# Inflation Based on the Tsallis Entropy

Zeinab<EMAIL_ADDRESS>Kazem
<EMAIL_ADDRESS>and Abasat<EMAIL_ADDRESS>1Department of Physics, University of Kurdistan, Pasdaran Street, P.O. Box
66177-15175, Sanandaj, Iran
2School of Physics, Institute for Research in Fundamental Sciences (IPM), P.O.
Box 19395-5531, Tehran, Iran
3Physics and Energy Engineering Department, Amirkabir University of
Technology, P.O. Box 159163-4311, Tehran, Iran

###### Abstract

We study the inflationary scenario in the Tsallis entropy-based cosmology. The
Friedmann equations in this setup can be derived by using the first law of
thermodynamics. To derive the relations of the power spectra of the scalar and
tensor perturbations in this setup, we reconstruct an $f(R)$ gravity model
which is thermodynamically equivalent to our model in the slow-roll
approximation. In this way, we find the inflationary observables, including
the scalar spectral index and the tensor-to-scalar ratio in our scenario.
Then, we investigate two different potentials in our scenario, including the
quadratic potential and the potential associated with the natural inflation in
which the inflaton is an axion or a pseudo-Nambu-Goldstone boson. We examine
their observational viability in light of the Planck 2018 CMB data. We show
that although the results of these potentials are in tension with the
observations in the standard inflationary setting, their consistency with the
observations can be significantly improved within the setup of the Tsallis
entropy-based inflation. Moreover, we place constraints on the parameters of
the considered inflationary models by using the current observational data.

Inflation, Tsallis entropy, the $f(R)$ gravity, Natural potential

###### pacs:

98.80.−k, 98.80.Cq, 04.50.Kd

##  I Introduction

The first inflationary model was proposed by Starobinsky Starobinsky (1980) in
1980 and it was based on the addition of the $R^{2}$ term in the Einstein-
Hilbert action with the motivation to include semi-classical quantum effects
to the gravity theory. It is interesting to point out that although this model
is the first inflationary model, it is in very good agreement with the current
observational data. Then, in 1981, Sato Sato (1981a, b), suggested a scenario
in which the Universe has undergone a rapid acceleration in the early stages
of its evolution, and afterward it turned into a fireball with a very high
temperature. Subsequently, in the same year, Guth Guth (1981) has shown that
by including an accelerating phase before the radiation-dominated era, one can
resolve the hot Big Bang cosmology problems such as the flatness problem, the
horizon problem, and the magnetic monopole problem. Guth’s model Guth (1981)
is known as the old inflation and it is based on a scalar field that goes from
a false vacuum towards a true vacuum through a first-order phase transition in
the form of a quantum-tunneling process. The old inflation Guth (1981) suffers
from some fundamental problems, and to resolve its problems, other
inflationary models were suggested later Linde (1982); Albrecht and Steinhardt
(1982); Linde (1983, 1986a, 1986b). For a nice review of the history of the
first 30+ years of inflation, see Sato and Yokoyama (2015).

Along with the evolution of the inflationary models, the theory of
cosmological perturbations was developed too Starobinsky (1979); Mukhanov and
Chibisov (1981); Hawking (1982); Starobinsky (1982); Guth and Pi (1982);
Starobinsky (1983). The calculation of the primordial tensor perturbations in
the early de Sitter stage was first elaborated by Starobinsky Starobinsky
(1979) in 1979, in terms of the related quantity which is the spectrum of
primordial GW background after the Hubble radius crossing at the radiation-
dominated stage. His original motivation was physically sound with the aim to
investigate the initial state of the Universe. Then, Mukhanov and Chibisov
Mukhanov and Chibisov (1981) calculated the spectrum of the perturbations in
the range of observable scales in the context of the Starobinsky $R^{2}$
inflation Starobinsky (1980). In addition, the quantitatively correct
expression for the GWs spectrum produced in the Starobinsky model Starobinsky
(1980) was first presented in Starobinsky (1983). The quantum fluctuations of
the scalar field during inflation lead to the generation of the perturbations
whose growth can seed the Large-Scale Structure (LSS) formation and the
anisotropies observed in the Cosmic Microwave Background (CMB) radiation
Starobinsky (1979); Mukhanov and Chibisov (1981); Hawking (1982); Starobinsky
(1982); Guth and Pi (1982); Starobinsky (1983). Thus, from the observations
corresponding to the LSS formation and CMB anisotropies, we can obtain
valuable information about the physics of the early Universe. Important
observational results about inflation have been presented by Planck
collaboration Akrami  _et al._ (2020), which are obtained from measurements of
the CMB anisotropies in both temperature and polarization. Using these
observational results, one can discriminate between different inflationary
models.

The simplest inflationary model is based on a single scalar field minimally
coupled to gravity Liddle and Lyth (2000); Lemoine  _et al._ (2007). The
scalar field responsible for inflation is known as the inflaton. If the
potential energy of the inflaton dominates over its kinetic energy, then the
inflaton rolls downward its potential slowly. In the chaotic inflationary
scenarios Linde (1983), the slow-roll motion of the inflaton is provided by
the Hubble friction term in the equation of motion of the scalar field, which
is predominant in the early stages of the inflaton, and it then suppresses as
the inflation goes towards its end. In the new inflation models Linde (1982),
however, the slow-roll phase of inflation is provided by a plateau-like
potential for the inflaton field.

Inflation has occurred in the GUT energy scale which is about two orders of
magnitude less than the Planck energy scale. Since inflation has occurred in
the regime of high-energy physics, therefore it is expected that quantum
gravitational effects had a decisive role in the dynamics of the early
Universe. In those energy scales, the gravity theory may be modified due to
the quantum gravitational effects. One way to determine the corrections to the
gravity theory is the use of the gravity-thermodynamics conjecture implying
that there is a deep connection between gravity and thermodynamics Jacobson
(1995); Frolov and Kofman (2003); Padmanabhan (2002, 2005); Eling  _et al._
(2006); Paranjape  _et al._ (2006); Kothawala  _et al._ (2007); Padmanabhan
(2010); Verlinde (2011). This conjecture implies that using the thermodynamics
laws for the Universe as a thermodynamical system, one can derive the
gravitational equations governing the evolution of the Universe. In
particular, the Friedmann equation can be derived from the first law of
thermodynamics Cai and Kim (2005); Cai  _et al._ (2008); Sheykhi (2010);
Sheykhi and Hendi (2011); Sheykhi (2018, 2021). If the entropy is assumed as
the Hawking-Bekenstein entropy Bekenstein (1973), one can derive the standard
Friedmann equation. If we consider some modifications to the entropy of the
system, then the gravitational equations will be modified accordingly. In the
early Universe, the entropy is expected to be different from the standard
Bekenstein-Hawking entropy Bekenstein (1973) due to the quantum gravitational
effects that appear in the regime of high-energy physics.

