# Reheating constraints on modified quadratic chaotic inflation

Sudhava Yadav1, Rajesh Goswami2, K.K Venkataratnam1,****Corresponding author
and Urjit A. Yajnik3,4
1 Department of Physics, Malaviya National Institute of Technology Jaipur,
Jaipur 302017, India
2 Department of Physics, National Institute of Technology Puducherry, Karaikal
609609, India
3 Department of Physics, Indian Institute of Technology Bombay, Mumbai 400076,
India
4 Department of Physics, Indian Institute of Technology Gandhinagar, Gujarat
382424, India

###### Abstract

The Reheating era of inflationary Universe can be parameterized by various
parameters like reheating temperature $T_{\text{re}}$, reheating duration
$N_{\text{re}}$ and average equation of state parameter
$\overline{\omega}_{\text{re}}$, which can be constrained by observationally
feasible values of scalar power spectral amplitude $A_{\text{s}}$ and spectral
index $n_{\text{s}}$. In this work, by considering the quadratic chaotic
inflationary potential with logarithmic-correction in mass, we examine the
reheating era in order to place some limits on model’s parameter space. By
investigating the reheating epoch using Planck’s 2018 data, we show that even
a small correction can make the quadratic chaotic model consistent with latest
cosmological observations. We also find that the study of reheating era helps
to put much tighter constraints on model and effectively improves accuracy of
model.

##  1 Introduction

The inflationary paradigm [1, 2, 3, 4, 5] is an exciting and influential epoch
of the cosmological universe. It has come up as an aid to resolve a range of
well-known cosmological problems like flatness, horizon and monopole problems
of famous cosmological big bang theory. The semi-classical theory of inflation
generates seeds for Cosmic Microwave Background anisotropy and Large Scale
Structures in the late universe [6, 7, 8]. Inflation predicts adiabatic,
gaussian and almost scale invariant density fluctuations, which are validated
by CMB observations like Cosmic Background Explorer (COBE) [9], Wilkinson
Microwave Anisotropy Probe (WMAP) [10, 11] and Planck space probe [12, 13, 14,
15, 16, 17].

In the realm of inflationary cosmology, a typical scenario involves the
presence of a scalar field, which is referred to as the inflaton $(\phi)$,
whose potential energy dominates the universe. In this picture, inflaton
slowly rolls through its potential, and the coupling of quantum fluctuations
of this scalar field with metric fluctuations is the source of primordial
density perturbations called scalar perturbations. The tensor part of the
metric has vacuum fluctuations resulting in primordial gravitational waves
called tensor perturbations. During inflation, power spectra for both these
perturbations depend on a potential called inflaton potential $V(\phi)$.

As Inflation ends, the universe reaches a highly nonthermal and frigid state
with no matter content in it. However, the universe must be thermalized at
extremely high temperature for big-bang nucleosynthesis (BBN) and
baryogenesis. This is attained by ‘reheating’[18, 19, 20, 21, 22, 23, 24],
transit between the inflationary phase and an era of radiation and matter
dominance.

There is no established science for reheating era and there is also a lack of
direct observational data in favor of reheating. However, recent CMB data
helped to obtain indirect bounds for various reheating parameters [25, 26, 27,
28, 29, 30, 31], and those parameters are: the reheating temperature
$\left(T_{\text{re}}\right)$, the effective equation of state (EoS) parameter
during reheating ($\omega_{\text{re}}$) and lastly, the reheating duration,
which can be written in the form of number of e-folds
$\left(N_{\text{re}}\right.$). It is challenging to bound the reheating
temperature by LSS and CMB observations. However, its value is assumed to be
higher than the electroweak scale for dark matter production at a weak scale.
A lower limit has been set on reheat temperature i.e. $\left(T_{\text{re}}\sim
10^{-2}GeV\right.$) for a successful primodial nucleosynthesis (BBN) [32] and
instantaneous reheating consideration allows us to put an upper bound i.e.
$\left(T_{\text{re}}\sim 10^{16}GeV\right.$) for Planck’s recent upper bound
on tensor-to-scalar ratio (r). The value of second parameter,
$\omega_{\text{re}}$, shifts from $-\frac{1}{3}$ to 1 in various scenarios. It
is 0 for reheating generated by perturbative decay of a large inflaton and
$\frac{1}{3}$ for instantaneous reheating. The next parameter in line is the
duration of reheating phase, $N_{re}$. Generally, it is incorporated by giving
a range of $N_{k}$, the number of e-foldings from Hubble crossing of a Fourier
mode $k$ to the termination of inflation. $N_{k}$ has value in the range 46 to
70 in order to work out the horizon problem. These bounds arise by considering
reheat temperature at electroweak scale and instantaneous reheating of the
universe. A comprehensive analysis of higher bound on $N_{k}$ is presented in
[33, 34].

The relation between inflationary parameters and reheating can be derived by
taking into consideration the progression of observable scales of cosmology
from the moment of their Hubble crossing during inflation to the current time.
We can deduce relations among $T_{\text{re}},N_{\text{re}}$ and
$\omega_{\text{re}}$, the scalar power spectrum amplitude
$\left(A_{s}\right.$) and spectral index $n_{s}$ for single-field inflationary
models. Further, the constraints on $T_{\text{re}}$ and $N_{\text{re}}$ can be
obtained from recent CMB data.

Although plenty of inflationary models have been studied in recent years[35]
and the inflationary predictions are in agreement with the recent CMB
observations, there is still a need for a unique model. The most famous
chaotic inflation with quadratic potential
$\left(\frac{1}{2}m^{2}\phi^{2}\right)$ is eliminated by recent cosmological
observations as it predicts large tensor perturbations due to large potential
energy it has during inflaton at large field amplitudes. Hence, lowering the
potential at higher field values can help getting rid of this obstacle.
Numerous hypotheses in this vein have been put forth [36, 37, 38, 39, 40, 41,
42, 43, 44, 45]. Radiative corrections provide an intriguing possibility [36,
37, 38] where, generally, the quadratic potential gets flatter as result of
running of inflaton’s quartic coupling. This article will rather examine a
straightforward scenario in which the mass exhibits a running behaviour
described as [46]:

