
the wrong sign, i.e. it would appear as $-({\partial\phi})^{2}$. It could
happen however that this function is negative in some region but at the
Minkowski vacuum is strictly positive. In this way ghosts are also avoided.
This case, interesting as might be, is not discussed and we prefer to keep a
rather conservative view point and take $K>0$ in the whole region. Then in
terms of the field $\phi$ the equation of motion (17) takes on the form

$\displaystyle\left(1+3\,R\,\dot{\phi}^{2}\right)\,\ddot{\phi}+3H\left(1+R\,\dot{\phi}^{2}\right)\,\dot{\phi}+\dfrac{dU_{eff}}{d\phi}+\,\dfrac{3}{4}\,\dfrac{d\,R}{d\phi}\,{\dot{\phi}}^{4}=0\,.$
(19)

Note the dependence of this equation on the ratio $R=L/K^{2}$ defined earlier
in Eq. (15). From this form it appears that the smallness of the ${\partial
h}^{4}$ terms in the action is quantified by the smallness of the ratio
$\frac{L}{K^{2}}\dot{\phi}^{2}\ll 1$, which is equivalent to
$\frac{L}{K}\dot{h}^{2}\ll 1$.

The equation governing the evolution as functions of time can be easily
converted to differential equation for the velocities as function of the
fields. This is done by noting that there is no explicit dependence on time in
either of (17) or (19). The velocity of $h$-field, $u(h)$ satisfies

$\displaystyle(K+3L\,u^{2})\,u\dfrac{du}{dh}+3H(K+L\,u^{2})\,u+U^{\prime}_{eff}(h)+\dfrac{1}{4}\,(2K^{\prime}+3L^{\prime}\,u^{2})\,u^{2}=0\,,$
(20)

which is a first order equation with respect the velocity $u(h)$. The velocity
$\dot{h}$ and the acceleration $\ddot{h}$, as functions of $h$, in Eq. (20)
are given by

$\displaystyle\dot{h}=u(h)\quad,\quad\ddot{h}=u(h)\dfrac{du(h)}{dh}\quad.$
(21)

This method is well-known in Mathematics 444 The points at which $u$ vanishes
correspond to ”cusp” points. There the acceleration $\dfrac{du}{dh}$ becomes
infinite. There are no cusp-points from the beginning of inflation up to
values of the field for which $u$ vanishes for first time. This covers the
whole inflation region.  . By the same token, for the normalized inflaton
field $\phi$, we have

$\displaystyle\left(1+3\,R\,\upsilon^{2}\right)\,\upsilon\frac{d\upsilon}{d\phi}+3H\left(1+R\,\upsilon^{2}\right)\,\upsilon+\dfrac{dU_{eff}}{d\phi}+\,\dfrac{3}{4}\,\dfrac{d\,R}{d\phi}\,\upsilon^{4}=0\quad,$
(22)

where, in this case, the velocity $\dot{\phi}$ and the acceleration
$\ddot{\phi}$, as functions of $\phi$, are given by

$\displaystyle\dot{\phi}=\upsilon(\phi)\quad,\quad\ddot{\phi}=\upsilon(\phi)\dfrac{d\upsilon(\phi)}{d\phi}\quad\quad.$
(23)

Equations (20) and (22 ), being first order equations, are more easily solved
for values of the fields lying in the inflationary regime.

###  III.2 Is slow-roll a valid scenario ?

The usual slow-roll solution is not be a valid approximate solution for any
values of the parameters involved. This can be exemplified in certain models,
as we shall see. To be more specific, in the class of models where
$M^{2}=constant$, which we study in this work, Eq. (19) takes on the form,
using Eq (15),

$\displaystyle\left(1+3\,R\,\dot{\phi}^{2}\right)\,\ddot{\phi}+3H\left(1+R\,\dot{\phi}^{2}\right)\,\dot{\phi}+\,(1+3R^{2}\,{\dot{\phi}}^{4})\,\dfrac{dU_{eff}}{d\phi}=0\,.$
(24)

where the Hubble function is given by,

$\displaystyle
3H^{2}=\left(1+\dfrac{3}{2}R\,\dot{\phi}^{2}\right)\dfrac{\dot{\phi}^{2}}{2}+U_{eff}\,.$
(25)

Neglecting $R\,\dot{\phi}^{2}$ in the equations above we recover the well-
known form of Friedmann equation for the canonically normalized field $\phi$.
In the regime $R\,\dot{\phi}^{2}<<1$ slow-roll evolution is realized. Using
the slow-roll expressions one can see, in a straightforward manner, that

$\displaystyle
R\,\dot{\phi}^{2}=\frac{2R\,U_{eff}}{3}\,\,\epsilon_{V}(\phi)\,,$ (26)

which holds provided $R\,\dot{\phi}^{2}$ is small enough. In this equation
$\epsilon_{V}=\frac{1}{2}\,({U_{eff}^{\prime}(\phi)}/{U_{eff}(\phi)})^{2}$.
However the smallness of $\epsilon_{V}(\phi)$ does not ensure smallness of
$R\,\dot{\phi}^{2}$. This is shown in Figure 1 for the minimally coupled
model, described by the potential $V=m^{2}h^{2}/2$ and $g=1,M^{2}=1/3\alpha$.
In this class of models both $\epsilon_{V}$ and $R\,\dot{\phi}^{2}$, depend
only on the combination $c=2\alpha m^{2}$. We have fixed $m$ by taking it to
be $m=6.5\times 10^{-6}$, or so, suggested by primordial scalar perturbations,
as we shall see later. The horizontal axis is $\sqrt{c}\phi$. For the case
shown on left, $c=0.845$, corresponding to $\alpha=10^{10}$, while on the
right $c=0.845\times 10^{2}$, corresponding to $\alpha=10^{12}$. Notice the
difference in the behaviour of the $\epsilon_{V}(\phi)$ (red solid line) and
$R\,\dot{\phi}^{2}$ (blue solid line) functions. For the lower $c$-case,
$R\,\dot{\phi}^{2}$ stays lower than unity all the way up to the point where
$\epsilon_{V}=1$, marked by the yellow horizontal line. In this case the usual
slow-roll scenario is trusted. The gray line is the exact solution for the
velocity $\upsilon(\phi)$, scaled by $1/\sqrt{a}$, that is
$\upsilon(\phi)/\sqrt{a}$, which has been derived numerically. The gray dot-
dashed line is the corresponding slow-roll approximation. These start
deviating significantly, from each other, as soon as $R\,\dot{\phi}^{2}$
starts becoming sizable. Note that for the larger $c$-case displayed, this
occurs well before $\epsilon_{V}(\phi)$ becomes unity, showing that slow-roll
ceases to be a good approximation although $\epsilon_{V}(\phi)$ is
significantly lower than unity. The conclusion is that, for lower $c$,
corresponding to lower $\alpha$, $R\,\dot{\phi}^{2}$ is small, and the slow-
roll solution is a good approximation. On the contrary for large $c$-values,
$R\,\dot{\phi}^{2}$ becomes large before $\epsilon_{V}=1$. Therefore slow-roll
in the usual sense can be only realized in the region of $\phi$-values for
which $R\,\dot{\phi}^{2}$ is small, and thus far from the point where
$\epsilon_{V}=1$. The evaluation of the end of inflation, in this case, can be
only achieved numerically since approximate slow-roll solutions, based on
$\epsilon_{V}=1$, cannot be trusted in the region of $\phi$ for which
$R\,\dot{\phi}^{2}$ starts approaching unity.

Concerning the end of inflation, this takes place when the parameter
$\epsilon_{1}=-\dot{H}/H^{2}$ becomes equal to unity. In the slow-roll regime,
where $R\,\dot{\phi}^{2}$ are negligible, $\epsilon_{V}$ and $\epsilon_{1}$
almost coincide, assuming de-Sitter expansion, but this is not the case when
$R\,\dot{\phi}^{2}$ terms start growing and the approximate slow-roll solution
no longer holds. Therefore we rely on $\epsilon_{1}=1$ as the only reliable
means to determine accurately the end of inflation. This is equivalent to
$\rho+3p=0$, corresponding to $\ddot{a}=0$ for the cosmic scale factor, and
yields the following relation between the velocity and field value, at time
$\epsilon_{1}$ reaches unity,

$\displaystyle\upsilon^{2}(\phi)\,=\,\dfrac{2}{3R}\,\left(-1+\sqrt{1+3RU_{eff}}\right)$
(27)

In this $\upsilon(\phi)$ is the velocity ${\dot{\phi}}$ expressed as function
of the field $\phi$, and can be extracted numerically by solving (22). In
ordinary inflation models where the $R$-terms are missing, that is there are
no terms quartic in the velocity in the action, the analog of Eq. (27) is
$\upsilon^{2}(\phi)=U_{eff}$, a well-known result, and this can be solved to
yield the value of the field, $\phi_{end}$ at end of inflation, if the slow-
roll solution is used. This is actually equivalent to $\epsilon_{V}=1$ 555
When the inflaton kinetic energy is ${\dot{\phi}}^{2}/2$, the condition
$\epsilon_{1}=1$ corresponds actually to $\epsilon_{H}(\phi)=1$, where
$\epsilon_{H}(\phi),\eta_{H}(\phi)$ are slow-roll parameters defined in
Liddle:1994dx . Thus $\phi_{end}$ extracted from $\epsilon_{V}=1$ can be
considered as a first order result. An improved value for $\phi_{end}$, using
the exact relation $\epsilon_{H}(\phi)=1$, can follow using
$\epsilon_{V}=\left(1+\sqrt{1-\eta_{V}/2}\right)^{2}$, Ellis:2015pla . To our
knowledge, there is no such a relation when $R$-terms. are present, which
would approximate (27) .  If the $R$-terms are sizable, before end of
inflation, we lack even an approximate solution for $\upsilon(\phi)$, and thus
(27) can be only tackled numerically, in order to know $\phi_{end}$. In fact
using the slow-roll parameter $\epsilon_{V}$ to determine the end of inflation
overestimates the value $\phi_{end}$, as this is extracted from (27) , leading
to erroneous results concerning cosmological observables, and in particular
the energy density at the end of inflation, which determines the reheating
temperature of the Universe.

Figure 1: Evolution of $\epsilon_{V}$, red line, and $R\,\dot{\phi}^{2}$, blue
line, in the slow-roll approximation in the Model- I . as functions of
$\sqrt{c}\phi$. On the left, $c=0.845$ corresponding to $\alpha=10^{10}$ and
on the right $c=0.845\times 10^{2}$ corresponding to $\alpha=10^{12}$. In both
case the parameter $m$ is $m=6.5\times 10^{-6}$.

It should be stressed that our numerical study duly takes into account the
contribution of these terms and no approximation, whatsoever, is made. We have
found numerically that they can be indeed small, Enckell:2018hmo ;
Antoniadis:2018ywb ; Antoniadis:2018yfq ; Tenkanen:2019jiq , however this
holds in a restricted range of the parameters and is not a general feature.
For instance in the non-minimal model discussed previously, with $V\sim
h^{2}$, the combination $c=2\alpha m^{2}$ has to be smaller than about unity,
for these terms to be small in the entire inflationary region.

##  IV Inflationary Evolution - End of inflation

The end of inflation is signaled when $\epsilon_{1}=1$, or equivalently when
acceleration ends, $\ddot{a}=0$. The determination of end of inflation
requires augmented accuracy, at least in some models, as we shall discuss in
the sequel. The slow-roll parameter $\epsilon_{1}$ is very small during de
Sitter phase and then starts increasing. Eventually becomes equal to unity
where Universe acceleration stops. In order to locate when this occurs we find
it useful to have an expression for the time derivative of it. $\epsilon_{1}$
can be expressed in the following way,

$\displaystyle\epsilon_{1}=\dfrac{3}{2}\dfrac{\rho+p}{\rho}=\dfrac{3}{2}\,\;\dfrac{V^{2}+R\,V^{4}}{\rho\;\;}\quad,\quad$
(28)

where $V$ is the velocity $V=\dot{\phi}$ and $R\equiv{L}/{K^{2}}$ . Note that
the numerator does not explicitly depend on the potential. Then the time
derivative of $\epsilon_{1}$ is of the form,

$\displaystyle\dot{\epsilon}_{1}\equiv\dfrac{d\epsilon_{1}}{dt}=\dfrac{\cal{N}}{\rho^{2}}$
(29)

where the numerator $\cal{N}$ is found to be

$\displaystyle{\cal{N}}=2V\dot{V}\,\left(-\dfrac{R\,V^{4}}{4}+(\,1+2\,R\,V^{2})\,U_{eff}\right)+\left(-\dfrac{V^{2}}{4}+\,U_{eff}\right)\,V^{5}\,R^{\prime}-(\,1+R\,V^{2})\,V^{3}\,U^{\prime}_{eff}$
(30)

In this all primed quantities are derivatives with respect $\phi$.