One form that can be considered for the entropy of the Universe in its
primordial stages is the Tsallis entropy $S_{h}\propto A^{\beta}$ Tsallis and
Cirto (2013) in which $A$ is the horizon area and $\beta$ is a constant
parameter known as the Tsallis parameter. This entropy is a generalization of
Boltzmann-Gibbs entropy Tsallis and Cirto (2013); Tsallis (2019), and it has
been suggested to solve a thermodynamic puzzle. The Boltzmann–Gibbs statistics
is not capable of describing the systems with divergent partition functions
such as the gravitational systems and requires a non-additive generalization
of the entropy definition Tsallis (1988); Lyra and Tsallis (1998); Tsallis
_et al._ (1998). Accordingly, Tsallis and Cirto Tsallis and Cirto (2013);
Tsallis (2019) introduced an entropy expression leading to non-extensive
statistics. The Tsallis entropy has attracted a high level of research
interest over the years, and so far, remarkable results have been found based
on this entropy in a lot of complex systems such as self-gravitating stellar
systems Plastino and Plastino (1993); Hamity and Barraco (1996), black holes
Tsallis and Cirto (2013); Tsallis (2019), background radiation Tsallis  _et
al._ (1995), neutrinos Kaniadakis  _et al._ (1996); Luciano and Blasone
(2021), holographic dark energy Tavayef  _et al._ (2018); Saridakis  _et al._
(2018); Nojiri  _et al._ (2021, 2022a); Pandey  _et al._ (2022) and dark
matter Guha  _et al._ (2019), thermodynamic gravity Sheykhi (2018); Geng  _et
al._ (2020); Luciano and Giné (2022), low-dimensional dissipative systems Lyra
and Tsallis (1998), and polymer chains Jizba  _et al._ (2017). In the special
case where the Tsallis parameter is taken as $\beta=1+\Delta/2$ with
$0\leq\Delta\leq 1$, the Tsallis entropy reduces to the Barrow entropy Barrow
(2020). The cosmological implications of the Barrow entropy have been regarded
in the literature extensively in recent years (see, e.g., Anagnostopoulos  _et
al._ (2020); Barrow  _et al._ (2021); Dabrowski and Salzano (2020); Saridakis
and Basilakos (2021); Jusufi  _et al._ (2022); Leon  _et al._ (2021); Nojiri
_et al._ (2022b); Luciano and Saridakis (2022); Luciano (2022); Luciano and
Giné (2023); Sheykhi and Farsi (2022); Luciano (2023)).

In this paper, we investigate the implications of the Tsallis entropy for the
inflationary phase of the early Universe. To provide the accelerated expansion
of the Universe in the inflationary phase, we assume the matter-energy content
of the Universe follows the form of a canonical scalar field that plays the
role of the inflaton. We take the inflaton potential to be in the form of a
simple quadratic potential $V(\phi)=m^{2}\phi^{2}/2$ which leads to a chaotic
inflationary scenario Linde (1983). In the framework of standard inflation,
this potential is not favored in light of the current CMB data provided by the
Planck 2018 collaboration Akrami  _et al._ (2020). We show that the results of
this potential for the inflationary observables can be improved significantly
in the context of Tsallis entropy-based inflation.

One another elegant inflationary scenario that provides a sensible mechanism
to generate a flat potential is natural inflation Freese  _et al._ (1990). In
this class of models, the scalar field $\phi$ enjoys a shift symmetry
$\phi\rightarrow\phi+{\rm const.}$, which is slightly broken explicitly or due
to the non-perturbative quantum effects to a discrete symmetry
$\phi\rightarrow\phi+2\pi$ Freese  _et al._ (1990). This feature gives rise to
a periodic potential, as appropriate for inflation Freese  _et al._ (1990);
Svrcek and Witten (2006). Note that the presence of this symmetry protects the
potential from radiative corrections. The scalar field with a flat potential
originating from the shift symmetry is known as an axion. In this regard, the
inflaton in the natural inflation is an axion or a pseudo-Nambu-Goldstone
boson. The original natural inflation is described by a cosine-type periodic
potential in the framework of a standard inflationary scenario based on the
Einstein gravity where the entropy of the horizon obeys the Bekenstein-Hawking
area law Bekenstein (1973). In this setup, the natural inflation is not very
favored by the latest observation of the Planck 2018 collaboration Akrami  _et
al._ (2020), and its results can satisfy only the 95% CL constraint of Planck
2018 data Akrami  _et al._ (2020). This point motivates us to study the
natural inflation in the setup of the Tsallis inflation and compare its
predictions with the Planck 2018 constraints Akrami  _et al._ (2020).

Although the Friedmann equations of the Friedmann-Robertson-Walker (FRW)
Universe in the framework based on the Tsallis entropy have already been
derived in Sheykhi (2018), the study of inflation in this context also
requires the equations of the power spectra of the primordial scalar and
tensor perturbations. To this aim, we need the action of the gravity theory.
In the absence of a unique action for our Tsallis inflation model, we
construct an $f(R)$ gravity model which is thermodynamically equivalent to the
Tsallis gravity in the slow-roll regime of inflation. The modified $f(R)$
gravity is a conceivable generalization of general relativity in which $f(R)$
is an arbitrary function of the Ricci scalar $R$. Over the years, the $f(R)$
gravity models have been extensively studied in the literature (see, e.g.,
Hwang and Noh (2001); Sotiriou and Faraoni (2010); De Felice and Tsujikawa
(2010); Kaneda  _et al._ (2010); Nojiri and Odintsov (2011); Huang (2014);
Nojiri  _et al._ (2017); Cheong  _et al._ (2020); Nojiri  _et al._ (2020a);
Aziz  _et al._ (2021); Parrilla and Escamilla-Rivera (2023)). One of the most
important motivations that led us to this choice is that it is possible in the
$f(R)$ gravity to construct a model whose entropy is proportional to the
powers of the horizon area, like the Tsallis entropy. Moreover, the $f(R)$
gravity models have phenomenological effective features that can describe
inflation in the regime of high-energy physics. On one side $f(R)$ actions are
general enough to cover some basic features of higher-order gravity, on the
other side they are sufficiently simple to be easy to work with Sotiriou and
Faraoni (2010). Besides, there is one other important reason that convinces us
that the $f(R)$ theories of gravity are good candidates to help us evaluate
the inflationary observables based on the Tsallis entropy. These models can
avoid serious problems such as negative energies and related instabilities
that are called the Ostrogradski instabilities Woodard (2007); Sotiriou and
Faraoni (2010); Ostrogradsky (1850). This feature made the $f(R)$ theories
unique compared to other higher-order gravity theories.

The paper is structured as follows. In Sec. II, we briefly study the
thermodynamics of the Tsallis cosmology and the background equations derived
from the first law of thermodynamics. In Sec. III, we review the inflationary
dynamics and spectra of primordial perturbations in the $f(R)$ gravity. The
thermodynamic behavior of the field equations in the $f(R)$ gravity will be
discussed in Sec. IV. Then, in Sec. V, we reconstruct an $f(R)$ model which is
thermodynamically equivalent to the Tsallis model. Using this equivalence, we
derive the equations of the inflationary observables in our Tsallis inflation
scenario. In Sec. VI, we examine the observational compatibility of the
quadratic and natural potentials with the Planck 2018 data in the framework of
the Tsallis inflation. Finally, we summarize our concluding remarks in Sec.
VII.