$m^{2}(\phi)=m^{2}\left(1-\text{K}\ln\left[\frac{\phi^{2}}{M^{2}}\right]\right),$
(1)

where M is large mass scale and K is some positive constant. The positive K
and the negative sign in above equation is a defining characteristic of
dominance of the coupling of inflaton field to fermion fields. Another
interesting way to make such models compatible with observations is by
extension of standard model as done in Ref. [47, 48].
Reheating is well known technique of constraining the inflationary models.
There are various ways to analyse the reheating phase as available in
literature e.g. one stage reheating study [49, 31], two stage reheating study
[50, 51]. In Ref. [52] reheating was analysed through perturvative decay of
inflaton to either bosonic or fermionic states through trilinear coupling
[52]. Considering one stage reheating technique of constraining the models, we
use various reheating parameters to put much tighter bounds on parameter space
of quadratic chaotic inflationary model with a logarithmic-correction in mass
in light of Planck’s 2018 data [16, 17]. By demanding $T_{\text{re}}>100$ GeV
for production of weak-scale dark matter and working in plausible range of
average equation of state (EoS) parameter
($-\frac{1}{3}\leq\overline{\omega}_{\text{re}}\leq 1$), we employ the derived
relation between inflationary and reheating parameters and observationally
feasible values of $A_{s}$, $n_{s}$ and r to place a limit on model’s
parameter space. It is a helpful and fairly new tool for putting relatively
tighter constraints on the model and reducing its viable parameter space,
providing significant improvement in accuracy of the model. Additionally, this
technique well differentiate various inflation models as they can have the
same forecasts for $n_{s}$ and r, but definitely not for the same
$\omega_{\text{re}}$, as the tightened constraints on $n_{s}$ will result in
an increasingly narrow permitted range of $\omega_{\text{re}}$ for a
particular inflationary model.

The organization of this paper is as follows: In Sec. 2 we discuss the
dynamics and predictions of slow-roll inflation. We also derived the
expressions for $T_{\text{re}}$ and $N_{\text{re}}$ as a function of
$\overline{\omega}_{\text{re}}$ and other inflationary parameters like
($\text{$\Delta$N}_{k}$ and $V_{\text{end}}$). In section 3, the Subsec. 3.1
has our recreated data for reheating scenario of simple quadratic chaotic
potential. In Subsec. 3.2, we discussed the various field domains within which
inflation can occur for quadratic chaotic potential with logarithmic
correction in mass and then we parameterized reheating for this model using
$T_{\text{re}}$ and $N_{\text{re}}$ as a function of the scalar spectral index
$n_{s}$ for different $\omega_{\text{re}}$. We have also examined the
observational limits and reheating parameters for both these models using
Planck 2018 data in Sec. 3. Sec. 4 is reserved for discussion and conclusions.

We will be working with $\hbar$ $=c=1$ units and the values of some standard
parameters used are reduced Planck’s mass $M_{P}=\sqrt{\frac{1}{8\pi G}}$
$=\text{}$2.435 $\times$ $10^{18}$ GeV, the redshift of matter radiation
equality $z_{\text{eq}}=$3402, $g_{\text{re}}$ $\approx$ 100 [27] and the
present value of Hubble parameter $H_{o}=100$h km $s^{-1}$ $\text{Mpc}^{-1}$
with h = 0.68 [16, 17]

##  2 Parameterizing reheating in slow-roll inflationary models

Reheating phase can be parameterized by assuming it been dominated by some
fluid [53] of energy density $\rho$ with pressure P and equation of state(EoS)
parameter $\omega_{\text{re}}=\frac{P}{\rho}$ where

$\rho=\frac{\dot{\phi}^{2}}{2}+V(\phi),\;\;\;\;\;\;\;\;\;P=\frac{\dot{\phi}^{2}}{2}-V(\phi).$
(2)

The continuity equation gives

$\dot{\rho}+3H(P+\rho)=0,$ (3)

$\dot{\rho}+3H\rho\left(\omega_{\text{re}}+1\right)=0.$ (4)

We analyze the dynamics of inflation by considering inflaton $\phi$ with
potential $V(\phi)$ evolving slowly with slow-roll parameters $\epsilon$ and
$\eta$. The approximation of Friedman equation using slow-roll conditions give

$3H\dot{\phi}+V^{\prime}(\phi)=0,$ (5) $H^{2}=\frac{V(\phi)}{3M_{P}^{2}},$ (6)

where prime$(^{\prime})$ denotes derivative w.r.t $\phi$ and H =
$\frac{\dot{a}}{a}$ is Hubble parameter. The definition of slow-roll parameter
give

$\epsilon=\frac{M_{P}^{2}}{2}\left(\frac{V^{\prime}}{V}\right)^{2},\;\;\;\;\;\;\;\;\;\;\;\eta=M_{P}^{2}\left(\frac{V^{\prime\prime}}{V}\right).$
(7)

The scalar spectral index $n_{s}$, tensor spectral index $n_{T}$ and tensor to
scalar ratio $r$ in terms of above slow-roll parameters satisfy the relations

$\text{
}n_{s}=1-6\epsilon+2\eta,\;\;\;\;\;\;\;n_{T}=-2\epsilon,\;\;\;\;\;\;\;r=16\epsilon.$
(8)

Now, the number of e-foldings in between Hubble crossing of mode $k$ and
termination of inflation denoted by subscript “end” can be given as

$\Delta
N_{k}=\ln\left(\frac{a_{\text{end}}}{a_{\text{k}}}\right)=\frac{1}{M_{P}^{2}}\int_{\phi_{\text{end}}}^{\phi_{k}}\frac{V}{V^{\prime}}\,d\phi,$
(9)

where $a_{k}$ and $\phi_{k}$ represents value of scale factor and inflaton at
the point of time when $k$ crosses the Hubble radius. The later part of eq.
(9) is obtained using the slow-roll approximations $\ddot{\phi}\ll
3H\dot{\phi}$ and $V(\phi)\gg\dot{\phi}^{2}$. Similarly,

$N_{\text{re}}=\ln\left(\frac{a_{\text{re}}}{a_{\text{end}}}\right),$ (10)

Here the quantity $N_{\text{re}}$ encrypts both, an era of preheating [22, 54,
55, 56, 57, 58] as well as later thermalization process. An energy density
controls the Universe’s subsequent evolution and can be written as

$\text{ }\rho_{\text{re}}=\frac{\pi^{2}}{30}g_{\text{re}}T_{\text{re}}^{4},$
(11)

where $g_{\text{re}}$ gives the actual count of relativistic species at
termination of reheating epoch and $T_{\text{re}}$ is the reheating
temperature. Now, in view of eq. (3)