We assume that the evolution starts from an initial position $h>0$ on the
plateau of the potential, or in terms of the canonically normalized inflaton
field $\phi>0$. Recall the two fields are related by
$\phi=\int\,\sqrt{K}\,dh$, and we have chosen the integration constant so that
$\phi=0$ corresponds to $h=0$ too. The potential is positive definite and
exhibits a zero at $h=0$ and as a consequence at $\phi=0$, if expressed in
terms of $\phi$. Due to its positivity the first derivative of the potential
vanishes at the minimum as well.

The inflaton start its journey with a very small, or even vanishing initial
velocity, which soon becomes negative ${V}<0$ and increases in magnitude, as
inflaton rolls towards the minimum of the potential. At some time its
direction is reversed. Therefore there is a time $t_{acc}$ for which its
acceleration vanishes, $\dot{V}(t_{acc})=0,$ while ${V}(t_{acc})<0$. Then
$\dot{V}>0$ for $t>t_{acc}$ and thus the velocity increases and eventually at
some time $t_{1}$ it vanishes for the first time, i.e $V(t_{1})=0$. Therefore
the picture is that $V<0$, as long as $t<t_{1}$, attaining its minimum value
at $t_{acc}$, and at the time $t_{1}$ it vanishes. At this time the
acceleration is still positive, i.e. the velocity continuous being increased
passed the time $t_{1}$. It takes some time after it vanishes, reversing its
direction, and start oscillating about the minimum of the potential.

From the equation of motion (19) for the field $\phi$, we have therefore that
at this time $t_{1}$,

$\displaystyle\left(\dot{V}+\dfrac{dU_{eff}}{d\phi}\right){\biggr{\rvert}}_{t_{1}}\,=\,0$
(31)

However since, as we have discussed, $\dot{V}(t_{1})>0$ it follows from the
equation above that $\frac{dU_{eff}}{d\phi}{\bigr{\rvert}}_{t_{1}}<0$. During
inflaton’s journey, up to time it reaches the minimum of potential, we have
that $\frac{dU_{eff}}{d\phi}\geq 0$, with the minimum being reached at time
$t_{0}$ and it for later times it becomes negative. Therefore $t_{0}$ is prior
to $t_{1}$, $t_{0}<t_{1}$. That is the minimum of potential, which is located
at $\phi=0$, or $h=0$, is reached before the velocity vanishes for the first
time during inflation, which is a rather expected behaviour.

From the previous discussion we concluded that the time $t_{acc}$, at which
the acceleration of $\phi$ vanishes, and hence the velocity $V$ takes its
minimum value, is prior to $t_{1}$. Its location relative to $t_{0}$, the time
inflaton reaches the minimum of the potential, may be derived as follows. The
acceleration is determined from the equations of motion. Using the fact that
$U_{eff},U^{\prime}_{eff}$ vanish at $t_{0}$ we have that

$\displaystyle\dot{V}(t_{0})\,=\,-\dfrac{1}{1+3\,R\,V^{2}}\;\left(3\,H\,(1+R\,V^{2})\,V+\dfrac{3}{4}\;R^{\prime}\,V^{4}\right){\biggr{\rvert}}_{t_{0}}\quad,\;\;\text{where}\quad
R^{\prime}=\dfrac{dR}{d\phi}$ (32)

Due to the fact that $V(t_{0})$ is negative, we have, from this equation, that
the acceleration $\dot{V}(t_{0})$ is positive if
$R_{0}^{\prime}=R^{\prime}(t_{0})\leq 0$. The sign of $R_{0}^{\prime}$ is
related to the derivative of the potential since,

$\displaystyle\dfrac{dU_{eff}}{d\phi}=\,\dfrac{3}{4}\,\dfrac{dM^{2}}{d\phi}+\dfrac{1}{4R^{2}}\,\dfrac{dR}{d\phi}\quad.$
(33)

This conclusion can be drawn using the form of the potential as given by eq
(15), which is valid for any $M^{2}$. From this it directly follows, using the
fact that the derivative of the potential vanishes at $\phi=0$, corresponding
to $t_{0}$,

$\displaystyle
R_{0}^{\prime}=-3\,R_{0}^{2}\,M_{0}^{2\;\prime}\quad\text{where}\quad
M_{0}^{2\;\prime}=\dfrac{dM^{2}}{d\phi}{\biggr{\rvert}}_{\phi=0}\quad.$ (34)

Therefore $R_{0}^{\prime}\leq 0$ in all models with $M_{0}^{2\;\prime}\geq
0\,$ and as a consequence $\dot{V}(t_{0})>0$. This states that $t_{0}$ lies in
a region the acceleration is positive, and this occurs after $t_{acc}$.
Therefore in models with $M_{0}^{2\;\prime}\geq 0\,$ we have that
$t_{acc}<t_{0}$. The models M1, M2 , which are characterized by a constant
$M^{2}$, fall within this class since $M^{2\;\prime}=0\,$. For models not
belonging to this category the condition $M_{0}^{2\;\prime}\geq 0\,$ has to be
checked as per case. Therefore when $M_{0}^{2\;\prime}\geq 0\,$ we have
$\dot{V}(t_{0})>0$ and $t_{0}$ lies in the region inflaton accelerates before
its velocity vanishes for the first time. This occurs for times larger than
$t_{acc}$. Concluding, there is a broad class of models, those with
$M_{0}^{2\;\prime}\geq 0\,$, for which the time ordering is
$\,t_{acc}<t_{0}<t_{1}\,$. The situation is shown in Figure 2 .

Figure 2: The ordering of times $t_{acc},t_{0},t_{1}$ is as depicted in this
figure for a wide class of models, including those with $M^{2}=constant$.
Between $t_{acc}$ and $t_{1}$ the acceleration of inflaton $\phi$ is positive.

Concerning the evolution of $\epsilon_{1}$, it initially has a very small
positive value and then starts increasing. This vanishes for first time when
the velocity does, as is evident from Eq. (28), that is at time $t_{1}$.
Therefore it ought develop a maximum at some intermediate time, say $t_{max}$
which is prior to $t_{1}$, i.e. $t_{max}<t_{1}$. Whether $t_{max}$ lies before
or after $t_{0}$ we do not know as yet. We shall show that in the class of
minimally coupled models M1, and also models M2, $t_{0}<t_{max}$ so that time
ordering is as $t_{0}<t_{max}<t_{1}$. That is the minimum of the potential is
reached for first time before $\epsilon_{1}$ attains its first maximum, and
later, at time $t_{1}$, $\epsilon_{1}$ vanishes for first time along with
velocity $V$. The proof relies on what is the sign of $\cal{N}$ in Eq. (30) at
$t_{0}$. Since the potential and its derivative vanish at $t_{0}$, due to the
fact that at this time $\phi=h=0$, the value of $\cal{N}$ is

$\displaystyle{\cal{N}}_{0}={\cal{N}}{\bigr{\rvert}}_{t_{0}}=-\dfrac{V^{5}}{2}\;\left(\,\dot{V}\,R+\dfrac{R^{\prime}}{2}\,V^{2}\right){\biggr{\rvert}}_{t_{0}}$
(35)

From this it is seen that the sign of ${\cal{N}}_{0}$ follows that of the
bracketed quantity in equation (35), due to the fact that $V(t_{0})<0$. Recall
that velocity is negative for $t<t_{1}$ and $t_{0}$ i.e. earlier than $t_{1}$
as we have shown before. Therefore for the sign of ${\cal{N}}_{0}$ we need
study the acceleration and also $R^{\prime}$ at $t_{0}$. For the models M1, M2
the acceleration is positive at the point $t_{0}$, as we have already shown (
see also Figure 2). For these models, on account of Eq. (33),
$R^{\prime}(t_{0})$ vanishes forcing the acceleration $\dot{V}(t_{0})$ at
$t_{0}$ to be positive, as we have already discussed. Therefore from (35) we
have that ${\cal{N}}_{0}>0$. This entails that $\dot{\epsilon}_{1}(t_{0})>0$,
using equation (29). Since $\epsilon_{1}$ is monotonically increasing until
this reaches its maximum at $t_{max}$, positivity of
$\dot{\epsilon}_{1}(t_{0})$ states that the maximum of $\epsilon_{1}$ is
reached later than $t_{0}$, i.e $t_{0}<t_{max}$, in the class of models having
$M^{2}=\text{constant}$, as shown on Figure 2. In the same figure the location
of $t_{max}$ relative to other critical times in the models M1, M2 is also
shown. As we have stated, this covers the class of minimally coupled models,
but also non-minimally coupled models in which $g(h)$, designating the
coupling of the Ricci term in (8), is not constant.

Concerning the maximum value of $\epsilon_{1}$ cannot exceed $3$. In fact from
(28) we have, writing explicitly the density $\rho$,

$\displaystyle\epsilon_{1}\,=\,3\,\dfrac{V^{2}+RV^{4}}{V^{2}+(3R/2)V^{4}+2U_{eff}}`\leq
3\,\dfrac{V^{2}+RV^{4}}{V^{2}+(3R/2)V^{4}}<3$ (36)

therefore a strict upper bound on $\epsilon_{1}$ can be established. At the
time $t_{0}$ we have, due to the fact that the potential vanishes,

$\displaystyle\epsilon_{1}(t_{0})=3\,\dfrac{1+R_{0}V_{0}^{2}}{1+(3R_{0}/2)V_{0}^{2}}\geq
2$ (37)

where the subscript $0$ means evaluation at $t_{0}$. This lower bound on the
value of $\epsilon_{1}(t_{0})$ combined with the fact that $\epsilon_{1}$ is
monotonically increasing for all times $t\leq t_{max}$ ensures that there is
certainly a time $t_{end}<t_{0}$ at which inflation ends, that is
$\epsilon_{1}(t_{end})=1$.

Figure 3: In model I, and for $\alpha=10^{8}$, we display the evolution of
$\epsilon_{1}$, in blue, and the velocity $\dot{\phi}$, dashed line, from some
initial $t_{ini}$ time. The horizontal axis is the rescaled cosmological time
$t/\sqrt{\alpha}$, and the velocity is actually $\dot{\phi}$ divided by its
values at $t_{ini}$. The horizontal amber line crosses the $\epsilon_{1}$
curve at where inflation ends, and this is prior to the minimum of the
potential, whose location is shown by the red circle. The maximum of
$\epsilon_{1}$ lies to the right of the minimum, but it is hardly visible on
the figure. $\epsilon_{1}$ vanishes for first time after the minimum is
reached. Its oscillatory behaviour after, is due to the fact that the velocity
oscillates, after inflaton passes the minimum of potential.

This behaviour is exemplified in Figure 3 for a model in which $M^{2}\equiv
1/3\alpha$ and $g(h)=1$, and quadratic potential
$V(h)=\frac{m^{2}\,h^{2}}{2}$. The values of the arbitrary parameters are
$\alpha=10^{8}$ and $m=6.5\times 10^{-5}$.

The results reached in this section are useful in our numerical treatment, in
order to locate with the required precision the end of inflation, in models
with $M^{2}=1/3\alpha$, especially when the parameter $\alpha$ is large. It is
in these cases that inflaton dynamics cannot be described by slow-roll and the
speed of sound, as we shall see, deviates from unity. This is equivalent to
having non-negligible contributions from the quartic in the velocity $L$-terms
appearing in the action, as inflaton approaches the end of inflatoin.