##  II Thermodynamics of the Tsallis cosmology

In Sheykhi (2018), it has been shown that by starting from the first law of
equilibrium thermodynamics, $dE=T_{h}dS_{h}+Wd\mathcal{V}$, at the apparent
horizon of the FRW Universe and taking the entropy associated with the
apparent horizon in the form of the Tsallis entropy Tsallis and Cirto (2013),
one can derive the modified Friedmann equations. Here, $E$ is the total energy
content of the Universe, $T_{h}$ is the temperature of the apparent horizon,
$W$ is the work density, and $\mathcal{V}$ is the volume inside the apparent
horizon. For these quantities, we have Akbar and Cai (2007a); Hayward  _et
al._ (1999); Sheykhi (2018)

$\displaystyle E$ $\displaystyle=\rho\,\mathcal{V}\,,$ (1) $\displaystyle
T_{h}$
$\displaystyle=-\frac{1}{2\pi\tilde{r}_{A}}\left(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\right)\,,$
(2) $\displaystyle W$ $\displaystyle=\frac{1}{2}(\rho-p)\,,$ (3)
$\displaystyle\mathcal{V}$ $\displaystyle=\frac{4}{3}\pi\tilde{r}_{A}^{3}\,,$
(4)

where $\rho$ and $p$ represent respectively the energy density and pressure of
all matter components in the Universe, and they satisfy the following
continuity equation

$\dot{\rho}+3H(\rho+p)=0\,.$ (5)

In the above equations, $H\equiv\dot{a}/a$ denotes the Hubble parameter,
$\tilde{r}_{A}$ indicates the radius of the apparent horizon, and the overdot
indicates the derivative with respect to the cosmic time $t$. The horizon
entropy is denoted by $S_{h}$, and it will be taken in the form of the Tsallis
entropy which is a non-extensive generalization of Boltzmann–Gibbs entropy
Tsallis (1988); Tsallis and Cirto (2013); Tsallis (2019)

$S_{h}=\gamma A^{\beta}\,.$ (6)

Here, $A=4\pi\tilde{r}_{A}^{2}$ is the area of the apparent horizon and
$\beta$ is a real parameter known as the Tsallis parameter that measures the
degree of non-extensivity Tsallis and Cirto (2013). In addition, $\gamma$ is
an unknown constant, and several definitions have been presented for it in
some literature for convenience Sheykhi (2018); Lymperis and Saridakis (2018);
Nojiri  _et al._ (2019, 2020b); Asghari and Sheykhi (2021). In our
investigation, however, this parameter will be treated as an unknown
parameter. It is clear that for $\beta=1$ and $\gamma=1/(4G)$, where $G$ is
the Newton gravitational constant, the Tsallis entropy (6) reduces to the
Bekenstein-Hawking entropy Bekenstein (1973).

On the flat FRW background, the modified Friedmann equations based on the
Tsallis entropy (6) can be derived as follows Sheykhi (2018)

$\displaystyle H^{2(2-\beta)}=\frac{8\pi\,G_{{\rm eff}}}{3}\rho\,,$ (7)
$\displaystyle(4-2\beta)\,\frac{\ddot{a}}{a}\,H^{2(1-\beta)}+(2\beta-1)H^{2(2-\beta)}=-8\pi\,G_{{\rm
eff}}\,p\,,$ (8)

where the effective gravitational constant is defined as

$G_{\rm
eff}\equiv\left(\frac{1}{\gamma}\right)\left(\frac{2-\beta}{4\beta}\right)(4\pi)^{1-\beta}\,.$
(9)

In this paper, we use the unit system in which $c=\hbar=\kappa_{B}=1$ where
$c$ is the speed of light, $\hbar$ is the reduced Planck constant, and
$\kappa_{B}$ is the Boltzmann constant. We define $\kappa^{2}=8\pi
G=8\pi/m_{P}^{2}=1/{M_{P}}^{2}$ where $m_{P}$ is the Planck mass with a
reduced value $M_{P}=m_{P}/\sqrt{8\pi}=(8\pi G)^{-1/2}$ which throughout this
paper we take it equal to unity, $M_{P}=1$. From Eqs. (7), (8), and (9), one
can easily show that the standard Friedmann equations based on the Einstein
gravity in which the entropy associated with the horizon follows the form of
Bekenstein-Hawking entropy Bekenstein (1973), are recovered for $\beta=1$ and
$\gamma=1/(4G)$, as expected. It is worth mentioning that although the author
of Sheykhi (2018) has defined $\gamma$ as
$\gamma\equiv(2-\beta)(4\pi)^{1-\beta}/(4\beta\,G$), we do not follow this
convention in the present work. For a spatially flat Universe, we have
$\tilde{r}_{A}=1/H$, and so one can rewrite the area of the apparent horizon
$A$ in terms of the Hubble parameter $H$ as

$A=\frac{4\pi}{H^{2}}\,.$ (10)

We assume that the matter-energy content of the Universe to be a scalar field
$\phi$ in the form of a perfect fluid with the energy-momentum tensor
$T^{\mu}{}^{(\phi)}_{\nu}=\mathrm{diag}\,(-\rho_{\phi},p_{\phi},p_{\phi},p_{\phi})$.
Here, $\rho_{\phi}$ and $p_{\phi}$ denote the energy density and pressure of
the scalar field, respectively, and they are given by

$\displaystyle\rho_{\phi}=\frac{1}{2}\dot{\phi}^{2}+V(\phi)\,,$ (11)
$\displaystyle p_{\phi}=\frac{1}{2}\dot{\phi}^{2}-V(\phi)\,.$ (12)

In these equations, $V(\phi)$ is the potential energy of the scalar field. The
energy density $\rho_{\phi}$ and pressure $p_{\phi}$ of the scalar field
fulfill the continuity equation

$\dot{\rho}_{\phi}+3H(\rho_{\phi}+p_{\phi})=0\,.$ (13)

From this equation together with Eqs. (11) and (12), the equation of motion of
the scalar field equation will be obtained as

$\ddot{\phi}+3H\dot{\phi}+V_{,\phi}=0\,,$ (14)

where $(,\phi)$ indicates the partial derivative with respect to the scalar
field $\phi$. As we see, Eq. (14) is the same as the one that is valid in the
standard scenario based on the Einstein gravity.

To study the inflationary epoch in the Tsallis entropy-based scenario, we need
to derive the fundamental relations governing the theory of cosmological
perturbations. For this purpose, we need the action of the model. But we don’t
know the action of the model, and to overcome this problem, we try to
reconstruct an $f(R)$ gravity model that is equivalent to our model from the
thermodynamic point of view. In the $f(R)$ gravity, the horizon entropy is
given by Akbar and Cai (2007b); De Felice and Tsujikawa (2010)

$S=\frac{AF(R)}{4G}\,,$ (15)

where $F\equiv df(R)/dR$. In the following, we review the basic formulas
governing the theory of cosmological perturbations in the $f(R)$ gravity, as
well as the thermodynamic behavior of field equations in this theory. Then we
apply the obtained results in our Tsallis entropy-based model and find the
necessary relations to study inflation.