$\rho_{\text{re}}=\rho_{\text{end}}e^{-3N_{\text{re}}\left(1+\overline{\omega}_{\text{re}}\right)},$
(12) $\text{where
}{}{}\overline{\omega}_{\text{re}}=<\omega>=\frac{1}{N_{\text{re}}}\int_{N_{\text{end}}}^{N}\omega_{\text{re}}(N)dN,$
(13)

Here $\overline{\omega}_{\text{re}}$ is average EoS parameter during reheating
[25, 49]. Now, eq. (12) can be recast as

$\frac{a_{\text{re}}}{a_{\text{end}}}=e^{N_{\text{re}}}=\left(\frac{\rho_{\text{re}}}{\rho_{\text{end}}}\right)^{-\frac{1}{3(1+\overline{\omega}_{\text{re}})}}.$
(14)

Using eq. (11) and eq. (14), reheating e-folds $N_{\text{re}}$ can be written
as

$N_{\text{re}}=\frac{1}{3\left(1+\overline{\omega}_{\text{re}}\right)}\left\\{\ln\left(\frac{3}{2}V_{\text{end}}\right)-\ln\left(\frac{\pi^{2}}{30}g_{\text{re}}\right)\right\\}-\frac{4}{3\left(1+\overline{\omega}_{\text{re}}\right)}\ln\left(T_{\text{re}}\right).$
(15)

For some physical scale $k$, the observed wavenumber ‘$\frac{k}{a}$’ can be
given in terms of above known quantities and the redshift during matter-
radiation equality epoch ($z_{\text{eq}}$) as [49]

$H_{k}=\frac{k}{a_{k}}=\left(1+z_{\text{eq}}\right)\frac{k}{a_{o}}\rho_{\text{re}}^{\frac{3~{}\overline{\omega}_{\text{re}}-1}{12\left(1+\overline{\omega}_{\text{re}}\right)}}\rho_{\text{eq}}{}^{-\frac{1}{4}}\left(\frac{3}{2}V_{\text{end}}\right)^{\frac{1}{3\left(1+\overline{\omega}_{\text{re}}\right)}}e^{\Delta
N_{k}}.$ (16)

Using eq. (16), $\Delta N_{k}$ can be given as

$\Delta N_{k}=\ln
H_{k}-\ln\left(1+z_{\text{eq}}\right)-\ln\left(\frac{k}{a_{o}}\right)-\frac{1}{\left.3(\overline{\omega}_{\text{re}}+1\right)}\ln\left(\frac{3}{2}V_{\text{end}}\right)-\frac{3~{}\overline{\omega}_{\text{re}}-1}{3\left(1+\overline{\omega}_{\text{re}}\right)}\ln\left(\rho_{\text{re}}^{\frac{1}{4}}\right)+\ln\left(\rho_{\text{eq}}^{\frac{1}{4}}\right).$
(17)

Inverting eq. (17), and using eq. (11) one can get a mutual relation among the
numerous parameters introduced,

$\ln(T_{\text{re}})=\frac{3~{}\left(1+\overline{\omega}_{\text{re}}\right)}{3~{}\overline{\omega}_{\text{re}}-1}\left\\{\ln
H_{k}-\ln\left(1+z_{\text{eq}}\right)-\ln\frac{k}{a_{o}}-\Delta
N_{k}+\ln\left(\rho_{\text{eq}}^{\frac{1}{4}}\right)\right\\}-\frac{1}{3~{}\overline{\omega}_{re}-1}\ln\left(\frac{3}{2}V_{end}\right)-\frac{1}{4}\ln\left(\frac{\pi^{2}}{30}g_{re}\right).$
(18)

The expression for $T_{\text{re}}$ from eq. (15) is substituted in eq. (18) to
get the expression for $N_{\text{re}}$ as mentioned below

$N_{\text{re}}=\frac{1}{3~{}\overline{\omega}_{\text{re}}-1}\ln\left(\frac{3}{2}V_{\text{end}}\right)+\frac{4}{3~{}\overline{\omega}_{\text{re}}-1}\left\\{\ln\left(\frac{k}{a_{o}}\right)+\Delta
N_{k}+\ln\left(1+z_{\text{eq}}\right)-\ln\left(\rho_{\text{eq}}^{\frac{1}{4}}\right)-\ln
H_{k}\right\\}.$ (19)

eq. (18) and eq. (19) are the two key relationships for parameterizing
reheating in slow-roll inflationary models.

##  3 Inflationary models

###  3.1 Quadratic Chaotic inflationary model

We are first considering simple quadratic chaotic potential before moving to
its modified form. The quadratic chaotic potential [4] has the form

$V=\frac{1}{2}m^{2}\phi^{2}.$ (20)

The reheating study of this potential was already done in [49] in light of
Planck’s 2015 data, we are recreating the data by doing the similar study
using Planck’s 2018 data. Using eq. (7) slow-roll parameters for this
potential can be given as

$\epsilon=\eta=\frac{2M_{P}^{2}}{\phi^{2}}.$ (21)

The Hubble parameter during the crossing of Hubble radius by scale $k$ for
this model can be written as

$H_{k}^{2}=\frac{1}{M_{P}^{2}}\left(\frac{V_{k}}{3-\epsilon_{k}}\right)=\frac{1}{2M_{P}^{2}}\left(\frac{m^{2}\phi_{k}^{2}}{3-2\frac{M_{P}^{2}}{\phi^{2}}}\right).$
(22)

where $\phi_{k}$, $\epsilon_{k}$ and $V_{k}$ respectively represent the
inflaton field, slow-roll parameter and potential during crossing of Hubble
radius by mode $k$.
Using the condition $\epsilon=1$, defining end of inflation, in eq. (21), we
obtained $\frac{\phi_{\text{end}}^{2}}{M_{P}^{2}}=2$
Now, corresponding to pivot scale $k_{*}$, used in Planck collaboration,
$\frac{k_{*}}{a_{o}}=0.05Mpc^{-1}$, consider the mode $k_{*}$ crossing the
hubble radius at a point where the field has achieved the value $\phi_{*}$
during inflation. The remaining number of e-folds persist subsequent to
crossing of hubble radius by $k_{*}$ are

$\Delta
N_{*}\simeq\frac{1}{M_{P}^{2}}\int_{\phi_{\text{end}}}^{\phi_{*}}\frac{V}{V^{\prime}}\,d\phi\\\
\\\ =\left[\left(\frac{\phi_{*}}{2M_{P}}\right)^{2}-\frac{1}{2}\right].$ (23)