##  V Universe’s Reheating Temperature

The reheating temperature the Universe reached after its thermalization has
been extensively studied and various mechanisms and models have been put under
theoretical scrutiny, Kofman:1994rk ; Shtanov:1994ce ; Kofman:1997yn ;
Chung:1998rq ; Kawasaki:1999na ; Davidson:2000er ; Giudice:2000ex ;
Martin:2010kz ; Allahverdi:2010xz ; Podolsky:2005bw ; Adshead:2010mc ;
Mielczarek:2010ag ; Easther:2011yq ; Dai:2014jja ; Harigaya:2013vwa ;
Munoz:2014eqa ; Cook:2015vqa ; Gong:2015qha ; Rehagen:2015zma ; Drewes:2015coa
; Cai:2015soa ; Ellis:2015pla ; deFreitas:2015xxa ; Lozanov:2016hid ;
Lozanov:2017hjm ; Dalianis:2016wpu ; DiMarco:2018bnw ; Hasegawa:2019jsa ;
He:2018mgb ; German:2020iwg ; Saha:2020bis ; Hamada:2020kuy ; He:2020ivk ;
He:2020qcb ; DiMarco:2021xzk ; Kawai:2021hvs ; Pareek:2021lxz . Reheating in
the framework of the Palatini gravity has been also studied in Rubio:2019ypq ;
Gialamas:2019nly ; Das:2020kff ; Karam:2021sno ; Cheong:2021kyc

As for the number of e-folds left, $\,N_{k}=ln\,\dfrac{a_{end}}{a}$, from time
some scale $k$ crossed the sound horizon to the end of inflation is given in
Liddle:2003as ; Dodelson:2003vq . See also Martin:2010kz ; Lozanov:2017hjm .
Note that the dependence on speed of sound $c_{s}$ should be included in
$N_{k}$, due to the fact that it may deviate from unity, as is the case in
$K$-inflation models.

The number of e-folds accrued during the reheating period, $\Delta N_{reh}$,
is given by

$\displaystyle\Delta N_{reh}\equiv
ln\,\dfrac{a_{reh}}{a_{end}}=-\dfrac{1}{3(1+w)}\,ln\,\dfrac{\rho_{reh}}{\rho_{end}}\,.$
(38)

The subscripts (reh), (end) in the cosmic scale factor and the energy
densities denote that these quantities are evaluated at the end of the
reheating period and inflation respectively. In terms of Hubble rate
$\rho_{end}=3m_{P}^{2}\,H_{end}$, and therefore this is known once $t_{end}$,
or the values of the fields and their velocities at end of inflation, are
known.

The effective equation of state parameter $w$ in the reheating period, is the
average

$\displaystyle w\,=\,\dfrac{1}{\Delta
N_{reh}}\,\int_{N_{end}}^{N_{reh}}\;w(N)\,dN\quad.$ (39)

The integration variable is the number of e-folds and $\Delta
N_{reh}=N_{reh}-N_{end}$, where $N_{end},N_{reh}$, are the number of e-folds
at end of inflation and reheating periods respectively. At the end of
inflation $w(N_{end})=-1/3$ while $w(N_{reh})=1/3$ corresponding to the onset
of radiation dominance. Lacking a particular reheating mechanism the value of
$w$ is largely unknown, therefore we shall consider it as a free parameter
taking values within some sensible range. In the canonical reheating scenario
$w=0$, but values in the range $\simeq 0.0-0.25$, or larger, right after
inflation, are also possible in some models Podolsky:2005bw ; Lozanov:2016hid
; Lozanov:2017hjm ; Dux:2022kuk .

In terms of $\Delta N_{reh}$, for given $w$, one has for the reheating
temperature,, see for instance Munoz:2014eqa ,

$\displaystyle
T_{reh}\,=\,\left(\dfrac{30}{\pi^{2}\,}\dfrac{\,\rho_{end}}{\,g^{*(reh)}\,}\,\,\right)^{1/4}\,exp\left({-\dfrac{3(1+w)\Delta
N_{reh}}{4}}\right)\,.$ (40)

In our numerical studies we shall adopt the common values
$g^{*(reh)}=g_{s}^{*(reh)}=106.75$, corresponding to the SM content, as
discussed before, for temperatures above $\sim 1\,TeV$. 666 With
$g^{*(reh)}=100$ Eq. (40) coincides with that given in Munoz:2014eqa .  . Note
that since ${a_{reh}}>{a_{end}}$ we have that $\Delta N_{reh}\geq 0$, and
therefore due to $w>-1$ the reheating temperature $T_{reh}$ is bounded from
above

$\displaystyle
T_{reh}\leq\left(\dfrac{30}{\pi^{2}\,}\dfrac{\,\rho_{end}}{\,g^{*(reh)}\,}\,\,\right)^{1/4}\,.$
(41)

The bound on the right hand side of this defines the instantaneous reheating
temperature, $T_{ins}$. The temperature $T_{reh}$ reaches this upper bound
when the reheating process is instantaneous, in which case $\Delta N_{reh}=0$.
Note that for rapid thermalization we have $\rho_{end}=\rho_{reh}$, from Eq.
(38). The reheating temperature should be larger than $\sim 1\,MeV$ so that
Big Bang Nucleosynthesis (BBN) is not upset. Lower values on $T_{reh}$ have
been established in Kawasaki:1999na  and more recently in Hasegawa:2019jsa  .

The maximum reheating temperature depends, as we shall see, on the value of of
the sound of speed parameter $c_{s}$ at end of inflation. This is not unity,
in general, due to the fact that in the Palatini formulation of
${\cal{R}}^{2}$ gravity higher in the velocity $\dot{h}$ terms are
unavoidable. In fact $c_{s}$ is defined by

$\displaystyle c_{s}^{2}\,=\,\dfrac{\partial p/\partial
X}{\partial\rho/\partial X}\,,$ (42)

where $X$, defined after Eq. (13), is half the velocity squared. In terms of
the fields $h$, or $\phi$, and their velocities $u$ and $\upsilon$
respectively, this receives the form

$\displaystyle
c_{s}^{2}\,=\,\dfrac{1+L\,u^{2}/K}{1+3L\,u^{2}/K}\,=\,\dfrac{1+R\,\upsilon^{2}}{1+3R\,\upsilon^{2}}$
(43)

Inverting this we get,

$\displaystyle\omega\,=\,\dfrac{1-c_{s}^{2}}{3\,c_{s}^{2}-1}$ (44)

where $\omega=L\,u^{2}/K$, which is also equivalent to
$\omega=R\,\upsilon^{2}$. Actually $\omega$ is the same combination that
appears in the equation of motion for the field $h$, or $\phi$ respectively.
The velocity $c_{s}$ is thus controlled by $\omega$, and it is seen from (43)
that $c_{s}^{2}$ is strictly less than unity and approaches unity only when
$\omega\ll 1\,$. Interestingly enough is also bounded from below by $1/3$.
Thus $c_{s}^{2}$ takes values in the range

$\displaystyle\dfrac{1}{3}<c_{s}^{2}\;\leq 1\quad.$ (45)

These are strict mathematical bounds. The value of $c_{s}^{2}$ at any epoch is
known if one solves the pertinent differential equation for $h$, or
equivalently $\phi$. The upper bound is reached for vanishing values of
$L\,u^{2}/K$, or $R\upsilon^{2}$, and the lower limit when these quantities
get values much larger than unity. As we shall prove the value of $c_{s}^{2}$
when inflation ends determines $\rho_{end}$, and as a consequence the maximum
reheating temperature.

At end of inflation $\rho+3p=0\;$ and the solution of this equation relates
the velocity and the position at end of inflation through Eq. (27), which in
terms of the velocity of the field $h$ takes the form

$\displaystyle
u^{2}(h)=\dfrac{2K}{3L}\,\left(-1+\sqrt{1+\dfrac{3LU_{eff}}{K^{2}}}\right)\;.$
(46)

Both (27), or (46), hold at end of inflation and cannot be treated
analytically. Only numerically we can solve the pertinent differential
equations and through these determine the end of inflation time, or
equivalently the value of the position at end of inflation. Approximate
solutions do exist but they are unreliable, in the present case, due to the
presence of the $L$-dependent terms. As we have already discussed, and
depending on the values of the parameters involved, the solutions may differ
substantially from slow-roll towards end of inflation. Therefore evaluation of
end of inflation period using approximations, relying on the use of the slow-
roll parameters $\epsilon_{V},\eta_{V}$, poorly determine when the end of
inflation actually occurred.

Using the expression for the density $\rho$ one can use (27), (46), to find,

$\displaystyle\rho_{end}\,=\,\frac{3}{2R}\;\omega\;(1+\omega)\;{\biggr{\rvert}}_{{end}}$
(47)

In this, $\omega$ has been defined previously and in terms of the sound of
speed is given by Eq. (44) 777 Obviously (47) holds provided $\omega$, as
defined by equation (44), is non-vanishing. The $\omega=0$ case corresponds to
$R=0$, and hence $c_{s}^{2}=1$ which is the case in the usual inflation
scenarios. In this case it is well-known that
$\rho_{end}\,=\frac{3}{2}\,U_{eff}$ at the end of inflation.  .

Note that all quantities in (47) are meant at the end of inflation. Combining
these we find

$\displaystyle\rho_{end}\,=\,\;\frac{\,3}{R}\;\dfrac{(1-c_{s}^{2})\,c_{s}^{2}}{(3\,c_{s}^{2}-1)^{2}}\;{\biggr{\rvert}}_{{end}}\quad.$
(48)

However the ratio $R=\frac{\,L}{K^{2}}$ can be expressed in terms of the
potential, using (15), from which using the equations (27), or (46), we
finally get

$\displaystyle\rho_{end}\,=\,\;\frac{\,9\,M^{2}}{4}\,(1-c_{s}^{2})\;{\biggr{\rvert}}_{{end}}\quad,$
(49)

a very handy relation. Note that all quantities are evaluated at end of
inflation. Using the lower bounds on $c_{s}^{2}$, see(45), we get a strict
upper bound on $\rho_{end}$,

$\displaystyle\rho_{end}\,<\,\dfrac{3\,M^{2}}{2}\,{\biggr{\rvert}}_{{end}}$
(50)

In Eqs. (49) and (50) the quantity $M^{2}$ is not constant, in general, but a
function of the position, $h$ ( or $\phi$), which should be replaced by its
end of inflation value, as well. The actual upper bounds, which are extracted
by solving the pertinent differential equations numerically, may be smaller
than those of Eq. (50). In fact $c_{s}^{2}$, at the end of inflation,
depending on the model and the values of the parameters involved, may be close
to unity. This is indeed the case when the effect of the $L$ \- terms is small
throughout the inflation evolution. It may also happen that the lower bound in
(45) may be larger due to the fact that $L\dot{h}^{2}/K$ can never exceed some
critical value. In that particular case the upper bound (50) is lowered. This
is the case, for instance, in the minimally coupled models to be discussed
below.

In general, there is a critical time, say $t_{crit}$, for which $\ddot{h}=0$,
that is the acceleration of $h$ vanishes. Note that this does not imply that
the corresponding acceleration for the field $\phi$ vanishes at $t_{crit}$ 888
In fact when $\ddot{h}=0$ we have,
$\displaystyle\ddot{\phi}=\dfrac{1}{2K}\,\dfrac{dK}{d\phi}\,\dot{\phi}^{2}\neq
0\quad.$ (51) . From initial stage of inflation till $t_{crit}$ the
acceleration, and also the velocity, of $h$ are negative, $\ddot{h}\leq 0$,
$\dot{h}<0$. Therefore in this time interval a solution of (17) exists
provided,

$\displaystyle S_{crit}\equiv
U^{\prime}_{eff}(h)+\dfrac{1}{4}\,(2K^{\prime}+3L^{\prime}\,\dot{h}^{2})\,\dot{h}^{2}\,\geq
0$ (52)

This puts an upper bound on the velocity $\dot{h}$ if $L^{\prime}<0$ which is
indeed the case in a variety of models. This is exemplified below for a class
of models that have attracted much interest.