##  III Inflationary dynamics and spectra of primordial perturbations in the
$f(R)$ gravity

The $f(R)$ gravity is described by the following action De Felice and
Tsujikawa (2010)

$S=\int{\rm
d}^{4}x\sqrt{-g}\left[\frac{1}{2\kappa^{2}}f(R)+X-V(\phi)\right]\,,$ (16)

where $g$ is the determinant of the metric $g_{{\mu}{\nu}}$, $f(R)$ is an
arbitrary function of the Ricci scalar $R$, and
$X\equiv-\frac{1}{2}g^{\mu\nu}\partial_{\mu}{\phi}\partial_{\nu}{\phi}$ is the
canonical kinetic term. For the flat FRW metric, the Ricci scalar $R$ is given
by De Felice and Tsujikawa (2010)

$R=6\left(2H^{2}+\dot{H}\right)\,.$ (17)

For the $f(R)$ gravity model with action (16), the Friedmann equations turn
into De Felice and Tsujikawa (2010)

$\displaystyle 3FH^{2}$
$\displaystyle=\frac{1}{2}\left(RF-f\right)-3H\dot{F}+\kappa^{2}\left[\frac{1}{2}\dot{\phi}^{2}+V(\phi)\right]\,,$
(18) $\displaystyle-2F\dot{H}$
$\displaystyle=\ddot{F}-H\dot{F}+\kappa^{2}\dot{\phi}^{2}\,.$ (19)

Following Hwang and Noh (2001); De Felice and Tsujikawa (2010), we introduce
the slow-roll parameters as

$\varepsilon_{1}\equiv-\frac{\dot{H}}{H^{2}}\,,\hskip
14.22636pt\varepsilon_{2}\equiv\frac{\ddot{\phi}}{H\,\dot{\phi}}\,,\hskip
14.22636pt\varepsilon_{3}\equiv\frac{\dot{F}}{2HF}\,,\hskip
14.22636pt\varepsilon_{4}\equiv\frac{\dot{E}}{2HE}\,,$ (20)

where

$E\equiv F\left(1+\frac{3\dot{F}^{2}}{2\kappa^{2}\dot{\phi}^{2}F}\right)\,.$
(21)

In the slow-roll regime, the quantities $\mid\varepsilon_{i}\mid\hskip
2.84544pt(i=1,2,3,4)$ are much smaller than unity. Under the slow-roll limit,
the Ricci scalar $R$ in Eq. (17) reduces to

$R\simeq 12H^{2}.$ (22)

The spectrum of the curvature perturbations generated during inflation and in
the slow-roll limit can be estimated as De Felice and Tsujikawa (2010)

${\cal P}_{s}\simeq\frac{1}{Q_{s}}\left(\frac{H}{2\pi}\right)^{2}\,,$ (23)

where

$Q_{s}\simeq\dot{\phi}^{2}\left(\frac{E}{FH^{2}}\right)\,.$ (24)

It is worth mentioning that ${\cal P}_{s}$ is computed at the time of horizon
exit at which $k=aH$, where $k$ is the comoving wavenumber. The observational
value of the amplitude of scalar perturbations at the CMB pivot scale
$k_{*}=0.05\,{\rm Mpc}^{{\rm-1}}$ has been constrained by the Planck 2018 CMB
observations to be ${\cal P}_{s}(k_{*})\simeq 2.1\times 10^{-9}$ Akrami  _et
al._ (2020).

The scalar spectral index $n_{s}$ during the slow-roll regime, in the
framework of the $f(R)$ gravity is given by De Felice and Tsujikawa (2010)

$n_{s}-1\simeq-4\varepsilon_{1}-2\varepsilon_{2}+2\varepsilon_{3}-2\varepsilon_{4}\,.$
(25)

The observational constraint from the Planck data on the scalar spectral index
at the CMB pivot scale $k_{*}=0.05\,{\rm Mpc}^{{\rm-1}}$ is $n_{s}=0.9657\pm
0.0044$ (68% CL, Planck 2018 TTTEEE+low$\ell$+lowE) Akrami  _et al._ (2020).

For $f(R)=R$, from Eqs. (20), (21), and (24), it follows that
$\varepsilon_{3}=0$, $\varepsilon_{4}=0$, and $Q_{s}=\dot{\phi}^{2}/H^{2}$.
Substitution of these results into Eqs. (23) and (25) leads to ${\cal
P}_{s}\simeq H^{4}/(4\pi^{2}\dot{\phi}^{2})$ and
$n_{s}-1\simeq-4\varepsilon_{1}-2\varepsilon_{2}$ which are the familiar
relations in standard inflation model based on the Einstein gravity Liddle and
Lyth (1992); Stewart and Lyth (1993).

The tensor power spectrum in the framework of the $f(R)$ gravity is given by
De Felice and Tsujikawa (2010)

${\cal
P}_{t}\simeq\frac{16}{\pi}\left(\frac{H}{m_{P}}\right)^{2}\frac{1}{F}\,.$ (26)

Also, the tensor spectral index $n_{t}$ in this setup is obtained as

$n_{t}\simeq-2\varepsilon_{1}-2\varepsilon_{3}\,.$ (27)

This parameter determines the scale dependence of the tensor power spectrum.
Currently, there is no precise measurement for this quantity and we hope the
future observations can provide some constraints on this observable.

Using Eqs. (23) and (26), one can find the tensor-to-scalar ratio in the
$f(R)$ gravity setting and in the slow-roll regime as De Felice and Tsujikawa
(2010)

$r\equiv\frac{{\cal P}_{t}}{{\cal
P}_{s}}\simeq\frac{64\pi}{m_{P}^{2}}\left(\frac{Q_{s}}{F}\right)\,.$ (28)

The Planck 2018 data sets an upper bound on the tensor-to-scalar ratio as
$r<0.0522$ at the CMB pivot scale $k_{*}=0.05\,{\rm Mpc}^{{\rm-1}}$ (68% CL,
Planck 2018 TTTEEE+low$\ell$+lowE) Akrami  _et al._ (2020). The most recent
upper limit on this parameter is $r_{0.01}<0.028$ at 95% CL which is obtained
at the CMB pivot scale $k=0.01\,{\rm Mpc}^{{\rm-1}}$ using 10 datasets from
the BICEP/Keck Array 2015 and 2018, Planck releases 3 and 4, and LIGO-Virgo-
KAGRA Collaboration Galloni  _et al._ (2023).

In the case of $f(R)=R$, using Eqs. (24), (26), (27), and (28), we easily find
that ${\cal P}_{t}\simeq 2H^{2}/\pi^{2}$, $n_{t}\simeq-2\varepsilon_{1}$ and
$r\simeq 16\varepsilon_{1}$. As we see, in this case, these results are
reduced to the well-known relations in the standard inflation.

##  IV Thermodynamic behavior of field equations in the $f(R)$ gravity

Let us discuss the relation between the first law of thermodynamics and field
equations in the $f(R)$ gravity. In Akbar and Cai (2007b); De Felice and
Tsujikawa (2010), it has been shown that the first law of equilibrium
thermodynamics, $dE=T_{h}dS_{h}+Wd\mathcal{V}$, does not hold at the apparent
horizon of the FRW Universe in the $f(R)$ gravity. Consequently, to derive the
Friedmann equations, we should apply the non-equilibrium thermodynamics and
write the first law of thermodynamics as
$dE=T_{h}dS_{h}+Wd{\mathcal{V}}+T_{h}d\bar{S}$. In this equation, $S_{h}$ is
the entropy associated with the apparent horizon and it is still given by Eq.
(15). Furthermore, $\bar{S}$ implies the non-equilibrium entropy and involves
the non-equilibrium thermodynamic effects of the $f(R)$ gravity Akbar and Cai
(2007b); De Felice and Tsujikawa (2010).