The spectral index for this model can be easily obtained using eq. (8) as

$n_{s}=1-8\left(\frac{M_{P}^{2}}{\phi_{*}^{2}}\right).$ (24)

Now, the formulation for tensor-to-scalar ratio from eq. (8) gives

$r=32\frac{M_{P}^{2}}{\phi_{*}^{2}}.$ (25)

Moreover, this model yields the relation

$H_{*}=\pi M_{P}\sqrt{16A_{s}\frac{M_{P}^{2}}{\phi_{*}^{2}}}.$ (26)

The relation of field $\phi$ and $H$ eq. (6), and the condition for
termination of inflation as used in eq. (23), along with eq. (26) gives
expression for $V_{\text{end}}$ as

$V_{\text{end}}(\phi)=\frac{1}{2}m^{2}\phi_{\text{end}}^{2}=\frac{3H_{*}^{2}M_{P}^{2}\phi_{\text{end}}^{2}}{\phi_{*}^{2}}=\frac{6H_{*}^{2}M_{P}^{4}}{\phi_{*}^{2}}.$
(27)

Figure 1: The plots for (a) $T_{\text{re}}$ and (b) $N_{\text{re}}$ versus
$n_{s}$ for the quadratic chaotic model $V(\phi)=\frac{1}{2}m^{2}\phi^{2}$ for
different values of $\overline{\omega}_{\text{re}}$ :
$\overline{\omega}_{\text{re}}$=$-\frac{1}{3}$(dotted orange),
$\overline{\omega}_{\text{re}}$= 0(medium dashed purple),
$\overline{\omega}_{\text{re}}$=$\frac{1}{6}$(solid green),
$\overline{\omega}_{\text{re}}$=$\frac{2}{3}$(large dashed red),
$\overline{\omega}_{\text{re}}$= 1(small dashed blue). The regions with light
and dark gray shades represent the Planck 2018 (TE+EE+TT+Low E+Lensing) [16,
17] 2$\sigma$ and 1$\sigma$ bounds on $n_{s}$ respectively.

Now, the expressions for $\Delta$$N_{*}$, $r$, $H_{*}$ and $V_{\text{end}}$ as
a function of $n_{s\text{ }}$ can be obtained by putting the value of
$\phi_{*}$ from eq. (24) in eqs. (23), (25), (26) and (27), and then these
expressions along with eqs. (18) and (19) gives number of reheating e-folds
$N_{\text{re}}$ and reheating temperature $T_{\text{re}}$. Planck’s 2018 value
of $\text{ }A_{s}=2.1\times 10^{-9}$ and computed value of
$\rho_{\text{eq}}^{\frac{1}{4}}=10^{-9}$GeV [16, 17] have been used for
calculation. The $N_{\text{re}}$ and $T_{\text{re}}$ versus $n_{s}$ plots,
along with Planck-2018 $1\sigma$ bound on $n_{s}$ i.e.
$n_{s}\text{=0.965$\pm$0.004}$ (dark gray) and $2\sigma$ bound on $n_{s}$ i.e.
$n_{s}\text{=0.965$\pm$0.008}$ (light gray), for this model are presented
graphically in figure 1 for a range of average EoS parameter during reheating.
By demanding $T_{\text{re}}\geq 100$ GeV for production of weak-scale dark
matter and solving eqs. (18) and (24), the bounds on $n_{s}$ are obtained and
are reflected on eq. (23) and eq. (25) to obtain bounds on $\Delta$$N_{*}$ and
r. All the obtained bounds are shown in table 1. For this model the bounds on
$n_{s}$ lies inside Planck-2018 $2\sigma$ bound demanding
$\overline{\omega}_{\text{re}}$ lies in the range
($-0.03\leq\overline{\omega}_{\text{re}}\leq 1$) and the corresponding range
for r is (0.172 $\geq$ r $\geq$ 0.117) while if we demand $n_{s}$ to lie
within $1\sigma$ bound by Planck 2018 observation then the allowed range of
$\overline{\omega}_{\text{re}}$ is ($0.09\leq\overline{\omega}_{\text{re}}\leq
0.67$) and the corresponding r values are (0.156 $\geq$ r $\geq$ 0.124).
Within these ranges of $\overline{\omega}_{\text{re}}$ the tensor-to-scalar
ratio (r) is greater than the combined BICEP2/Keck and Planck’s bound
($r<0.06)$ [59].

Table 1: The permissible range for values of $n_{s}$, $\Delta N_{*}$ and $r$ for Quadratic Chaotic inflationary potential $\left(\frac{1}{2}m^{2}\phi^{2}\right)$ by demanding $T_{re}\geq 100GeV$ $\left(\frac{1}{2}m^{2}\phi^{2}\right)$ | Average Equation of state | $n_{s}$ | $\Delta N_{*}$ | $r$
---|---|---|---|---
$-\frac{1}{3}\leq\overline{\omega}_{re}\leq 0$ | $0.926\leq n_{s}\leq 0.958$ | $26.47\leq\Delta N_{*}\leq 47.45$ | $0.297\geq r\geq 0.166$
$0\leq\overline{\omega}_{re}\leq\frac{1}{6}$ | $0.958\leq n_{s}\leq 0.963$ | $47.45\leq\Delta N_{*}\leq 53.38$ | $0.166\geq r\geq 0.148$
$\frac{1}{6}\leq\overline{\omega}_{re}\leq\frac{2}{3}$ | $0.963\leq n_{s}\leq 0.969$ | $53.38\leq\Delta N_{*}\leq 63.99$ | $0.148\geq r\geq 0.124$
$\frac{2}{3}\leq\overline{\omega}_{re}\leq 1$ | $0.969\leq n_{s}\leq 0.971$ | $63.99\leq\Delta N_{*}\leq 68.10$ | $0.124\geq r\geq 0.117$

From figure 1(a), we can see for Planck’s 2018 $1\sigma$ bound on
$n_{s}\text{(0.965$\pm$0.004}$), curves
($-\frac{1}{3}\leq\overline{\omega}_{\text{re}}\leq 0$) and
($\frac{2}{3}\leq\overline{\omega}_{\text{re}}\leq 1$) predicts
$T_{\text{re}}>2.082\times 10^{6}$ GeV and $T_{\text{re}}>86.08$ GeV
respectively while all values of reheating temperature are possible for
$0.135\leq\overline{\omega}_{re}\leq 0.577$.
From table 1, we can see that all the r values are greater than the combined
BICEP2/Keck and Planck’s bound ($r<0.06)$ [59]. Hence, this model is
incompatible with the data for any choice of $\overline{\omega}_{re}$ taken.