Minimally coupled $h^{n}$ \- models

These models belong to the class  M1, discussed previously, characterized by
constants $g$ and $M^{2}$, and a potential $V(h)$ which is a monomial,

$\displaystyle g=1\quad,\quad M^{2}=1/3\alpha\quad,\quad
V(h)=\dfrac{\lambda}{n}\,h^{n}\quad(n=\text{ positive integer })$ (53)

Then the functions $K,L$ are given by,

$\displaystyle K(h)=(1+ch^{n})^{-1}\quad,\quad L=\alpha\,K\quad.$ (54)

with the constant $c$ being defined by

$\displaystyle c=\dfrac{4\lambda\alpha}{n}\quad.$ (55)

The case $n=2$ belongs to the class of the cosmological attractors
Carrasco:2015pla ; Carrasco:2015rva , which is clearly seen if one uses the
canonically normalized field $\phi$ of (18), see reference Antoniadis:2018ywb
. Thus $L$ is linearly dependent on $K$ through the constant parameter
$\alpha$. The potential $U_{eff}$ is given by

$\displaystyle
U_{eff}(h)=\dfrac{1}{4\,\alpha}\,\dfrac{ch^{n}}{1+ch^{n}}=\dfrac{1}{4\,\alpha}\,(1-K)\quad.$
(56)

In this model the terms $S_{crit}$, defined in (52), receive the form

$\displaystyle
S_{crit}=U^{\prime}_{eff}(h)\,(1-2\alpha\dot{h}^{2}-3(\alpha\dot{h}^{2})^{2})\quad.$
(57)

Due to the fact that the derivative of the potential stays positive until $h$
vanishes for first time, at $t_{0}$, positivity of $S_{crit}$, in the interval
$t\leq t_{crit}$, entails

$\displaystyle\alpha\dot{h}^{2}<\dfrac{1}{3}\quad.$ (58)

which surely applies at $t_{crit}$. Since at $t_{crit}$, where $\ddot{h}=0$,
the velocity gets its minimum value, and it is negative until it vanishes for
first time at $t_{1}$, we have,

$\displaystyle\dot{h}(t)>-\,\dfrac{1}{\sqrt{3\alpha}}\quad,\text{or}\quad\;u(h)>-\,\dfrac{1}{\sqrt{3\alpha}}$
(59)

from begin of inflation until the velocity vanishes for the first time. This
region includes the whole inflationary period and therefore these bounds apply
to end of inflation ! Whether the lower bound above is reached in inflationary
era depends on inputs. Using (46) and the bound (59) one can derive

$\displaystyle h_{end}^{n}\leq\dfrac{5}{3\,c}\quad,$ (60)

in this class of models. For large $\alpha$, corresponding to large $c$, when
$\lambda$ is fixed, the end of inflation value $h_{end}$, is small due to
(60). Note that the bound (59) yields a more stringent lower bound on
$c_{s}^{2}$ than the one given by Eq. (45). In fact one has, using (43),

$\displaystyle c_{s}^{2}>\dfrac{2}{3}$ (61)

which on account of (49) results to

$\displaystyle\rho_{end}<\dfrac{3M^{2}}{4}=\dfrac{1}{4\alpha}$ (62)

which is actually half of the bound given by (50). As we have already stated
the actual upper bound is even less and depends on the precise value of
$c_{s}^{2}$ at the end of inflation. The right hand side of (62) sets the
maximum value the energy density $\rho_{end}$ can reach. Obviously, this is
almost saturated if $c_{s}^{2}\simeq\dfrac{2}{3}$ at end of inflation, which
using (44), yields $R\upsilon^{2}\simeq 1/3$, or $\alpha u(h)^{2}\simeq 1/3$.
That is, at end of inflation the velocity of the $h$-field should be close to
its lowest bound, as this is set by Eq (59), for the energy density to reach
its maximum allowed value in this type of models. Then knowing the velocity at
end of inflation, one can use (27), or same (46), to derive the value of
$h_{end}$. This is found to approach its upper bound (60),
$h_{end}^{n}\simeq\frac{5}{3\,c}$. For large values of the parameter $\alpha$
this is indeed the case in this type of models. That is the bounds derived
previously are saturated for sufficiently large values of $\alpha$. This will
be exemplified in the following, when discussing the bounds set on the
reheating temperature.

The aforementioned bound on $\rho_{end}$ yields in turn bounds for the
instantaneous reheating temperature $T_{ins}$ which is the largest reheating
temperature allowed in any inflation model. From Eq (41) we actually have,

$\displaystyle
T_{ins}=\left(\dfrac{30}{\pi^{2}\,}\dfrac{\,\rho_{end}}{\,g^{*(reh)}\,}\,\,\right)^{1/4}\,\leq\,\left(\dfrac{15}{2\pi^{2}\,\,g^{*(reh)}}\,\,\right)^{1/4}\,\alpha^{-1/4}$
(63)

To convert it to $GeV$ this should be multiplied by the reduced Planck mass
$m_{P}=(8\pi G_{N})^{-1/2}\simeq 2.435\times 10^{18}\,GeV$. With
$g^{*(reh)}=106.75$ this yields the bound

$\displaystyle T_{ins}\leq\,\dfrac{0.290\times
m_{P}}{\,\alpha^{1/4}}\,=\,\dfrac{0.7073\,\times
10^{18}}{\,\alpha^{1/4}}\,GeV$ (64)

In Figure 4 we display the actual upper bound on $T_{ins}$ ( blue solid line
), derived numerically, and the strict mathematical upper bound of Eq. (64), (
dashed red line ), which is based on (62) . The cases displayed correspond to
the minimally coupled model with $n=2$ ( left pane) , with the parameter
$\lambda$ of the scalar potential in Eq. (53) given by $\lambda\equiv m^{2}$,
with $m=6.2\times 10^{-6}$, and the case $n=4$ ( right pane ), corresponding
to $\lambda=2.025\times 10^{-13}$ . These values are consistent with scalar
perturbations, as we shall see.

Note that when $\alpha\lesssim 10^{10}$ the actual bound is lower than the
mathematical upper bound set on $T_{ins}$ and is almost independent of
$\alpha$, depending however on the values of $m$ or $\lambda$. However, the
two bounds coincide for values $\alpha>10^{10}$. The reason is that in this
region of $\alpha$ the sound of speed squared $c_{s}^{2}$, at the end of
inflation, approaches $2/3$, as we have already discussed, and the velocity
approaches its lowest allowed limit, as this is set by (59), resulting to
$\rho_{end}\simeq 1/4\alpha$. Note that in this case the contribution of the
$L$-terms is important in extracting the correct value of the field $h$ at the
end of inflation, $h_{rend}$, and hence the correct values for $\rho_{end}$.
In the regime of small $\alpha$, smaller than $10^{10}$ or so, the sound of
speed squared $c_{s}^{2}$ at the end of inflation, is very close to unity,
that is the contribution of the $L$-terms is indeed negligible and the
dynamics is the same as in ordinary models, that is models which quartic in
the velocity terms are absent. It is only in this case that the usual slow-
roll approximation schemes for extracting $h_{rend}$ can be employed.

The CMB observations restrict considerably the predictions of all inflationary
models. The first calculations were performed in Starobinsky:1979ty ;
Mukhanov:1985rz ; Mukhanov:1988jd  and since then there has been an intense
activity towards improving the calculations, by considering also higher order
corrections, demanded by the precise measurements of the cosmological
parameters, or tackle theories with variable speed of sound, Lucchin:1984yf ;
Stewart:1993bc ; Gong:2001he ; Schwarz:2001vv ; Martin:2002vn ; Habib:2002yi ;
Leach:2002ar ; Habib:2004kc ; Casadio:2004ru ; Wei:2004xx ; Casadio:2005xv ;
Kinney:2007ag ; Lorenz:2008je ; Lorenz:2008et ; Agarwal:2008ah ;
Martin:2013uma ; Jimenez:2013xwa ; Alinea:2015gpa . Therefore the bounds
discussed previously on the instantaneous reheating temperature are narrowed
if the observational constraints, from various astrophysical sources are used.
These actually impose bounds on the parameter $\alpha$, or equivalently on the
scale of inflation. For the derivation of the cosmological parameters, we
shall make use of the number of e-folds $N_{*}$, corresponding to a pivot
scale $k^{*}$, which can be written as Liddle:2003as ; Dodelson:2003vq ,

$\displaystyle N_{*}\,=$
$\displaystyle\,66.89-lnc_{s}^{*}-ln\left(\dfrac{k^{*}}{a_{0}H_{0}}\right)+\dfrac{1}{4}\left(ln\,\dfrac{\,3H_{*}^{2}}{m_{P}^{2}}+ln\,\dfrac{3H_{*}^{2}m_{P}^{2}}{\rho_{end}}\right)-\dfrac{1}{12}\,ln\,g_{s}^{*(reh)}\,$
(65)
$\displaystyle+\dfrac{1-3w}{12(1+w)}\,ln\left(\dfrac{\rho_{reh}}{\rho_{end}}\right)\,.$

See also Martin:2010kz ; Lozanov:2017hjm . The advantage of using this, among
other, is that incorporates information on the reheating temperature through
the last term, the so-called reheating parameter. This does not contribute
when reheating is instantaneous, in which case $\rho_{reh}=\rho_{end}$. It
does not contribute either when $w=1/3$, a well-known result. We have included
in Eq. (65) the dependence on the variable speed of sound and the reduced
Hubble parameter is taken equal to $h=0.676$, for details see see
Gialamas:2019nly . Given a pivot scale $k^{*}$ the end of inflation affects
the $N^{*}$ through $\rho_{end}$. Note that in our approach we solve
numerically the pertinent background equations, to find the values of the
cosmic scale factor and inflaton velocity, as functions of the field values,
and we use (27), or (46), to find where inflation ends. To this goal slow-
roll approximation has not been invoked. In the sequel we solve (65) to find
the pivot values, $\phi^{*}$, corresponding to first horizon crossing of the
given scale $k^{*}$, through which the cosmological parameters are determined.
In the slow-roll approximation this is implemented by using
$N^{*}=\int_{\scriptsize{\phi_{end}}}^{\tiny{\phi^{*}}}\frac{U\,d\phi}{U^{\prime}}$,
where $U$ is the inflaton potential. Concerning the cosmological observables,
for a pivot scale, $k^{*}$ that exited the sound horizon at $t^{*}$, that is
$k^{*}c_{s}(t^{*})=a(t^{*})H(t^{*})$, the scalar and tensor power spectra can
be expanded about this pivot. Keeping the first order terms in the Hubble flow
functions (HFF), one has that the corresponding amplitudes are given by,

$\displaystyle
A_{s}\,=\,\dfrac{H_{*}^{2}}{8\pi^{2}m_{P}^{2}\,\epsilon_{1}^{*}c_{s}^{*}}\,\left(1-2(D+1)\,\epsilon_{1}^{*}-D\epsilon_{2}^{*}-(2+D)\,s_{1}^{*}\right)\,\,,\,A_{t}\,=\,\dfrac{2\,H_{*}^{2}}{\pi^{2}m_{P}^{2}}\,\left(1-2(D+1-lnc_{s}^{*})\,\epsilon_{1}^{*}\right)\,.$
(66)

These can be found in Martin:2013uma , where techniques similar to the WKB
approximation has been used for their derivation. The constant $D$ is given by
$D=7/19-ln3$. From these one has for the tensor-to- scalar ratio,

$\displaystyle
r\equiv\dfrac{A_{t}}{A_{s}}=16\,\epsilon_{1}^{*}\,c_{s}^{*}\,(1+2\,lnc_{s}^{*}\,\epsilon_{1}^{*}+D\epsilon_{2}^{*}+(2+D)\,s_{1}^{*})\quad.$
(67)

Concerning the spectral index, following standard definitions, to same
approximation, we have

$\displaystyle n_{s}\,=\,1-2\,\epsilon_{1}^{*}-\epsilon_{2}^{*}-s_{1}^{*}\,.$
(68)

In the equations given before $\epsilon_{2},s_{1}$, whenever they appear, are
given by $\epsilon_{2}=\dot{\epsilon}_{1}/\epsilon_{1}H$ and
$s_{1}=\dot{c}_{s}/c_{s}H$. For the primordial power spectrum we consider
scales $k$ sampled by Planck CMB observations, lying in the range $\sim
10^{-3}-10^{-1}\,Mpc^{-1}$. The spectrum features small amplitudes $A_{s}\sim
10^{-9}$ and is almost almost scale invariant. 999 Starobinsky-type models in
metric formulation, with non-minimal coupled scalar field, can yield enhanced
primordial curvature perturbations on small scales Pi:2017gih . Such a study
is however outside the scope of this article.