From the non-equilibrium relation
$dE=T_{h}dS_{h}+Wd{\mathcal{V}}+T_{h}d\bar{S}$, we find

$\frac{d\bar{S}}{dS_{h}}=-1+\frac{dE-Wd{\mathcal{V}}}{T_{h}dS_{h}}\,.$ (29)

Taking the differentiation of Eq. (1), and then using Eq. (4) and also the
relation $d{\mathcal{V}}=4\pi\tilde{r}_{A}^{2}d\tilde{r}_{A}$, we obtain

$dE=4\pi\tilde{r}_{A}^{2}\rho
d\tilde{r}_{A}+\frac{4}{3}\pi\tilde{r}_{A}^{3}\dot{\rho}dt\,.$ (30)

With the help of Eq. (5), we get

$dE=4\pi\tilde{r}_{A}^{2}\rho d\tilde{r}_{A}-4\pi
H\tilde{r}_{A}^{3}(\rho+p)dt\,.$ (31)

It is assumed that the entropy of the horizon $S_{h}$ is in the form of Eq.
(15). Differentiating the entropy (15), it follows that

$dS_{h}=\frac{\pi}{G}\left(\tilde{r}_{A}^{2}\dot{F}+2F\tilde{r}_{A}\dot{\tilde{r}}_{A}\right)dt\,.$
(32)

where we have used $A=4\pi\tilde{r}_{A}^{2}$ in deriving the above equation.

Now, using Eqs. (2), (3), (4), (31), and (32), in the right side of Eq. (29),
we find

$\frac{d\bar{S}}{dS_{h}}=\frac{8\pi
GH\tilde{r}_{A}^{3}(\rho+p)-\tilde{r}_{A}\dot{F}-2F\dot{\tilde{r}}_{A}}{\tilde{r}_{A}\dot{F}+2F\dot{\tilde{r}}_{A}}\,.$
(33)

Supposing the matter-energy content of the Universe to be a scalar field
$\phi$ with the energy density $\rho=\rho_{\phi}$ and the pressure
$p=p_{\phi}$, we use Eqs. (11) and (12), and then we can rewrite Eq. (33) as

$\frac{d\bar{S}}{dS_{h}}=\frac{-H\dot{F}+2F\dot{H}+8\pi
G\dot{\phi}^{2}}{H\dot{F}-2F\dot{H}}\,.$ (34)

Note that in deriving Eq. (34), we have used the relation $\tilde{r}_{A}=1/H$.
Finally, with the help of the second and third relations in Eq. (20), we find

$\frac{d\bar{S}}{dS_{h}}=-1+\frac{4\pi
G\,\dot{\phi}^{2}}{FH^{2}(\varepsilon_{1}+\varepsilon_{3})}\,.$ (35)

From Eq. (29), it is clear that in the absence of the term $d\bar{S}/dS_{h}$
the first-law of equilibrium thermodynamics $dE=T_{h}dS_{h}+Wd\mathcal{V}$ on
the apparent horizon holds. Therefore, Eq. (35) is an important relation in
our examination. In the following, we show that in the framework based on the
Tsallis entropy and under the slow-roll approximation, the term
$d\bar{S}/dS_{h}$ vanishes, and hence $d\bar{S}=0$. Therefore, this point
allows us to use the obtained relations in the $f(R)$ gravity to examine the
inflationary models in the Tsallis entropy-based setting.

##  V Inflationary dynamics in the Tsallis entropy-based model

In this section, we assume that the Tsallis entropy-based model and the $f(R)$
gravity are equivalent thermodynamically in the slow-roll regime. With the
help of Eq. (22), one can easily express the horizon area $A$ in Eq. (10) in
terms of Ricci scalar $R$ as

$A\simeq\frac{48\pi}{R}\,.$ (36)

Since we have supposed that the Tsallis entropy-based model and the $f(R)$
gravity are equivalent in the slow-roll limit, we set Eqs. (6) and (15) equal
to each other. Next, by using Eq. (36), we reach the following differential
equation

$\frac{12\pi}{G}F(R)-\gamma(48\pi)^{\beta}R^{1-\beta}\simeq 0\,.$ (37)

Since $F(R)\equiv df(R)/dR$, solving this differential equation analytically,
the function $f(R)$ is obtained as the following form

$f(R)=\left(\frac{4^{-1+2\beta}(3\pi)^{-1+\beta}\gamma\,G}{2-\beta}\right)R^{2-\beta}\,.$
(38)

As we see, $f(R)$ is a power-law function of $R$ as $f(R)=\mu R^{n}$ where
$\mu\equiv(4^{-1+2\beta}(3\pi)^{-1+\beta}\gamma\,G)/(2-\beta)$ and $n\equiv
2-\beta$. For $\beta=1$ and $\gamma=1/(4G)$, we find $f(R)=R$, and then Eq.
(15) reduces to the Bekenstein-Hawking area-law of entropy Bekenstein (1973)
in the Einstein gravity.

Here, it is worthwhile to point out that the $R^{n}$ Lagrangian was originally
regarded in the context of higher derivative theories Schmidt (1989); Maeda
(1989), and then applied to inflation Muller  _et al._ (1990); Gottlober  _et
al._ (1992); Motohashi (2015), which provides a simple and practical
generalization of the Starobinsky $R^{2}$ inflation Starobinsky (1980). In
particular, in Motohashi (2015), this Lagrangian with $n\approx 2$ has been
investigated to establish a way to measure a deviation from the $R^{2}$
inflation Starobinsky (1980). However, our methodology in the present paper
differs from the approach of Motohashi (2015) in several aspects which are as
follows. The action in our model is completely different from the action of
Motohashi (2015) because, in our model, the gravitational part of the action
consists of only the $R^{n}$ term, but the action of Motohashi (2015) contains
the term $R+R^{n}$ which includes also the standard Einstein-Hilbert term.
Furthermore, in our model, the contribution of a scalar field has been
included in the action beside the $R^{n}$, but such a contribution is absent
in the action of Motohashi (2015). In the analysis of Motohashi (2015), a
conformal transformation from the Jordan frame to the Einstein frame has been
performed, and the calculations of the inflationary observables are
accomplished in the Einstein frame. In contrast, in our analysis, all
calculations are performed in the Jordan frame. Finally, the form of the
potentials that we consider in our work differs from the potentials that are
regarded in Motohashi (2015) for the Einstein-frame scalar field.

Using Eq. (22), one can easily rewrite Eq. (38) as the following form

$f=\left(\frac{3\,\pi^{-1+\beta}4^{1+\beta}\gamma\,G}{2-\beta}\right)H^{2(2-\beta)}\,.$
(39)

From the slow-roll parameters (20), we find that the field equations (7) and
(14) in the slow-roll limit lead to

$\displaystyle H^{2(2-\beta)}\simeq\frac{8\pi\,G_{{\rm eff}}}{3}V(\phi)\,,$
(40) $\displaystyle 3H\dot{\phi}+V_{,\phi}\simeq 0\,.$ (41)

It can be shown that in the slow-roll approximation, the same equation as Eq.
(40) can also be derived by using the first Friedmann equation (18) in the
$f(R)$ gravity. This arises from the fact that in the slow-roll regime, the
non-equilibrium entropy can be neglected in front of the equilibrium entropy,
as we will show explicitly at the end of this section. Applying Eq. (40), we
can rewrite the function $f$ in Eq. (39) in terms of the scalar field $\phi$
as

$f=\left(\frac{32\,G_{{\rm
eff}}(4\pi)^{\beta}\gamma\,G}{2-\beta}\right)V(\phi)\,.$ (42)

Taking the time derivative of Eq. (40) and then applying Eq. (41), we can
rewrite $\varepsilon_{1}$ in Eq. (20) as

$\varepsilon_{1}\simeq\frac{1}{2(2-\beta)}\left(\frac{V_{,\phi}^{2}}{3H^{2}V(\phi)}\right)\,.$
(43)

For $\beta=1$ and $\gamma=1/(4G)$, from Eqs. (9), (40), and (43), one can
easily verify that the corresponding relation in the standard slow-roll
inflationary model will be recovered. It is known that inflation ends when the
first slow-roll parameter $\varepsilon_{1}$ reaches unity Martin  _et al._ .