###  3.2 Modified quadratic chaotic inflation

The quadratic chaotic inflationary potential with logarithmic-correction in
mass term has the form [38, 46]

$V(\phi)=\frac{1}{2}m^{2}\left(1-\text{K}\ln\frac{\phi^{2}}{M_{P}^{2}}\right)\phi^{2}=(M^{\prime})^{4}\left(1-\text{K}\ln\frac{\phi^{2}}{M_{P}^{2}}\right)\frac{\phi^{2}}{M_{P}^{2}},$
(28)

where $(M^{\prime})^{4}=m^{2}M_{P}^{2}/2$ and K is some positive constant. The
positive K is a defining characteristic of dominance of fermion couplings.
This work is inspired by Ref. [46], where the inflationary scenario of this
potential was studied. We are considering this potential in context of
reheating in light of Planck’s 2018 data.

Figure 2: The plot of potential versus $\frac{\phi}{M_{P}}$ for quadratic
chaotic model with corrected mass
$V(\phi)=\frac{1}{2}m^{2}\left(1-\text{K}\ln\frac{\phi^{2}}{M_{P}^{2}}\right)\phi^{2}$
for different values of K : K=0.13 (large dashed red), K=0.14 (solid green),
K=0.15 (dot-dashed blue), K=0.16 (small dashed orange). The small circle on
each curve signifies the maxima of potential in each case and the
corresponding field value at maxima is $\frac{\phi_{Top}}{M_{P}}$.

We will start our discussion with various field domains [35] within which
inflationary phenomena may occur for above potential. It is evident that the
above-mentioned potential eq. (28) does not exhibit positive definiteness for
all values of the field (${\phi}$). The value of this potential becomes
negative after a specific point

$\frac{\phi_{V=0}}{M_{P}}=\sqrt{e^{\frac{1}{\text{K}}}}.$ (29)

The model can only be defined within a specific regime i.e.,
$\phi<\phi_{V=0}$. On the contrary, the highest point of the potential
function, where $V^{\prime}=0$ (or can say $\epsilon=0)$, corresponds to field
value given as:

$\frac{\phi_{V^{\prime}=0}}{M_{P}}=\frac{\phi_{Top}}{M_{P}}=\sqrt{e^{\frac{1-\text{K}}{\text{K}}}},$
(30)

The model has a sense provided the correction term doesn’t have its dominance
on the potential, hence the suitable regime is $\phi<\phi_{Top}<\phi_{V=0}$.
The potential versus $\frac{\phi}{M_{P}}$ plot for four different values of K
is depicted in figure 2. From figure 2 it can be seen that each K has specific
viable regime in which the model is defined and have a sense and we will be
working in these regions only.
Now moving further, the slow-roll parameters for this potential can be given
as

$\epsilon=2M_{P}^{2}\left(\frac{1-\text{K}\left(1+\ln\frac{\phi^{2}}{M_{P}^{2}}\right)}{\phi(1-\text{K}\ln\frac{\phi^{2}}{M_{P}^{2}}}\right)^{2},$
(31)

$\eta=\frac{2M_{P}^{2}\left(-3\text{K}+1-\text{K}\ln\frac{\phi^{2}}{M_{P}^{2}}\right)}{\phi^{2}\left(1-\text{K}\ln\frac{\phi^{2}}{M_{P}^{2}}\right)}.$
(32)

After substituting the values of $\epsilon$ eq. (31) and $\eta$ eq. (32) in
eq. (8), we can write scalar spectral index as

$n_{s}=\frac{-4\left(2-3\text{K}+3\text{K}^{2}\right)+\frac{\phi^{2}}{M_{P}^{2}}-2\text{K}\left(-8+6\text{K}+\frac{\phi^{2}}{M_{P}^{2}}\right)\ln\frac{\phi^{2}}{M_{P}^{2}}+\text{K}^{2}\left(-8+\frac{\phi^{2}}{M_{P}^{2}}\right)\ln\left[\frac{\phi^{2}}{M_{P}^{2}}\right]^{2}}{\frac{\phi^{2}}{M_{P}^{2}}\left(-1+\text{K}\ln\frac{\phi^{2}}{M_{P}^{2}}\right)^{2}}.$
(33)

The Hubble parameter during the crossing of Hubble radius by scale $k$ can be
written as

$H_{k}^{2}=\frac{1}{M_{P}^{2}}\left(\frac{V_{k}}{3-\epsilon_{k}}\right)=\left(\frac{\frac{1}{2}m^{2}\left(1-\text{K}\ln\frac{\phi_{k}^{2}}{M_{P}^{2}}\right)\frac{\phi_{k}^{2}}{M_{P}^{2}}}{3-2M_{P}^{2}\left(\frac{1-\text{K}\left(1+\ln\frac{\phi^{2}}{M_{P}^{2}}\right)}{\phi\left(1-\text{K}\ln\frac{\phi^{2}}{M_{P}^{2}}\right)}\right)^{2}}\right).$
(34)

Using the condition $\epsilon=1$ defining end of inflation, we have obtained
$\frac{\phi_{\text{end}}}{M_{P}}$ for different values of K. The remaining
number of e-folds persist subsequent to crossing of hubble radius by $k_{*}$
till the termination of inflationary epoch can be given as

$\text{$\Delta$}N_{*}\simeq\frac{1}{M_{P}^{2}}\int_{\phi_{\text{end}}}^{\phi_{*}}\frac{V}{V^{\prime}}\,d\phi_{*}=\frac{1}{2M_{P}^{2}}\int_{\phi_{\text{end}}}^{\phi_{*}}\frac{\left(1-\text{K
{ln}}\frac{\phi^{2}}{M_{P}^{2}}\right)\phi}{1-\text{K}\left(1+\text{ln}\frac{\phi^{2}}{M_{P}^{2}}\right)}\,d\phi.$
(35)

Defining $\frac{\phi_{*}}{M_{P}}=x$. The spectral index $n_{s}$ eq. (33), at
$\phi=\phi_{*}$ in terms of $x$ will have the form

$n_{s}=\frac{-4\left(2-3\text{K}+3\text{K}^{2}\right)+x^{2}-2\text{K}\left(-8+6\text{K}+x^{2}\right)\ln
x^{2}+\text{K}^{2}\left(-8+x^{2}\right)\ln\left[x^{2}\right]^{2}}{x^{2}\left(-1+\text{K}\ln
x^{2}\right)^{2}}.$ (36)

Figure 3: The plots of (a) $\frac{\phi_{*}}{M_{P}}$ and (b)
$\text{$\Delta$N}_{*}$ versus $n_{s}$ for quadratic chaotic model with
corrected mass
$V(\phi)=\frac{1}{2}m^{2}\left(1-\text{K}\ln\frac{\phi^{2}}{M_{P}^{2}}\right)\phi^{2}$
for different values of K. The shaded regions and color codings are same as in
figure 1 and 2 respectively.