In Figure 5, we display the instantaneous temperature $T_{ins}$ and the bounds
set by other observables, assuming that reheating is instantaneous, that is
$T_{reh}=T_{ins}$. This is drawn in solid blue line with values shown on the
right vertical axis. The gray dashed line is the bound on the instantaneous
temperature discussed previously, see Figure 4. These figures are the
predictions for the minimally coupled model model with potential
$V(h)={m^{2}}\,h^{2}/2$, and slightly different values $m=6.2\times 10^{-6}$
(left pane) and $m=6.8\times 10^{-6}$ (right pane). In these figures, the
amplitude of scalar perturbations $A_{s}$ is displayed ( in magenta ) as
function of the parameter $\alpha$, with values shown on left vertical axis,,
for a pivot scale $k^{*}=0.05\,Mpc^{-1}$. The horizontal stripe, in yellow,
marks the region allowed by Planck 2018 observations on $A_{s}=(2.10\pm
0.03\,)\times 10^{-9}$, Planck:2018jri ; Planck:2018vyg . One notices that
$A_{s}$ puts severe constraints on the parameter $\alpha$, and hence on the
instantaneous temperature as well, given the value of the parameter $m$.

Along we show the lower experimental bound on the spectral index $n_{s}$, as a
red dash-dotted vertical line. The region to the left of this line is within
the Planck 2018 limits $n_{s}=0.9649\pm 0.0042$. The bound $r<0.06$ on the
tensor to scalar ratio $r$, set by various measurements, is also shown as a
vertical blue-dashed line. The allowed region by the $r$-bound stands to the
right of this line, designated by an arrow. The aforementioned bound on $r$ is
pretty close to that established by Planck 2018 data, when combined with the
BICEP2/Keck Array BK15 measurements, see Planck:2018jri ; Ade:2018gkx . The
BICEP/Keck collaboration BICEP:2021xfz  has further put a more tightened bound
$r<0.036$. However we shall prefer to use the more conservative bound
$r<0.06$. Actually using the bound $r<0.036$ will shift very little the blue-
dashed line to the right and therefore the predictions are almost intact.

The combined $n_{s},r$ bounds further narrow the range of $\alpha$-values
allowed by $A_{s}$. We remark that these bounds, unlike those stemming from
$A_{s}$, are rather insensitive to small changes of the parameter $m$ and for
this reason the location of the $r$ and $n_{s}$ bound are almost same in in
the two figures. However $A_{s}$ predictions depend sensitively on $m$. In the
displayed figures the $A_{s}$-curves have a quite different shape, although
the parameter $m$ has only slightly changed. Notice that the case $m=6.8\times
10^{-6}$, corresponding to the figure on the right, can be considered as the
largest allowed by the $A_{s}$ constraint, if the bound on $n_{s}$ is
observed, assuming instantaneous reheating. Actually for larger $m$-values the
allowed by $A_{s}$ region lies to the right of the $n_{s}$ lower observational
limit, shown by the dashed-dotted red line in the figures, and hence excluded.
On the other hand the case $m=6.2\times 10^{-6}$ is the minimum allowed, as
lower values move the allowed $A_{s}$-region to the left of the $r<0.06$
bound. Therefore combining all bounds we have that the allowed window for $m$
is in the range $m\simeq(6.2-6.8)\times 10^{-6}$ with corresponding limits for
$\alpha$ in the range $\alpha\simeq 10^{8}-10^{14}$. These induce the bounds
$T_{ins}\simeq 2.2\times 10^{14}GeV$ and $T_{ins}\simeq 2.3\times 10^{15}GeV$
respectively, with the larger (smaller) bound corresponding to the smaller (
larger ) allowed value of $\alpha$. We remark that for $m$-values in the range
given before, most of the the allowed $\alpha$ range, actually
$\alpha>10^{10}$, is within the regime where the quartic in the velocity terms
are sizable and $c_{s}^{2}\simeq 2/3$ at end of inflation. Note however that
the first horizon crossing of the CMB scales occurs when $c_{s}^{2}\simeq 1$,
that is when the quartic terms are negligible. Therefore, when
$\alpha>10^{10}$, the previously discussed bounds could not have been
predicted in the usual slow-roll approximation schemes. In fact inflaton
leaves the slow-roll regime well before the end of inflation, for such high
values of $\alpha$. As a result, a more delicate study is needed in deriving
the end of inflation energy density $\rho_{end}$ which determines the
instantaneous reheating temperature, which we do in the present work. It may
not have been passed unnoticed that for $\alpha\lesssim 10^{9}$ the amplitude
$A_{s}$, and also the temperature $T_{ins}$, are insensitive to changes in
$\alpha$. This is due to the fact that for such values of $\alpha$ the quartic
in the velocity terms are small and slow-roll is a good approximation scheme.
Then using the slow-roll function $\epsilon_{V}$ to find the the pivot value
$h^{*}$, or $\phi^{*}$,and end of inflation $h_{end}$, or $\phi_{end}^{*}$,
yields results that are accurate enough. That done one finds that
$A_{s},T_{ins}$ depend explicitly on the parameter $m$ and only implicitly on
$\alpha$. That was shown analytically in Gialamas:2019nly . This is the reason
for fixed $m$-values these quantities are almost constant with changes in
$\alpha$. The same holds for other models, as well, namely the Higgs model,
which we shall discuss in the next section.

Therefore, in the framework of Palatini ${\cal{R}}^{2}$ inflation, in the
context of the minimal models with monomial potential $V(h)=m^{2}h^{2}/2$, and
$g=1,M^{2}=1/3\alpha$, the scalar power spectrum amplitude $A_{s}$ constrains
the parameter $m$, which in conjunction with $n_{s},r\,$-data leads to bounds
on $\alpha$. Upper bounds on the instantaneous temperature can be established,
which are saturated for large $\alpha$. For such values of $\alpha$ need go
beyond slow-roll to derive reliable cosmological predictions. Note that by
reinstating units, the plateau values for the potential can be cast in the
form $U_{eff}(h)=3M_{s}^{2}m_{P}^{2}/4$, for a comparison with Starobinsky
inflation, where for the case at hand the inflation scale $M_{s}$ is defined
by $M_{s}=m_{P}/\sqrt{3\,\alpha}$. Then the previously discussed bounds on
$\alpha$ translate to $M_{s}\simeq(10^{-4}-10^{-7})\,m_{P}$. For comparison,
recall that CMB amplitude restricts the Starobinsky scale to
$M_{s}=M_{Starobinsky}\simeq 1.3\times 10^{-5}\,m_{P}$.

So far we have assumed that reheat is instantaneous and derived bounds arising
from cosmological observations. By the same token, similar bounds can be also
established for lower temperatures, as well. The analysis, in this case,
depends on the effective equation of state parameter $w$ . We shall assume
that the latter takes values in the range $w\simeq 0.0-0.25$, with $w=0$
corresponding to the canonical reheating scenario. This range of $w$ is
favoured in some inflationary scenarios, as we have already discussed. For the
minimally coupled model, with potential $V(h)={m^{2}}\,h^{2}/2$, we have found
that the bounds established for the parameter $m$, $\,6.2\times
10^{-6}\lesssim m\lesssim 6.8\times 10^{-6}$, hold for lower temperatures as
well. Values outside this range are ruled out by the cosmological data. In
fact lower $m$-values lead to $10^{9}A_{s}\lesssim 2.0$, while for larger
values, either $A_{s}$ is small or, even when $A_{s}$ is within observational
limits, $n_{s}$ is too low. Hence $m$-values outside the aforementioned range
are discarded. For any $m$, within this range, the amplitude $A_{s}$ and/or
$r$, the tensor to scalar ratio, put lower bounds on $\alpha$, while upper
bounds on $\alpha$ are set by $A_{s}$ and/or $n_{s}$. Given the parameter $m$,
a range of allowed $\alpha$ is therefore established, and for any $\alpha$, in
this range, the reheating temperature $T_{reh}$ takes values within an
interval whose lower/upper limits correspond to the minimum/maximum allowed
$A_{s}$. These limits are larger (smaller) the larger (smaller) the value of
$\alpha$ is, within its allowed range. This is exemplified in Figure 6, where
for the largest and smallest allowed values of $m$, $m_{max}\simeq 6.8\times
10^{-6}$ and $m_{min}\simeq 6.2\times 10^{-6}$, we have drawn the amplitude
$A_{s}$, as function of the reheat temperature, for different values of the
parameter $\alpha$. Each line shown is marked by an integer which denotes the
value of $\log_{10}\alpha$. The right end of each $A_{s}$ \- line stops at the
instantaneous reheating temperature. Whenever a star symbol appears on a line
it is there to indicate the boundary $n_{s}=0.9607$. On the other hand
whenever a red bullet appears it designates the location of the upper bound
$r=0.06$. Points on the line lying to the left of these symbols are not
acceptable. Whenever any of these symbols is absent there is no restriction
stemming from the corresponding bound.

On the left panes of Figure 6, the equation of state parameter is $w=0.0$ and
on the right panes $w=0.25$. For each $\alpha$, the segment of each line
within the yellow stripe, which designates the allowed range $A_{s}=2.10\pm
0.07$, projected onto the $T_{reh}$ \- axis locates the allowed range of
temperatures for this value of $\alpha$. For instance, for the case shown on
top left of this figure, when $\alpha=10^{10}$ the temperature range is
$\simeq 10^{11}-10^{12}\,GeV$. As far as values of $\alpha$ are concerned, for
the same case, the allowed range of $\alpha$ is $\alpha\simeq 10^{8}-10^{14}$.
Lower values are excluded since they violate the $r<0.06$ bound. As a
demonstration of it, we have displayed the $\alpha=10^{7}$ case, by a blue
dashed line, which while being in agreement with $n_{s}$ \- data, and also
$A_{s}$ in some temperature range, the bound on $r$ is violated for any
temperature. This is indicated by the red bullet on the far right of it. The
same holds for lower $\alpha$-values. Therefore only values $\alpha\gtrsim
10^{8}$ are allowed. For the figure at the bottom right, the allowed range is
$\alpha\simeq 10^{10}-10^{14}$. Only values within this range can be
compatible with $A_{s}$ observational limits, as is clearly seen. Note that,
although the $\alpha$ \- range has shrunk, in comparison with the $w=0$ case,
it allows for lower temperatures. This will be discussed in the sequel.