Since $d/dt=\dot{\phi}d/{d\phi}$ and $d/dR=(1/{R_{,\phi}})d/d\phi$, we can
also rewrite the slow-roll parameters $\varepsilon_{2}$, $\varepsilon_{3}$,
and $\varepsilon_{4}$ in Eq. (20) in the following forms

$\displaystyle\varepsilon_{2}\simeq\frac{H_{,\phi}V_{,\phi}-HV_{,\phi\phi}}{3H^{3}}\,,$
(44)
$\displaystyle\varepsilon_{3}\simeq\frac{V_{,\phi}\left(f_{,\phi}R_{,\phi\phi}-R_{,\phi}f_{,\phi\phi}\right)}{6H^{2}f_{,\phi}R_{,\phi}}\,,$
(45)
$\displaystyle\varepsilon_{4}\simeq\frac{V_{,\phi}\left(f_{,\phi}R_{,\phi\phi}-R_{,\phi}f_{,\phi\phi}\right)\Big{(}R_{,\phi}^{3}+6f_{,\phi}R_{,\phi\phi}^{2}+3R_{,\phi}^{2}f_{,\phi\phi\phi}-3R_{,\phi}(2f_{,\phi\phi}R_{,\phi\phi}+f_{,\phi}R_{,\phi\phi\phi})\Big{)}}{3H^{2}R_{,\phi}\Big{(}3R_{,\phi}^{2}f_{,\phi\phi}^{2}+3f_{,\phi}^{2}R_{,\phi\phi}^{2}+2f_{,\phi}(R_{,\phi}^{3}-3R_{,\phi}f_{,\phi\phi}R_{,\phi\phi})\Big{)}}.$
(46)

In deriving Eqs. (44), (45), and (46), we have also used Eqs. (21) and (41).
Besides, applying Eqs. (21), (24), and (41), the scalar power spectrum ${\cal
P}_{s}$ in Eq. (23) takes the form

${\cal
P}_{s}\simeq\left(\frac{9H^{6}}{2\pi^{2}V_{,\phi}^{2}}\right)\left(\frac{f_{,\phi}R_{,\phi}^{3}}{2f_{,\phi}R_{,\phi}^{3}+3(R_{,\phi}f_{,\phi\phi}-f_{,\phi}R_{,\phi\phi})^{2}}\right).$
(47)

Substituting Eqs. (43), (44), (45), and (46) into Eq. (25), the scalar
spectral index $n_{s}$ reads

$\displaystyle n_{s}\simeq$ $\displaystyle
1+\frac{1}{3H^{3}}\Big{[}-2H_{,\phi}V_{,\phi}+\frac{2HV_{,\phi}^{2}}{(\beta-2)V}+HV_{,\phi}\left(\frac{R_{,\phi\phi}}{R_{,\phi}}-\frac{f_{,\phi\phi}}{f_{,\phi}}\right)+2HV_{,\phi\phi}$
$\displaystyle+\Big{(}2HV_{,\phi}(R_{,\phi}f_{,\phi\phi}-f_{,\phi}R_{,\phi\phi})(R_{,\phi}^{3}+6f_{,\phi}R_{,\phi\phi}^{2}+3R_{,\phi}^{2}f_{,\phi\phi\phi}$
$\displaystyle-3R_{,\phi}(2f_{,\phi\phi}R_{,\phi\phi}+f_{,\phi}R_{,\phi\phi\phi}))\Big{)}\Big{]}\Big{/}\Big{[}R_{,\phi}(3R_{,\phi}^{2}f_{,\phi\phi}^{2}+3f_{,\phi}^{2}R_{,\phi\phi}^{2}$
$\displaystyle+2f_{,\phi}(R_{,\phi}^{3}-3R_{,\phi}f_{,\phi\phi}R_{,\phi\phi}))\Big{]}\,.$
(48)

We can also rewrite the tensor-to-scalar ratio $r$ in Eq. (28) in terms of the
scalar field $\phi$ as

$r\simeq\left(\frac{4V_{,\phi}^{2}}{9H^{4}}\right)\left(\frac{R_{,\phi}\big{(}2f_{,\phi}R_{,\phi}^{3}+3(R_{,\phi}f_{,\phi\phi}-f_{,\phi}R_{,\phi\phi})^{2}\big{)}}{f_{,\phi}^{2}R_{,\phi}^{3}}\right)\,,$
(49)

where we have used Eqs. (21), (24), and (41) together with the relations
$d/dt=\dot{\phi}d/{d\phi}$ and $d/dR=(1/{R_{,\phi}})d/d\phi$.

It is convenient to evaluate the inflationary observations in terms of the so-
called $e$-fold number $N$ which measures the growth of the scale factor $a$
during inflation. It is defined as

$N\equiv\ln\left(\frac{a_{\rm e}}{a}\right),$ (50)

where the subscript “${\rm e}$” refers to end of inflation. The definition
(50) leads to

$dN=-Hdt=-\frac{H}{\dot{\phi}}d\phi\,.$ (51)

Note that the anisotropies observed in the CMB exit the Hubble horizon around
$N_{*}\approx 50–60$ $e$-folds before the end of inflation Dodelson and Hui
(2003); Liddle and Leach (2003). The precise value of horizon exit $e$-fold
number $N_{*}$ depends on the energy scale of inflation and also on the
details of the reheating process after inflation Dodelson and Hui (2003);
Liddle and Leach (2003). In our model, like most conventional inflationary
models, the features of the reheating mechanism after inflation are unknown to
us, and therefore it is not possible to determine the precise value of
$N_{*}$.

Using the last equality in Eq. (51) and also applying Eqs. (40) and (41), we
reach the following differential equation

$\frac{d\phi}{dN}\simeq\left(3^{1-\beta}8\pi G_{\rm
eff}\right)^{\frac{1}{\beta-2}}\left(\frac{V_{,\phi}}{V^{1/(2-\beta)}}\right)\,.$
(52)

One can solve Eq. (52) to obtain the scalar field $\phi$ as a function of the
$e$-fold number $N$ in the slow-roll approximation. In this way, we find the
inflationary observables in terms of $N$.