;

Figure 4: The plots for $T_{\text{re}}$ and $N_{\text{re}}$ versus spectral
index ($n_{s}$) for quadratic chaotic model with corrected mass
$V(\phi)=\frac{1}{2}m^{2}\left(1-\text{K}\ln\frac{\phi^{2}}{M_{P}^{2}}\right)\phi^{2}$
for different K and $\overline{\omega}_{\text{re}}$ values. The shaded regions
and color codings are same as in figure 1

The variation of $\frac{\phi_{*}}{M_{P}}$ and $\text{$\Delta$N}_{*}$ with
$n_{s}$ using eq. (36) and eq. (35) for 4 different values of K are shown in
figure 3a and 3b respectively. Further in this model, we can write the tensor
- to - scalar ratio and $H_{*}$ as

$r=32\left(\frac{1-\text{K}\left(1+\ln x^{2}\right)}{x\left(1-\text{K}\ln
x^{2}\right)}\right)^{2}.$ (37)

$H_{*}=4\pi M_{P}\sqrt{A_{s}}\left(\frac{1-\text{K}\left(1+\ln
x^{2}\right)}{x\left(1-\text{K}\ln x^{2}\right)}\right).$ (38)

Defining $\frac{\phi_{\text{end}}}{M_{P}}=y$. The relation of field $\phi$ and
$H$, and the condition for termination of inflation, along with eq. (38) gives
expression for $V_{\text{end}}$ in terms of $x$ and $y$ as

$V_{\text{end}}(\phi)=\frac{1}{2}m^{2}\left(1-\text{K}\ln\left(\frac{\phi_{\text{end}}^{2}}{M_{P}^{2}}\right)\right)\phi_{\text{end}}^{2}\\\
\\\ \text{ }=\frac{3H_{*}^{2}M_{P}^{2}\left(1-\text{K}\ln
y^{2}\right)y^{2}}{x^{2}\left(1-\text{K}\ln x^{2}\right)}\text{ }.$ (39)

Now, the expressions for $\Delta$$N_{*}$, $r$, $H_{*}$ and $V_{\text{end}}$ as
function of $n_{s\text{ }}$ can be obtained by putting the value of $x$ from
eq. (36) and the value of y obtained using the condition for termination of
inflation ($\epsilon=1$) in eqs. (35), (37), (38) and (39), and then these
expressions along with eqs. (18) and(19) gives number of reheating e-folds
$N_{\text{re}}$ and reheating temperature $T_{\text{re}}$. The $N_{\text{re}}$
and $T_{\text{re}}$ versus $n_{s}$ plots, along with Planck-2018 bounds, for 4
different K values for this model are presented graphically in figure 4.
By demanding $T_{\text{re}}>100$ GeV for production of weak-scale dark matter
and solving eqs. (18) and (36), the bounds on $n_{s}$ are obtained and are
reflected on eq. (35) and eq. (37) to obtain bounds on $\Delta$$N_{*}$ and r.
All the obtained bounds for various choices of K are shown in table (2). The r
versus $n_{s}$ plots, along with Planck-2018 bounds, for a range of K values
are presented graphically in figure 5. The figure 5 shows that the tensor- to
scalar ratio is greater than the viable range ($r<0.06)$ for ${\text{K}}\leq
0.13$ while for ${\text{K}}>0.16$ the $n_{s}$ value is outside Planck’s 2018
bound for any choice of $\overline{\omega}_{\text{re}}$. The value
$\text{K}=0$ gives us the normal quadratic chaotic potential. The allowed
range of K satisfying Planck’s 2018 constraints on both $n_{s}$ and r for this
model is found to be ($0.13<{\text{K}}\leq 0.16$). Individually, the range of
$\overline{\omega}_{\text{re}}$ for which our obtained data is compatible with
Planck-2018 $2\sigma$ bounds on $n_{s}$ and the combined BICEP2/Keck and
Planck’s bound on r, i.e. ($r<0.06)$ gives
($0.352<\overline{\omega}_{\text{re}}\leq 1$) for K=0.14 and
($0.009\leq\overline{\omega}_{\text{re}}\leq 1$) for K=0.15. For
${\text{K}}\leq 0.13$, there is no compatibility with data for any value of
$\overline{\omega}_{\text{re}}$ taken and K=0.16 is compatible with data for
only $\overline{\omega}_{\text{re}}$=1 . Similarly, the $1\sigma$ bound on
$n_{s}$ and $r<0.06$ gives ($0.352<\overline{\omega}_{\text{re}}\leq 0.640$)
for K=0.14 and ($0.212\leq\overline{\omega}_{\text{re}}\leq 1$) for K=0.15
while K=0.13 and K=0.16 are completely outside the combined $1\sigma$ and r
bounds.