Concerning the highest temperature attainable, for any given value of the
parameter $m$, there is always a narrow range of $\alpha$’s for which their
corresponding instantaneous temperature are cosmologically acceptable, in the
sense of giving predictions compatible with all data. See for instance Fig. 5.
Since the instantaneous temperature $T_{ins}$ drops with increasing $\alpha$,
the lowest of these $\alpha$’s, yields the larger $T_{ins}$ for this
particular value of $m$. On the other hand, the minimum value of the parameter
$m$, $m_{min}\simeq 6.2\times 10^{-6}$, yields the lowest possible $\alpha$.
Therefore, applying this reasoning for the minimum $m=m_{min}$, we pick the
higher possible instantaneous temperature which is the highest possible
reheating temperature. Phrased differently, and with reference Figure 6, for
any given value of the parameter $m$ the higher temperature is the
instantaneous corresponding to the lowest $\alpha$ whose $A_{s}$ \- line ends
up within the yellow band $A_{s}=2.10\pm 0.03$, shown in the figure, provided
all other available data are observed. Thus, the highest possible reheat
temperature is the instantaneous corresponding to the minimum value ,
$m=m_{min}\simeq 6.2\times 10^{-6}$ and the lowest in this case $\alpha$,
having its $A_{s}$ \- line ending inside the yellow stripe, satisfying, also,
the $n_{s},r$ data. This is clearly shown at the bottom panes of Fig. 6 from
which we see that the lowest $\alpha$ lies in a very narrow range around
$\alpha\simeq 10^{8}$, yielding $T_{ins}\simeq 2.30\times 10^{15}\,GeV$. This
result is $w$ \- independent, and for this reason the values of $T_{ins}$ are
same for both left and right figures presented. Therefore the maximal
attainable temperature within the context of this model is $2.30\times
10^{15}\,GeV$, quoted before, as being the largest instantaneous temperature,
of all, for which all data are satisfied. In order to complete the picture,
for the maximum value $m_{max}\simeq 6.8\times 10^{-6}$ the situation is
displayed at the top of Fig. 6. In this case the lowest allowed $\alpha$, with
the desired properties, is $\alpha\simeq 10^{14}$, or somehow lower than it.
This yields as instantaneous temperature $2.23\times 10^{14}\,GeV$, obviously
lower than the one corresponding to $m_{min}$ by an order of magnitude or so.

As far as the lowest temperatures are concerned. It should not have passes
unnoticed that when $w=0$, the available temperatures have lower bounds. From
the left panes in Fig. 6, at both top and bottom, this is clearly seen, and
this is the case for any allowed value of $m$ in the canonical reheating
scenario, i.e. when the equation of state parameter is vanishing. In the $w=0$
case the lowest temperature is reached when $m$ gets its maximum value. From
the top left figure in Fig. 6, we see that this cannot be lower than
$T_{reh}\simeq 10^{8}\,GeV$, obtained when $\alpha\simeq 10^{8}$, the minimum
allowed. For the minimum $m$, the lowest temperature is higher, $T_{reh}\simeq
10^{14}\,GeV$, as seen at the left bottom figure. Thus in this model and for
canonical reheat, lower bounds on $T_{reh}$ are imposed, and temperatures
below $T_{reh}\simeq 10^{8}\,GeV$ are unattainable. However the situation
drastically changes if $w$ is allowed to take non-vanishing values. For
$w=0.25$ and for the maximum $m$, at the top right panel of figure, the lowest
acceptable $\alpha$ is $\simeq 10^{10}$, and temperatures in the range $\simeq
0.01\,MeV-0.6\,GeV$ are allowed 101010 As a general remark, concerning $w$,
the closer the $w$ is to the radiation value $1/3$ the more extended is the
range of the allowed temperatures. Evidently for $w=1/3$, there are no
restrictions imposed on the reheat temperature. . Recall however that, low
temperatures are constrained by BBN, which imposes lower bounds $\simeq 1MeV$
on the reheating temperature. For the minimum $m$ \- case , see right pane at
bottom, lower temperatures, as compared to $w=0$, can be obtained but not
lower than$\simeq 10^{11}\,GeV$.

Figure 4: On the left, we display the bound set on the instantaneous reheating
temperature as function of $\alpha$ for the minimally coupled model with
potential $V(h)={m^{2}}\,h^{2}/2$, and $m=6.2\times 10^{-6}$. On the right we
show the same bound for the quartic potential $V(h)={\lambda}\,h^{4}/4$, and
$\lambda=2.025\times 10^{-13}$. In both figures, the solid (blue) line is the
actual bound as derived numerically and the dashed red line is the bound of
Eq. (64). These coincide when $\alpha$ exceeds $\sim 10^{10}$.

Figure 5: The bound set on the instantaneous reheating temperature, and other
observables, as function of $\alpha$ for the minimally coupled model with
potential $V(h)={m^{2}}\,h^{2}/2$, for $m=6.2\times 10^{-6}$ (left) and
$m=6.8\times 10^{-6}$ (right). In these figures, we display the amplitude of
scalar perturbations $A_{s}$, in magenta, with values on left vertical axis,
as function of the parameter $\alpha$, for $k^{*}$ shown in the figure,
assuming instantaneous reheating, $T_{ins}$, drawn in solid blue line, with
values shown on the right vertical axis. The horizontal stripe ( in yellow )
marks the region allowed by Planck 2018 observations on $A_{s}$. Along we show
the lower observational bound on the spectral index $n_{s}$, red dash-dotted
vertical line line, and the bound $r<0.06$ set on the tensor to scalar ratio,
blue dashed line, in each of the figures.

Figure 6: The amplitude $A_{s}$ as function of the reheat temperature, for
various values of $\alpha$, marked by an integer on each line denoting the
value of $\log_{10}\alpha$, for the minimally coupled model with potential
$V=m^{2}h^{2}/2$. The equation of state parameter equals to $w=0.0$, left
pane, and $w=0.25$, right pane. At top, the parameter $m$ has its highest
allowed value, $m=6.8\times 10^{-6}$ and at bottom $m$ gets its lowest allowed
value, $m=6.2\times 10^{-6}$ The star symbol, whenever appears, indicates the
point for which $n_{s}=0.9607$, the lowest allowed observational bound. The
red bullet, if it appears, marks the boundary $r=0.06$. Points on the line
lying to the left of these symbols are not allowed.  Figure 7: The bound set
on the instantaneous reheating temperature, and other observables, as function
of $\alpha$ for the minimally coupled model with potential
$V(h)=\lambda\,h^{4}/2$, for ${\lambda}\equiv m^{2}$ with $m=4.1\times
10^{-7}$. Along we show the bound $r<0.06$ set on the tensor to scalar ratio,
as a blue dashed line. In the whole range for $\alpha$ the spectral index is
below observational limits.

For the minimally coupled model with quartic potential
$V(h)={\lambda}\,h^{4}/2$ the situation is similar as far as $A_{s}$ and $r$
constraints are concerned. However there is no agreement with $n_{s}$ data. As
an example we display a representative case, in Figure 7, for ${\lambda}\equiv
m^{2}$ with $m=4.1\times 10^{-7}$. As in the $V(h)={m^{2}}\,h^{2}/2$ case
there is a lower bound on $\alpha>10^{8}$, imposed by $r<0.06$, and for the
case displayed there is a also complete agreement with the amplitude $A_{s}$
for $\alpha$ slightly below $10^{10}$. However the spectral index for any
$\alpha$, allowed by $A_{s}$, is $n_{s}<0.9490$, well below the observational
limits. Therefore the quartic potential fails to satisfy all observational
data. The situation changes if the model is promoted to include non-minimal
couplings, as is the case of the Higgs model to be studied in the next
section.

Non-minimally coupled models

A particularly interesting model, belonging to the class  M2 , is the one for
which

$\displaystyle g=1+\xi h^{2}\quad,\quad M^{2}=1/3\alpha\quad,\quad
V(h)=\dfrac{\lambda}{4}\,h^{4}\quad$ (69)

This model is well-known to arise from the Higgs coupling to Palatini gravity
in the unitary gauge when the electroweak scale is considered small. The
parameter $\xi$ sets the coupling of the Higgs field to the curvature terms
and $\alpha$ is the coefficient of the quadratic in the curvature term as
defined in (8).

The Higgs coupling to gravity and its role as the inflaton, in the metric
formulation, has been proposed in Bezrukov:2007ep ; Bezrukov:2008ej  and it
has been widely studied, since then, Barbon:2009ya ; Barvinsky:2009fy ;
Barvinsky:2009ii ; Germani:2010gm ; Germani:2010ux ; Lerner:2010mq ;
Karam:2018mft ; Jinno:2019und ; Bezrukov:2010jz ; Kamada:2010qe ;
Kamada:2012se ; Bezrukov:2013fka ; Allison:2013uaa ; Bezrukov:2014bra ;
Hamada:2014iga ; Hamada:2014wna ; Salvio:2015kka ; Saltas:2015vsc ;
Calmet:2016fsr ; Jinno:2017lun ; Bezrukov:2017dyv ; Antoniadis:2018ywb ;
Antoniadis:2018yfq ; Enckell:2018kkc ; He:2018gyf ; Gundhi:2018wyz ;
Rubio:2018ogq ; Karam:2021sno ; Karam:2021wzz ; Tenkanen:2020cvw ;
Lykkas:2021vax ; Gialamas:2019nly ; Gialamas:2021enw ; Dioguardi:2021fmr ;
He:2018mgb ; Steinwachs:2019hdr ; Rubio:2019ypq ; Takahashi:2018brt ;
Tenkanen:2019jiq ; Cheong:2021vdb ; Hamada:2020kuy ; Rigouzzo:2022yan ;
Durrer:2022emo ; Dux:2022kuk ; He:2022xef  both in the context of the metric
and Palatini formulation. The importance of the ${\cal{R}}^{2}$ coupling, in
the Palatini approach, has been discussed in Antoniadis:2018ywb ;
Antoniadis:2018yfq ; Takahashi:2018brt ; Tenkanen:2019jiq ; Tenkanen:2020cvw ;
Lykkas:2021vax ; Karam:2021sno ; Gialamas:2019nly ; Gialamas:2021enw ;
Dioguardi:2021fmr ; Cheong:2021vdb . Quintessential inflation in the framework
of Palatini ${\cal{R}}^{2}$ gravity has been considered in Dimopoulos:2020pas
; Dimopoulos:2022tvn ; Dimopoulos:2022rdp

In this work we shall show that the quartic coupling $\lambda$, as in the
minimally coupled quartic model studied previously, is constrained
considerably by cosmological data, especially by the power spectrum amplitude
$A_{s}$. Combining all data, further limits are imposed restricting the
available options. The pertinent functions $K,L$ are given by,

$\displaystyle K(h)=\dfrac{1+\xi h^{2}}{(1+\xi h^{2})^{2}+ch^{4}}\quad,\quad
L=\dfrac{\alpha}{(1+\xi h^{2})^{2}+ch^{4}}\quad.$ (70)

while the potential $U_{eff}$ is given by,

$\displaystyle U_{eff}(h)=\dfrac{1}{4\,\alpha}\,\dfrac{ch^{4}}{(1+\xi
h^{2})^{2}+ch^{4}}.$ (71)

The bounds set on the instantaneous temperature are more difficult to derive
in this case due to the dependence on the parameter $\xi$. Although the
analysis is the same, in this case the term in Eq. (52) is not as simple as
that given by (57). For the case at hand we had better used
$\omega=L\dot{h}^{2}/K$ as dependent variable n Eq (20), instead of $u$. Thus
is the same quantity used in (44). That done the pertinent equation takes on
the form,

$\displaystyle(1+3\,\omega)\,\dfrac{d\omega}{dh}-3H\sqrt{\frac{4L}{K}}(1+\omega)\,\omega^{1/2}+{\tilde{S}}_{crit}=0\,,$
(72)

where

$\displaystyle{\tilde{S}}_{crit}=\frac{2L}{K^{2}}\,U^{\prime}_{eff}(h)\left(1-2\omega-3\omega^{2}\right)\quad.$
(73)

Modulo the factor $2L/K^{2}$, this is reminiscent of (57). In (72) we have
anticipated the fact that in the region of interest, from begin of inflation
until the velocity $u$ vanishes for the first time, $u$ is negative, and hence
the negative sign in the Hubble term. Note that $\omega$ starts increasing, as
$h$ decreases, while it vanishes when the velocity $u$ vanishes for the first
time, and therefore a maximum of $\omega_{max}$ is developed at a critical
value $h_{crit}$. At this point $d\omega/dh=0$, while $d\omega/dh\leq 0$ for
any $h\geq h_{crit}$, since $\omega$ increases with decreasing $h$, in this
region. Therefore from (72) we deduce that ${\tilde{S}}_{crit}>0$ for any
$h\geq h_{crit}$, which entails $\omega<1/3$, in this region, and therefore
the maximum value of $\omega$ is bounded, $\omega_{max}<1/3$. Due to the fact
that $\omega_{max}$ is the maximum value we have

$\displaystyle\omega<1/3$ (74)

for any $h$ from begin of inflation until $u$ it vanishes. This region
certainly includes the end of inflation $h_{end}$ and this puts an upper bound
on the value of $\omega_{end}\equiv\omega(h_{end})$ at end of inflation, i.e.