Using Eqs. (40) and (41), we can rewrite Eq. (35) in the slow-roll limit as

$\frac{d\bar{S}}{dS_{h}}\simeq-1+3^{\frac{2}{2-\beta}}\left(\frac{4\pi
G\,}{9F(8\pi G_{\rm
eff})^{\frac{2}{2-\beta}}(\varepsilon_{1}+\varepsilon_{3})}\right)\left(\frac{V_{,\phi}}{V^{1/(2-\beta)}}\right)^{2}\,.$
(53)

Using Eq. (22), we obtain $R_{,\phi}=24HH_{,\phi}$, and since
$F=df/dR=f_{,\phi}/R_{,\phi}$, we have $F=f_{,\phi}/(24HH_{,\phi})$.
Substituting this relation into Eq. (53), we get

$\frac{d\bar{S}}{dS_{h}}\simeq-1+3^{\frac{2}{2-\beta}}\left(\frac{96\pi
G\,H_{,\phi}H}{9f_{,\phi}(8\pi G_{\rm
eff})^{\frac{2}{2-\beta}}(\varepsilon_{1}+\varepsilon_{3})}\right)\left(\frac{V_{,\phi}}{V^{1/(2-\beta)}}\right)^{2}\,.$
(54)

Now substituting Eqs. (43) and (45) into Eq. (54), and then using Eqs. (9),
(40), and (42), we will have

$\frac{d\bar{S}}{dS_{h}}\simeq 0\hskip 14.22636pt\Rightarrow\hskip
14.22636ptd\bar{S}\simeq 0\,,$ (55)

which means that in the slow-roll Tsallis entropy-based inflation, the first
law of equilibrium thermodynamics $dE=T_{h}dS_{h}+Wd\mathcal{V}$ on the
apparent horizon holds. This point makes us sure to apply the results derived
in the $f(R)$ gravity to find the inflationary observables in the Tsallis
entropy-based cosmology.

##  VI Observational constraints

In this section, we apply the obtained results in the previous section to
investigate the observational consistency of two different inflationary
potentials in the framework of the Tsallis entropy-based inflation. These
potentials are the quadratic and natural potentials that are not in good
agreement with the current CMB data in the setting of standard inflation.

###  VI.1 Quadratic potential

Let us continue studying the inflationary scenario in the Tsallis entropy-
based setting by considering the quadratic potential Linde (1983)

$V(\phi)=\frac{1}{2}m^{2}\phi^{2}\,,$ (56)

where $m$ is the inflaton mass. Applying the potential (56) and also using
Eqs. (9), (22), (40), and (42) , the slow-roll parameters in Eqs. (43), (44),
(45), and (46) take the form

$\displaystyle\varepsilon_{1}\simeq\left(\frac{3^{\frac{-1+\beta}{2-\beta}}}{2-\beta}\right)\left(\frac{4^{1-\beta}\pi^{2-\beta}(2-\beta)}{\beta}\phi^{2}\right)^{\frac{1}{\beta-2}}\Theta\,,$
(57)
$\displaystyle\varepsilon_{2}=\varepsilon_{3}\simeq(-1+\beta)\left(\frac{3^{\frac{-1+\beta}{2-\beta}}}{2-\beta}\right)\left(\frac{4^{1-\beta}\pi^{2-\beta}(2-\beta)}{\beta}\phi^{2}\right)^{\frac{1}{\beta-2}}\Theta\,,$
(58)
$\displaystyle\varepsilon_{4}\simeq\left(\frac{3^{\frac{-1+\beta}{2-\beta}}}{2-\beta}\right)(-1+\beta)\left(\frac{4^{1-\beta}\pi^{2-\beta}(2-\beta)}{\beta}\phi^{2}\right)^{\frac{1}{\beta-2}}\Theta\Bigg{\\{}(12\pi)^{\beta}(-1+\beta)\beta\Theta+4\pi^{2}(-2+\beta)^{2}$
$\displaystyle\times
3^{-\frac{(1-\beta)^{2}}{2-\beta}}\left(\frac{4^{1-\beta}\pi^{2-\beta}(2-\beta)}{\beta}\phi^{2}\right)^{\frac{-1+\beta}{2-\beta}}\phi^{2}\Bigg{\\}}\Big{/}\Bigg{\\{}(12\pi)^{\beta}(-1+\beta)^{2}\Theta+4\pi^{2}(-2+\beta)^{2}\,3^{-\frac{(1-\beta)^{2}}{2-\beta}}$
$\displaystyle\times\left(\frac{4^{1-\beta}\pi^{2-\beta}(2-\beta)}{\beta}\phi^{2}\right)^{\frac{-1+\beta}{2-\beta}}\phi^{2}\Bigg{\\}}.$
(59)

The parameter $\Theta$ in these equations is defined as

$\Theta\equiv m^{2}\left(\frac{m^{2}}{\gamma}\right)^{\frac{1}{\beta-2}}\,.$
(60)

It is easy to show that for $\beta=1$, we have $\Theta=\gamma$. Therefore, for
$\beta=1$ and $\Theta=1/(4G)$, we find the same relations in the Einstein
gravity.

Using Eqs. (9), (22), (40), (42), and (56), the scalar power spectrum (47)
turns into

$\displaystyle{\cal P}_{s}\simeq$
$\displaystyle\frac{3^{\frac{1-2\beta}{2-\beta}}m^{2}(2-\beta)^{2}\Theta^{-3}\left(\frac{4^{1-\beta}\pi^{2-\beta}(2-\beta)}{\beta}\,\phi^{2}\right)^{\frac{3}{2-\beta}}}{3^{\frac{1}{2-\beta}}(4\pi)^{\beta}(1-\beta)^{2}\Theta\left(\frac{4^{1-\beta}\pi^{2-\beta}(2-\beta)}{\beta}\,\phi^{2}\right)^{\frac{1-\beta}{2-\beta}}+4\pi^{2}(2-\beta)^{2}\phi^{2}}\,.$
(61)

With the help of Eqs. (9), (22), (40), (42), and (56), one can rewrite the
scalar spectral index in Eq. (V) and the tensor-to scalar ratio in Eq. (49) as

$\displaystyle n_{s}\simeq$ $\displaystyle
1+\,2\,\Theta^{-\beta+2}\,3^{\frac{1-\beta}{-2+\beta}}\Big{[}2^{1+2\beta}3^{1+\beta}\pi^{\beta}+3^{1+\beta}(4\pi)^{\beta}(-3+\beta^{2})\beta+(4\pi^{2})(2-\beta)^{2}\,3^{\frac{-1-\beta+\beta^{2}}{-2+\beta}}$
$\displaystyle\times(1+\beta)\,\Theta^{-1}\left(\frac{4^{1-\beta}\pi^{2-\beta}(2-\beta)}{\beta}\,\phi^{2}\right)^{\frac{\beta-1}{2-\beta}}\phi^{2}\Big{]}\Big{/}(\beta-2)\Big{[}3^{1+\beta}(4\pi)^{\beta}(1-\beta)^{2}$
$\displaystyle\times\Theta^{-\beta+1}\left(\frac{4^{1-\beta}\pi^{2-\beta}(2-\beta)}{\beta}\,\phi^{2}\right)^{\frac{1}{2-\beta}}+(4\pi^{2})\,3^{\frac{1-\beta+\beta^{2}}{-2+\beta}}(2-\beta)^{2}\Theta^{-\beta}$
$\displaystyle\times\left(\frac{4^{1-\beta}\pi^{2-\beta}(2-\beta)}{\beta}\,\phi^{2}\right)^{\frac{\beta}{2-\beta}}\phi^{2}\Big{]}\,,$
(62) $\displaystyle r\simeq$
$\displaystyle\,4\beta\,(2-\beta)^{-3}\,3^{\frac{2-\beta^{2}}{\beta-2}}\,\Theta\left(\frac{4^{1-\beta}\pi^{2-\beta}(2-\beta)}{\beta}\,\phi^{2}\right)^{-\frac{\beta}{2-\beta}}$
$\displaystyle\times\left(3^{1+\beta}(4\pi)^{\beta}\pi^{-2}(1-\beta)^{2}\Theta\,\phi^{-2}+4(2-\beta)^{2}\,3^{\frac{\beta(1-\beta)+1}{2-\beta}}\left(\frac{4^{1-\beta}\pi^{2-\beta}(2-\beta)}{\beta}\,\phi^{2}\right)^{\frac{\beta-1}{2-\beta}}\right)\,.$
(63)