Table 2: The permissible range for $n_{s}$, $\Delta N_{*}$ and $r$ for different K values for modified Quadratic Chaotic inflation by demanding $T_{re}\geq 100GeV$ K = $0.13$ | Average Equation of state | $n_{s}$ | $\Delta N_{*}$ | $r$
---|---|---|---|---
$-\frac{1}{3}\leq\overline{\omega}_{re}\leq 0$ | $0.932\leq n_{s}\leq 0.962$ | $26.07\leq\Delta N_{*}\leq 47.03$ | $0.195\geq r\geq 0.096$
$0\leq\overline{\omega}_{re}\leq\frac{1}{6}$ | $0.962\leq n_{s}\leq 0.966$ | $47.03\leq\Delta N_{*}\leq 52.95$ | $0.096\geq r\geq 0.083$
$\frac{1}{6}\leq\overline{\omega}_{re}\leq\frac{2}{3}$ | $0.966\leq n_{s}\leq 0.971$ | $52.95\leq\Delta N_{*}\leq 63.56$ | $0.083\geq r\geq 0.066$
$\frac{2}{3}\leq\overline{\omega}_{re}\leq 1$ | $0.971\leq n_{s}\leq 0.973$ | $63.56\leq\Delta N_{*}\leq 67.68$ | $0.066\geq r\geq 0.060$
4 K = $0.14$ | $-\frac{1}{3}\leq\overline{\omega}_{re}\leq 0$ | $0.931\leq n_{s}\leq 0.960$ | $26.02\leq\Delta N_{*}\leq 46.94$ | $0.175\geq r\geq 0.080$
$0\leq\overline{\omega}_{re}\leq\frac{1}{6}$ | $0.960\leq n_{s}\leq 0.964$ | $46.94\leq\Delta N_{*}\leq 52.86$ | $0.080\geq r\geq 0.068$
$\frac{1}{6}\leq\overline{\omega}_{re}\leq\frac{2}{3}$ | $0.964\leq n_{s}\leq 0.969$ | $52.86\leq\Delta N_{*}\leq 63.46$ | $0.068\geq r\geq 0.052$
$\frac{2}{3}\leq\overline{\omega}_{re}\leq 1$ | $0.969\leq n_{s}\leq 0.971$ | $63.46\leq\Delta N_{*}\leq 67.56$ | $0.052\geq r\geq 0.047$
4 K = $0.15$ | $-\frac{1}{3}\leq\overline{\omega}_{re}\leq 0$ | $0.929\leq n_{s}\leq 0.957$ | $25.95\leq\Delta N_{*}\leq 46.84$ | $0.152\geq r\geq 0.063$
$0\leq\overline{\omega}_{re}\leq\frac{1}{6}$ | $0.957\leq n_{s}\leq 0.960$ | $46.84\leq\Delta N_{*}\leq 52.74$ | $0.063\geq r\geq 0.052$
$\frac{1}{6}\leq\overline{\omega}_{re}\leq\frac{2}{3}$ | $0.960\leq n_{s}\leq 0.965$ | $52.74\leq\Delta N_{*}\leq 63.31$ | $0.052\geq r\geq 0.037$
$\frac{2}{3}\leq\overline{\omega}_{re}\leq 1$ | $0.965\leq n_{s}\leq 0.966$ | $63.31\leq\Delta N_{*}\leq 67.40$ | $0.037\geq r\geq 0.033$
4 K = $0.16$ | $-\frac{1}{3}\leq\overline{\omega}_{re}\leq 0$ | $0.925\leq n_{s}\leq 0.950$ | $25.88\leq\Delta N_{*}\leq 46.71$ | $0.129\geq r\geq 0.044$
$0\leq\overline{\omega}_{re}\leq\frac{1}{6}$ | $0.950\leq n_{s}\leq 0.953$ | $46.71\leq\Delta N_{*}\leq 52.60$ | $0.044\geq r\geq 0.034$
$\frac{1}{6}\leq\overline{\omega}_{re}\leq\frac{2}{3}$ | $0.953\leq n_{s}\leq 0.956$ | $52.60\leq\Delta N_{*}\leq 63.11$ | $0.034\geq r\geq 0.022$
$\frac{2}{3}\leq\overline{\omega}_{re}\leq 1$ | $0.956\leq n_{s}\leq 0.957$ | $63.11\leq\Delta N_{*}\leq 67.18$ | $0.022\geq r\geq 0.019$
Figure 5: The r versus $n_{s}$ plot for range of K value with different
$\overline{\omega}_{\text{re}}$:
$-\frac{1}{3}\leq\overline{\omega}_{re}<0$(dot-dashed red),
$0\leq\overline{\omega}_{re}\leq\frac{1}{6}$(dotted blue),
$\frac{1}{6}<\overline{\omega}_{re}\leq 1$(solid green). The region with
bluish shade corresponds to ($r<0.06$), the combined BICEP2/Keck and Planck’s
bound [59, 16] on r while the gray regions are same as figure 1. Table 3: The
range of $T_{re}$ and $N_{re}$ for potential
$V(\phi)=\frac{1}{2}m^{2}\left(1-\text{K}\ln\frac{\phi^{2}}{M_{P}^{2}}\right)\phi^{2}$
which shows compatibility with Planck’s 2018 1$\sigma$ bound on $n_{s}$.

| K $=0.13$ | K $=0.14$ | K $=0.15$
---|---|---|---
|  $T_{re}$(GeV) | $N_{re}$ |  $T_{re}$(GeV) | $N_{re}$ |  $T_{re}$(GeV) | $N_{re}$
$\overline{\omega}_{re}=-\frac{1}{3}$ | $T_{re}\geq 5.51\times 10^{10}$ | $N_{re}\leq 22.09$ | $T_{re}\geq 3.58\times 10^{11}$ | $N_{re}\leq 18.30$ | $T_{re}\geq 1.38\times 10^{14}$ | $N_{re}\leq 5.86$
$\overline{\omega}_{re}=0$ | $T_{re}\geq 14.06$ | $N_{re}\leq 44.18$ | $T_{re}\geq 4.04\times 10^{3}$ | $N_{re}\leq 36.60$ | $T_{re}\geq 3.92\times 10^{11}$ | $N_{re}\leq 11.72$
$\overline{\omega}_{re}=\frac{1}{6}$ | $10^{16}\geq T_{re}\geq 10^{-2}$ | $N_{re}\leq 88.36$ | $10^{16}\geq T_{re}\geq 10^{-2}$ | $N_{re}\leq 73.21$ | $T_{re}\geq 3.19\times 10^{6}$ | $N_{re}\leq 23.44$
$\overline{\omega}_{re}=\frac{2}{3}$ | $T_{re}\geq 7.12\times 10^{11}$ | $N_{re}\leq 6.67$ | $T_{re}\geq 249.61$ | $N_{re}\leq 23.95$ | $10^{16}\geq T_{re}\geq 10^{-2}$ | $N_{re}\leq 94.11$
$\overline{\omega}_{re}=1$ | $T_{re}\geq 2.03\times 10^{13}$ | $N_{re}\leq 3.35$ | $T_{re}\geq 3.95\times 10^{7}$ | $N_{re}\leq 11.97$ | $10^{16}\geq T_{re}\geq 10^{-2}$ | $N_{re}\leq 47.05$