$\displaystyle\omega_{end}<1/3\quad.$ (75)

Note that this is the analog of (58) for the minimally coupled models studied
earlier. The bound (74) yields again the bound of (61), $c_{s}^{2}>2/3$, which
results to (62), $\rho_{end}<1/4\alpha$ . Therefore the bound (64) on the
instantaneous temperature holds in the Higgs case as well.

For sufficiently large $\alpha$, as in the case $\xi=0$, the bound
$\omega_{end}\simeq 1/3$ is saturated. Then $\rho_{end}=1/4\alpha$ and the
upper bound on $T_{ins}$ of Eq. (64) is reached. With the aid of end of
inflation relation (46) we get in a straightforward manner $RU_{eff}=5/12$, at
end of inflation, or using the form of the potential as given in Eq. (15),
$R_{end}=8\alpha/3$. This relation can be solved for $h_{end}$, to derive the
value of $h_{end}$, when the upper bound on $T_{ins}$ is reached. The solution
is,

$\displaystyle(\sqrt{3c/5}-\xi)\,h_{end}^{2}=1\;.$ (76)

Note that this holds for sufficiently large $\alpha$. We have verified that Eq
(76) indeed reproduces very accurately the numerical results for $h_{end}$,
for values $\alpha>\alpha_{crit}$, with $\alpha_{crit}$ given by where the
coefficient of $h_{end}^{2}$ on the left hand side of (76) vanishes. This
results to a critical value
$\alpha_{crit}=\frac{5}{3}\,{\left(\frac{\xi^{2}}{\lambda}\right)}$. A last
comment concerns the value of $\xi\,h_{end}^{2}$ which turns out to be much
smaller than unity in the regime $\alpha\gg\alpha_{crit}$. In this case
$g(h_{end})\simeq 1$ and hence predictions are expected to be same as with the
$\xi=0$ case. Anticipating the fact that the Higgs model can be in agreement
with cosmological observations, as we shall discuss in the sequel, this by no
means should lead us to the wrong conclusion that the simple quartic
potential, $\xi=0$, seen as the limiting case of the Higgs model, when
$\xi\,h_{end}^{2}\ll 1$, can lead to successful inflation, In fact
$\xi\,h_{end}^{2}={\left(\sqrt{\alpha/\alpha_{crit}}-1\right)}^{-1}$, and one
needs $\alpha\geq 10^{4}\,\alpha_{crit}$, or so, to obtain indeed small values
$\xi\,h_{end}^{2}\leq 10^{-2}$. Such values for $\alpha$ outstrip the lower
observational bound on $n_{s}$, as shown in Figures 8,9, in which some
representative outputs are displayed, and therefore are not acceptable. This
is in perfect consistency with the statement made towards the end of previous
sub-section that the simple quartic potential, $\xi=0$ case, is in tension
with $n_{s}$, predicting too low values for the spectral index.

Predictions of the Higgs Model when the parameter $\xi$ is small, $\xi=0.1$,
are shown at top of Figure 8. Denoting the quartic coupling by
$\lambda=m^{2}$, the cases shown correspond to the lowest, $m=2.85\times
10^{-6}$ , and largest, $m=3.05\times 10^{-6}$ allowed, when all observational
data are observed and reheating is instantaneous. At the bottom of the same
figure the case $\xi=1.0$ is displayed. In this case the minimum and maximum
allowed $m$\- values are $m=9.10\times 10^{-6}$ and $m=9.70\times 10^{-6}$
respectively.

Given $\xi$, the min/max values of $m$ and the ranges for the parameter
$\alpha$ are shown in Table 1. The corresponding ranges for the instantaneous
temperature and the cosmological observables $A_{s},n_{s},r,$ and the number
of efolds $N_{*}$, are also shown corresponding to a pivot scale
$k^{*}=0.05\,Mpc^{-1}$. In all cases the value of $r$ is small enough to
satisfy the more stringent bound $r<0.036$ put by BICEP/Keck observations
BICEP:2021xfz . For the maximum allowed value of $m$, the spectral index
$n_{s}$ is close to its lowest value, allowed by cosmological observations,
while $\alpha$ gets its largest value. At the same time, the instantaneous
reheating temperature gets its lowest value. These $\alpha$-values are in the
regime where the quartic in the velocity terms contribute substantially and
slow-roll approximation is not applicable. Thus need go beyond slow-roll to
derive these predictions, which has been done numerically. The lowest
$m$-values allow for a broad range $\alpha\lesssim 10^{9}$ and instantaneous
temperature is larger. In this case, the inflationary dynamics can be
successfully described by the slow-roll mechanism. The constancy of $A_{s}$
for the lowest allowed $m$-values actually follows from slow-roll. Using slow-
roll approximation $A_{s}$ is found to depend on the ratio $c/\alpha$, that is
on the parameter $m$ alone and not on $\alpha$. Note that for the lowest $m$
case the tensor to scalar ratio is of the order of $r\sim 10^{-3}$, much
larger, by almost three orders of magnitude, from the largest $m$-case. These
values will be therefore within reach by future missions aiming to probe small
values of $r$ as small as $r\sim 10^{-3}$, or so, Kogut:2011xw ;
Matsumura:2016sri .

One observes that the instantaneous temperature indeed touches the bound (64)
for sufficiently large values of $\alpha$, larger than some critical value
$\alpha_{crit}$ that indeed coincides with the one estimated previously, after
Eq. (76), for each of the cases considered. Actually, $\alpha_{crit}$
separates the two regimes. The slow-roll and the region of $\alpha$ for which
the contribution of the quartic in the velocity terms is important in
determining the end of inflation parameters. For low $\alpha<\alpha_{crit}$
the instantaneous temperature stays well below the bound (64) and is rather
insensitive to changes with $\alpha$, for fixed values of $\xi,m$. This region
belongs to the slow-roll regime and all data, including $T_{ins}$, can be
derived accurately enough employing the the well-known methodology based on
the slow-roll functions. For high $\alpha>\alpha_{crit}$ slow-roll is not a
valid approximation as we approach the end of inflation, and one should rely
on a numerical treatment.

Lastly, in Figure 9, we display the bound set on the instantaneous reheating
temperature, and other observables, for the Higgs model, for a large value
$\xi=10^{5}$. The case displayed corresponds to $m=3.05\times 10^{-3}$ which
is the lowest allowed by all data for this $\xi$. This is actually fine-tuned
since by slightly increasing the value of $m$ will move the $A_{s}$
predictions off its experimental bounds. Note that this is a case where the
entire allowed region falls within the slow-roll regime.

Concluding this section, we state that for the Higgs model, as for the
quadratic potential, $V\sim h^{2}$ discussed in previous sections, and under
the same conditions, the bounds derived when reheat is instantaneous, hold for
lower temperatures, as well. In fact, for given $\xi$ the range of the quartic
coupling $\lambda$, which we derived assuming instantaneous reheating, hold
true for lower temperatures, as well. The situation concerning bounds on the
reheat temperature $T_{reh}$, and on the parameter $\alpha$, are similar to
those of the minimally coupled model studied in the previous section, and thus
need not be discussed in detail. However there is an important difference,
regarding the bound $r<0.06$ which is much weaker in the non-minimal Higgs
case ( $\xi\neq 0$ ), allowing for much lower values of $\alpha$, as low as
$\alpha\simeq 10$, or even lower. Recall that in the case of the quadratic
potential $V\sim h^{2}$, discussed in previous sections, $\alpha$ cannot be
lower than about $10^{8}$, due mainly to the aforementioned bound on the
tensor to scalar ratio.

Figure 8: As in Figure 5, we display the bound set on the instantaneous
reheating temperature, and other observables, for the Higgs model,
$V(h)={\lambda}\,h^{4}/4$. We have set $\lambda=m^{2}$ and the figures on top
correspond to $\xi=0.1$. For the top left pane $m=2.85\times 10^{-6}$ and for
the right $m=3.05\times 10^{-6}$. These are the lowest and highest allowed
$m$-values when $\xi=0.1$. The figures at bottom correspond to $\xi=1.0$. For
the bottom left pane $m=9.10\times 10^{-6}$ and for the right $m=9.70\times
10^{-6}$. These are the lowest and highest allowed when $\xi=1.0$. Note that
in all cases displayed the bound for $r$ has been moved to the far left so
that all $\alpha$-region . is consistent with the bound $r<0.06$ set on the
tensor to scalar ratio.  Higgs Model ( pivot scale $k^{*}=0.05\,Mpc^{-1}$ )
---
Value of $\quad\xi=0.1$
min/max value of $\;\;m$ : |  |  $m=2.85\times 10^{-6}$ |  |  $m=3.05\times 10^{-6}$
$\;\;\;\quad\alpha\,$ |  $10$ \- $10^{9}$ |  | $\simeq 10^{13}$
$\;\;\;\;T_{ins}\,$ |  $2.05\times 10^{15}$ \- $1.97\times 10^{15}$ |  | $3.82\times 10^{14}$
$\;\;\;10^{9}\,A_{s}\,$ |  $2.12\;$ \- $2.07$ |  | $2.08$
$\;\;\;\;\;n_{s}\,$ |  $0.9638$ \- $0.9636$ |  | $0.9609$
$\;\;\;\;\;r\,$ |  $0.006$ \- $0.004$ |  | $\sim 10^{-6}$
$\;\;\;\;N_{*}\,$ |  $56.11\;$ \- $\;55.86$ |  | $53.31$
Value of $\quad\xi=1.0$
min/max value of $\;\;m$ : |  |  $m=9.10\times 10^{-6}$ |  |  $m=9.70\times 10^{-6}$
$\;\;\;\quad\alpha\,$ |  $10$ \- $10^{10}$ |  | $\simeq 10^{14}$
$\;\;\;\;T_{ins}\,$ |  $1.70\times 10^{15}$ \- $1.53\times 10^{15}$ |  | $2.21\times 10^{14}$
$\;\;\;10^{9}\,A_{s}\,$ |  $2.10\;$ \- $2.07$ |  | $2.12$
$\;\;\;\;\;n_{s}\,$ |  $0.9632$ \- $0.9630$ |  | $0.9610$
$\;\;\;\;\;r\,$ |  $0.0007$ \- $0.0004$ |  | $\sim 10^{-7}$
$\;\;\;\;N_{*}\,$ |  $55.17\;$ \- $\;54.98$ |  | $52.71$
Table 1: Predictions of the Higgs Model , for $\xi=0.1$ and $\xi=1.0$ , when
$m$ gets its minimum and maximum allowed value allowed by all data, when
reheating is instantaneous. The quartic coupling is $\lambda=m^{2}$. These are
actually the cases displayed in Figure 8. For the lowest $m$ values allowed, a
broad range of $\alpha$ is allowed, which is shrunk when $m$ gets its maximum
value. The corresponding ranges for the instantaneous temperature and the
cosmological observables $A_{s},n_{s},r,$, as well as the number of efolds
$N_{*}$, are also shown corresponding to a pivot scale $k^{*}=0.05\,Mpc^{-1}$.
Figure 9:  As in in Figure 8, we display the bound set on the instantaneous
reheating temperature, and other observables, for the Higgs model, for a large
value $\xi=10^{5}$. The case displayed corresponds to $m=3.05\times 10^{-3}$
which is the lowest allowed by all data for this value of $\xi$. A slight
increase of $m$ will move the $A_{s}$ predictions off experimental bounds.
Note that in this case the allowed region belongs to the slow-roll regime.

Summary of results -Discussion

The main results reached, for the models studied in this work, are summarized
in the following.