We have also found the viable range of the reheat temperature and number of
e-foldings for each case which shows compatibility with Planck’s 2018
1$\sigma$ bound on $n_{s}$ using figure 4 and the findings have been clearly
presented in a tabular format in table 3. The table 3 shows that the curve
corresponding to $\overline{\omega}_{\text{re}}=\frac{1}{6}$ for K=0.13 and
0.14 and the curves corresponding to
($\frac{2}{3}\leq\overline{\omega}_{\text{re}}\leq 1$) for K=0.15, give every
possible value of reheating temperature ($10^{-2}$ GeV to $10^{16}$ GeV) while
K=0.16 shows incompatibility with data for all $\overline{\omega}_{\text{re}}$
taken. The $n_{s}$ values for these $\overline{\omega}_{\text{re}}$ ranges are
0.966 and 0.964 for K=0.13 and 0.14 while it is (0.965 $<$ $n_{s}$ $\leq$
0.966) for K=0.15 which sets limit on tensor to scalar ratio(r) and the
obtained values of r are 0.083 and 0.068 for K=0.13 and 0.14 while it is
(0.037 $\leq$ r $\leq$ 0.033) for K=0.15 and only the r values for K=0.15 are
satisfying the condition ($r<0.06)$.

##  4 Discussion and conclusion

In this work, we have considered a modified form of quadratic chaotic
inflation. Our primary goal is to study the reheating phase in light of
Planck’s 2018 observations. For that, we have considered two parameters,
namely duration of reheating $N_{\text{re}}$ and reheating temperature
$T_{\text{re}}$ and obtained their variation as function of scalar spectral
index $n_{\text{s}}$ by considering a suitable range of effective equation of
state $\overline{\omega}_{\text{re}}$. By demanding $T_{\text{re}}>100$ GeV
for production of weak-scale dark matter and allowing
$\overline{\omega}_{\text{re}}$ to vary in the range
($-\frac{1}{3}\leq\overline{\omega}_{\text{re}}\leq 1$), we tried to find the
permissible ranges for $n_{\text{s}}$, $\Delta N_{*}$ and tensor-to-scalar
ratio(r) for our models.
We first restudied the simple quadratic chaotic inflation using the most
recent Planck’s 2018 data and found that the condition $T_{\text{re}}$ $>$ 100
GeV gives ($-0.03\leq\overline{\omega}_{\text{re}}\leq 1$) for $n_{s}$ to lie
inside Planck-2018 $2\sigma$ bounds while if we demand $n_{s}$ to lie within
$1\sigma$ bounds than the allowed range of $\overline{\omega}_{\text{re}}$ is
($0.09\leq\overline{\omega}_{\text{re}}\leq 0.67$). Within these ranges of
$\overline{\omega}_{\text{re}}$, r is greater than the combined BICEP2/Keck
and Planck’s bound on r, i.e. ($r<0.06)$.
Since the normal quadratic chaotic potential is not favoring the observational
data. We have considered a modified form of quadratic chaotic potential where
a logarithmic correction containing a model parameter K is added to the mass
term. We have found that for each value of model parameter K of the modified
model, there is only a specific range of inflaton field $(\phi)$ within which
the model is defined and the correction part is not dominant over the actual
quadratic term of potential. We have constrained ourself to only those regions
for our analysis. By imposing the reheating conditions on this model, we found
that the constraints on $n_{\text{s}}$ and r are consistent with Planck’s 2018
data for only a particular range of K values and is found to be
($0.13<{\text{K}}\leq 0.16$), where each K value has a different range of
$\overline{\omega}_{\text{re}}$ in which it is compatible with the data. The
ranges of $\overline{\omega}_{\text{re}}$ for which our obtained data is
compatible with Planck-2018 $2\sigma$ bounds on $n_{s}$ and the combined
BICEP2/Keck and Planck’s bound on r, i.e. ($r<0.06)$ gives
($0.352<\overline{\omega}_{\text{re}}\leq 1$) for K=0.14 and
($0.009\leq\overline{\omega}_{\text{re}}\leq 1$) for K=0.15. For
${\text{K}}\leq 0.13$, there is no compatibility with data for any value of
$\overline{\omega}_{\text{re}}$ taken and K=0.16 is compatible with data for
only $\overline{\omega}_{\text{re}}$=1 . Similarly, the $1\sigma$ bound on
$n_{s}$ and $r<0.06$ gives ($0.352<\overline{\omega}_{\text{re}}\leq 0.640$)
for K=0.14 and ($0.212\leq\overline{\omega}_{\text{re}}\leq 1$) for K=0.15
while K=0.13 and K=0.16 are completely outside the combined $1\sigma$ and r
bounds.
Also, from the plots showing the variation of $T_{\text{re}}$ with $n_{s}$, we
have found that different values of K and $\overline{\omega}_{\text{re}}$ give
different ranges of reheating temperature as compatible with Planck’s
1$\sigma$ bounds on $n_{\text{s}}$, but if we allow $T_{\text{re}}$ to vary
over the whole range ( $10^{-2}$ GeV to $10^{16}$ GeV) , then
$\overline{\omega}_{\text{re}}$ is restricted to
($0.047\leq\overline{\omega}_{\text{re}}\leq 0.391$) for K=0.13,
($0.087\leq\overline{\omega}_{\text{re}}\leq 0.559$) for K=0.14 and
($0.237\leq\overline{\omega}_{\text{re}}\leq 1$) for K=0.15 while K=0.16 shows
incompatibility with Planck’s 2018 1$\sigma$ bounds on $n_{\text{s}}$ for all
$\overline{\omega}_{\text{re}}$ taken.
To conclude, the reheating study shows that the values of K close to 0.15 are
more favorable ones and the $\overline{\omega}_{\text{re}}$ range satisfying
the observational data for K=0.15 suggests the possible production of Feebly
Interacting Massive Particle(FIMP) and Weakly Interacting Massive
Particle(WIMP)-like dark matter particles [60, 61] and primordial black holes
[62]. Elaborated study of possible particle production will be done in our
future publications. The findings of the reheating study prove that even a
small correction in mass term can help quadratic chaotic potential to favour
Planck-2018 observations. Also, we have found that considering the reheating
constraints, the average equation of state parameter
$\overline{\omega}_{\text{re}}$ plays a vital role in defining the compatible
range of reheating parameters, which effectively narrows the model’s viable
parameter space and significantly increases the model’s accuracy.

## Acknowledgments

SY would like to acknowledge the Ministry of Education, Government of India,
for providing the JRF fellowship. UAY acknowledges support from an Institute
Chair Professorship of IIT Bombay.

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