For the Model I we have found that the allowed values for the mass parameter
$m$ lie within rather tight limits

$\displaystyle 6.2\times 10^{-6}\lesssim m\lesssim 6.8\times 10^{-6}\,,$ (77)

while the range of acceptable $\alpha$ is

$\displaystyle 10^{8}\lesssim\alpha\lesssim 10^{14}\,.$ (78)

Values of $\alpha<10^{7}$ are not allowed, violating the bound set on $r$, and
higher values $\alpha>10^{14}$ are incompatible with $n_{s}>0.9607$. We remark
that within the range (78), a critical value exists, $\alpha_{crit}\simeq
10^{10}$, and for values of $\alpha$ beyond that the contribution of the
quartic in the velocity terms are important in the determination of the end of
inflation parameters. This is beyond the slow - roll regime. For such values
of $\alpha$ the speed of sound approaches $c_{s}^{2}\simeq 2/3$, its lowest
mathematical bound in this kind of theories, and the upper bound set on the
instantaneous temperature is saturated.

For any given $m$, within the range (77), there exists a range of
$\alpha$-values, within (78), which are compatible with cosmological data.
Assuming instantaneous reheating, and when $m$ gets its lowest value, in range
(77), only values $\alpha\simeq 10^{8}$ are allowed. In this case we have the
largest possible instantaneous temperature $T_{ins}=2.30\times 10^{15}\,GeV$.
For this case $r\simeq 0.05$, that is close to its upper bound, and the
spectral index is $n_{s}\simeq 0.9648$, that is it approaches its highest
observational value. On the other hand, when $m$ receives its largest value,
within the range (77), the allowed $\alpha$ are near the upper limit of (78),
$\alpha\simeq 10^{14}$. In this case we have the lowest possible instantaneous
temperature $T_{ins}\simeq 2.23\times 10^{14}\,GeV$ while the value of $r$ is
very tiny $r\simeq 3\times 10^{-8}$, and $n_{s}\simeq 0.9610$, that is close
to its lowest observational bound.

Concerning the value of the tensor to scalar ratio $r$, this model is known to
be in tension with $r$, predicting unacceptably large values in metric
formulation. However the situation is rescued in the Palatini formulation
thanks to the appearance of $\alpha\,{\cal{R}}^{2}$ terms in the action.

For the Higgs model, an additional parameter $\xi$ exists, which controls the
non-minimal coupling of inflaton to gravity. Given the parameter $\xi$, bounds
are set on the quartic Higgs coupling $\lambda$, defined in (69), in the same
way as for the Model I. For the benchmark values $\xi=0.1,1.0$ and
$\xi=10^{5}$, considered in this work, the allowed ranges for the parameter
$m$, where $\lambda=m^{2}$, are as follows

$\displaystyle\xi=0.1\quad$ $\displaystyle:$ $\displaystyle\quad 2.85\times
10^{-6}\lesssim m\lesssim 3.05\times 10^{-6}$ $\displaystyle\xi=1.0\quad$
$\displaystyle:$ $\displaystyle\quad 9.10\times 10^{-6}\lesssim m\lesssim
9.70\times 10^{-6}$ (79) $\displaystyle\xi=10^{5}\quad$ $\displaystyle:$
$\displaystyle\quad\quad\simeq 3.05\times 10^{-3}$

while the range of acceptable $\alpha$ is

$\displaystyle\alpha\lesssim 10^{14}\,.$ (80)

Values of $\alpha>10^{14}$ are incompatible with $n_{s}>0.9607$. Note that
there is no lower bound on $\alpha$ arising from $r<0.06$, that is the
$r$-bound weakens in the Higgs case. For the case of Higgs, a critical value
exists, as well, given by $\alpha_{crit}=\frac{5}{3}\,(\frac{\xi}{m})^{2}$,
and for values of $\alpha$ larger than $\alpha_{crit}$ the contribution of the
quartic in the velocity terms are important, as in the case of Model I. In
fact, for $\alpha>\alpha_{crit}$ the speed of sound approaches
$c_{s}^{2}\simeq 2/3$, and the upper bound set on the instantaneous
temperature is saturated. Note however that $\alpha_{crit}$ exceeds the upper
limit (80) for the case $\xi=10^{5}$. This means that the contribution of the
quartic in the velocity terms are negligible for values of $\alpha$ that are
of physicsl interest to us, for this value of $\xi$.

For any $m$ within (79), ranges of $\alpha$-values exist that are compatible
with the cosmological data. For $\xi=0.1$, the range of allowed $\alpha$ is
$10-10^{9}$, when $m$ gets its lowest value $m=2.85\times 10^{-6}$, with
ranges of instantaneous temperature $T_{ins}=2.05\times 10^{15}-1.97\times
10^{14}\,GeV$. The value $T_{ins}=2.05\times 10^{15}\,GeV$ is the largest
possible. The tensor to scalar ratio $r$ is in the range $r\simeq
0.006-0.004$, the highest values obtained in the Higgs model, while
$n_{s}\simeq 0.9638-0.9636$. For the largest value $m=3.05\times 10^{-6}$ we
can only have $\alpha\simeq 10^{13}$ and in this case $T_{ins}=3.82\times
10^{14}\,GeV$, while $r\simeq 8.0\times 10^{-7}$, which is pretty small, and
$n_{s}\simeq 0.9609$.

Passing to $\xi=1.0$ case, for the smallest $m$, $m=9.10\times 10^{-6}$, the
allowed range of $\alpha$ is $10-10^{10}$ with $T_{ins}=1.70\times
10^{15}-1.53\times 10^{15}\,GeV$ and values of $r$ in the range $r\simeq
7.0\times 10^{-4}-4.0\times 10^{-4}$. As for the spectar index $n_{s}\simeq
0.9632-0.9630$. For the largest allowed $m$, $m=9.70\times 10^{-6}$ the
$\alpha$ parameter is around $\alpha\simeq 10^{14}$, and in this case
$T_{ins}=2.21\times 10^{14}\,GeV$, while $r$ is very low $r\simeq 8.0\times
10^{-8}$, and $n_{s}\simeq 0.9610$

Finally for the case $\xi=10^{5}$, $m$ is almost fine tuned to $m\simeq
3.05\times 10^{-3}$. The allowed $\alpha$ \- values span the region
$\alpha\simeq 10-10^{14}$, with $T_{ins}=1.23\times 10^{14}-1.21\times
10^{13}\,GeV$, while $r$ is extremely low $r\simeq 7.5\times 10^{-9}$ and
$n_{s}\simeq 0.9608$.

Before leaving this section, we shall present a brief account of the metric
and Palatini formulations of gravity. In Palatini gravity the affine
connection $\Gamma$ is an independent variable and it is through the equations
of motion that is connected to the metric $g_{\mu\nu}$. In the absence of
higher-$R$ terms in the action, and if additional scalars are present, that
are minimally coupled to gravity the affinity $\Gamma$ becomes the well-known
Levi-Civita connection $\Gamma=\Gamma(g)$ ( Christoffel symbols ). In this
case the two theories yield identical results. However when non-minimal
couplings exist and/or higher in the curvature $R$ terms are present the two
theories differ.

In the present work we have considered ${\cal{R}}^{2}$-terms coupled to
Palatini gravity as $\alpha\,{\cal{R}}^{2}$ with $\alpha$ a constant. In the
framework of metric gravity this theory describes, besides gravitons, the
dynamics of a scalar propagating degree of freedom, the scalaron with mass
$\sim 1/\sqrt{\alpha}$. This is best seen in the Einstein frame where the
scalaron appears as a scalar field, which plays the role of the inflaton,
moving under the influence of the Starobinsky potential, which is predicted
and not put in by hand. In the framework of Palatini gravity the inclusion of
$\alpha\,{\cal{R}}^{2}$ terms differs from the Starobinsky case of metric
formulation. There are no extra propagating degrees of freedom and the would
be inflaton, as well as its potential, have to be introduced explicitly. Note
that the inclusion of additional scalars in the metric formulation, when
$\alpha\,{\cal{R}}^{2}$ is present, leads to multifield inflation in the
Einstein frame.

The inclusion of $\alpha\,{\cal{R}}^{2}$ in Palatini action has two important
consequences. It flattens the scalar potential in the Einstein frame, even for
a steep original potential $V(h)$, and a plateau is created, for large field
values, which can sustain inflation. Besides, it induces quadratic terms in
the velocity which affect the inflaton evolution towards the end of inflation
for large values of the coupling $\alpha$, larger than some critical value.
Both of these features are absent in the limit of $\alpha$ tending to zero,
that is for sufficiently small couplings, and thus are expected to play little
role in the small $\alpha$-regime. Note that the absence of
$\alpha\,{\cal{R}}^{2}$ terms does not imply that metric and Palatini
formulations lead to equivalent theories. In fact, if non-minimal couplings
are present, as in the Higgs case for instance, the two formalisms lead to
different actions in the Einstein frame and thus to different inflationary
models.

On the phenomenological side, the inclusion of $\alpha\,{\cal{R}}^{2}$ terms
is well known to lower the value of $r$ for any inflationary model
Antoniadis:2018ywb ; Antoniadis:2018yfq ; Enckell:2018hmo  . The quadratic
potential $\sim h^{2}$ belongs to this class which given unacceptably values
for $r$ in metric formulation but it survives in the Palatini gravity, as we
have seen. Also in the Higgs case the values of $r$ are systematically lower
than the Higgs inflation in metric formulation. Also the values of the non-
minimal coupling $\xi$ can take much lower values in the Palatini formulation.

##  VI Conclusions

We have studied models of ${\cal{R}}^{2}$-inflation in the framework of
Palatini Gravity, where the coupling of the ${\cal{R}}^{2}$ to gravity is of
the form $\sim\alpha\,{\cal{R}}^{2}$, with $\alpha$ is constant. The
appearance of terms which are quartic in the velocity of inflaton is
unavoidable in these theories. These play little role, as being very small,
during first horizon crossing, however they may play an important role in
determining the end of inflation dynamics, and in particular the instantaneous
reheating temperature, when the scale $\alpha$ is large. We have found that
there is some critical value of $\alpha$ below which the mechanism of
inflation follows slow-roll during the whole inflation era. However above
this, the inflationary evolution deviates from slow-roll, as inflaton
approaches the end of inflation, defined by where the acceleration $\ddot{a}$
of the cosmic scale factor vanishes. In these case the determination of end of
inflaton is inaccurate when slow-roll is employed. In a class of popular
models, the speed of sound is bounded by $c_{s}^{2}\geq 2/3$ putting upper
bounds on the instantaneous reheating temperature $T_{ins}$, given by
$T_{ins}\leq{0.290\,m_{Planck}}/{\,\alpha^{1/4}}$. These bounds are saturated
for large values of $\alpha$.

Assuming instantaneous reheating, we have derived bounds on the parameters of
the models studied in this work, arising from the amplitude of the scalar
power spectrum $A_{s}$, the spectral index $n_{s}$ and the tensor to scalar
ratio $r$. The instantaneous reheating temperature cannot be arbitrarily small
since the scale $\alpha$ is bounded from above by observations, imposing in
turn lower bounds on the inflationary scale. For the models considered it is
found that $\alpha$ can not exceed $\alpha\simeq 10^{14}$, arising from
$n_{s}$-data, which for the largest $\alpha$ touches its lowest observational
limit.

When reheating is not instantaneous predictions depend on the value of the
effective equation of state parameter $w$. However, the constraints on the
coupling of the scalar potential remain the same. For reasonable values of
$w$, in the range $w\simeq 0.0-0.25$, the allowed values of $\alpha$ lie in a
certain range, constraining the reheat temperature $T_{reh}$ to be within a
range dictated by the value of $\alpha$ taken. When the coupling of the scalar
potential gets its minimum allowed value we obtain the highest possible
temperature. This is independent of the value of $w$ taken. In fact, this is
the instantaneous temperature corresponding to the smallest allowed value of
$\alpha$, which receives its lowest possible value allowed by all data in this
case. Then temperatures as large as $\sim 10^{15}\,GeV$ can be reached, in
principle, for the models studied in this work. Low values can be also
obtained, as low as $\simeq MeV$, when the equation of state parameter is
$w\simeq 0.25$, or higher. The acceptable values of $\alpha$ in this case are
larger than $\sim 10^{10}$, within the regime where slow-roll is not
applicable. In this case need go beyond slow-roll for a reliable cosmological
study.

Acknowledgments A.B.L. wishes to thank I. D.Gialamas for discussions at the
initial stages of this work. He also thanks V. C. Spanos and K. Tamvakis for
illuminating discussions.

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