Using Eqs. (9) and (56) in Eq. (42), we find

$f=\frac{2m^{2}}{\beta}\phi^{2}\,.$ (64)

We can also find the Hubble parameter $H$ by substituting Eqs. (9) and (56) in
Eq. (40), as the following form

$H=3^{\frac{1}{2(-2+\beta)}}m\Theta^{-\frac{1}{2}}\left(\frac{2^{2(1-\beta)}\pi^{2-\beta}(2-\beta)\phi^{2}}{\beta}\right)^{\frac{1}{2(2-\beta)}}\,.$
(65)

With the help of Eq. (57) and solving the equation $\varepsilon_{1}(\phi_{\rm
e})=1$, we can obtain the value of the scalar field at the end of inflation as

$\phi_{\rm
e}^{2}\simeq\frac{12^{-1+\beta}\,\pi^{-2+\beta}\beta\,\Theta^{2-\beta}}{(2-\beta)^{3-\beta}}\,.$
(66)

If we use Eqs. (9) and (56) in Eq. (52), we will have

$\frac{d\phi}{dN}\simeq
3^{\frac{-1+\beta}{2-\beta}}\,\Theta\left(\frac{4^{1-\beta}\pi^{2-\beta}(2-\beta)}{\beta}\,\phi^{2}\right)^{\frac{1}{\beta-2}}\phi.$
(67)

Solving the differential equation (67), we find the scalar field in terms of
the $e$-fold number $N$ as

$\phi(N)\simeq(2\pi)^{\frac{\beta}{2}}\left(\frac{\beta}{\pi^{2}(2-\beta)}\right)^{\frac{1}{2}}\left(\frac{(2-\beta)}{3^{\frac{1-\beta}{\beta-2}}N\,\Theta+\left(\frac{(-1)^{1+\beta}\,2^{\beta}\pi^{-2+\beta}\,(\beta-2)^{\beta-3}\beta}{\phi_{\rm
e}^{2}}\right)^{\frac{1}{\beta-2}}}\right)^{\frac{\beta-2}{2}}.$ (68)

Substituting $\phi_{e}$ from Eq. (66) into Eq. (68), we find

$\phi(N)\simeq(2\pi)^{\frac{\beta}{2}}\left(\frac{\beta}{\pi^{2}(2-\beta)}\right)^{\frac{1}{2}}\left(\frac{3^{\frac{-1+\beta}{-2+\beta}}(2-\beta)}{\frac{1}{2}(2N+1)\Theta}\right)^{\frac{\beta-2}{2}}.$
(69)

When $\beta=1$ and $\Theta=1/(4G)$, Eq. (69) reduces to the obtained result in
conventional inflation, i.e. $\phi(N)=\sqrt{2(2N+1)}$.

With the help of Eq. (69), we can rewrite Eq. (64) in terms of the $e$-fold
number $N$ and the parameters $\beta$, $\Theta$, and $m$ as

$f=\frac{(2-\beta)^{-3+\beta}}{6\pi^{2}}(12\pi)^{\beta}m^{2}\Big{(}\Theta(1+2N)\Big{)}^{2-\beta}\,.$
(70)

Applying Eq. (69) in Eq. (65), we take

$H=\frac{m\sqrt{1+2N}}{\sqrt{3(2-\beta)}}\,.$ (71)

Now, we can substitute Eq. (69) into Eqs. (57), (58), and (VI.1), and obtain
the slow-roll parameters as the following forms

$\displaystyle\varepsilon_{1}\simeq\frac{1}{1+2N},$ (72)
$\displaystyle\varepsilon_{2}=\varepsilon_{3}\simeq\frac{-1+\beta}{1+2N}\,,$
(73)
$\displaystyle\varepsilon_{4}\simeq\frac{\beta(-1+\beta)\Big{(}3(-1+\beta)+(1+2N)\Big{)}}{(1+2N)\Big{(}3(-1+\beta)^{2}+\beta(1+2N)\Big{)}}.$
(74)

Using Eq. (69) into Eq. (61), we find the scalar power spectrum as

${\cal
P}_{s}\simeq\frac{4\pi^{-\beta}(6-3\beta)^{-\beta}m^{2}(1+2N)^{2+\beta}}{3(1-\beta)^{2}+\beta(1+2N)}\,\Theta^{\beta-2}.$
(75)

As we see from Eq. (75), the power spectrum of the curvature perturbation is a
function of the $e$-fold number $N$ and three free parameters $\beta$,
$\Theta$, and $m$. One can use Eq. (75) and then impose the CMB normalization
at the observable scale to find a constraint on the parameter $m$.

Applying Eq. (69), the scalar spectral index (62) and the tensor-to-scalar
ratio (63) take the forms

$\displaystyle n_{s}$ $\displaystyle\simeq
1-\frac{2\Big{(}6+3\beta(-3+\beta^{2})+\beta(1+\beta)(1+2N)\Big{)}}{(1+2N)\Big{(}\beta(1+2N)+3(1-\beta)^{2}\Big{)}}\,,$
(76) $\displaystyle r$
$\displaystyle\simeq\frac{16\Big{(}3(1-\beta)^{2}+\beta(1+2N)\Big{)}}{(1+2N)^{2}}\,.$
(77)

In the limit of standard inflation where $\beta=1$, Eqs. (76) and (77) reduces
to $n_{s}=1-4/(2N+1)$ and $r=16/(2N+1)$, respectively. These results are the
same equations that we find in the setup of the standard inflation. Let us now
consider the limits where $\beta\ll 1$ and $N\gg 1$. From Eqs. (76) and (77),
the leading contributions to $n_{s}$ and $r$ become

$\displaystyle n_{s}\simeq$ $\displaystyle 1-\frac{2}{N}\,,$ (78)
$\displaystyle r\simeq$ $\displaystyle\frac{12}{N^{2}}\,.$ (79)

This means that in the regime $\beta\ll 1$ and $N\gg 1$, our theory reduces to
the $R^{2}$ inflationary model proposed by Starobinsky Starobinsky (1980). In
this regime, the power spectrum (75) reduces to the form

${\cal P}_{s}\simeq\frac{m^{2}(2N)^{2}}{3\Theta^{2}}\,.$ (80)

Using Eq. (80) and then impose the CMB normalization at the pivot scale
$k_{*}=0.05\,{\rm Mpc}^{{\rm-1}}$ Akrami  _et al._ (2020), we find

$\displaystyle m\simeq 7.937\times 10^{-7}\Theta\hskip 28.45274pt{\rm
for}\hskip 28.45274ptN_{*}=50\,,$ (81) $\displaystyle m\simeq 6.614\times
10^{-7}\Theta\hskip 28.45274pt{\rm for}\hskip 28.45274ptN_{*}=60\,.$ (82)

As we see the value of the parameter $m$ depends on what value the parameter
$\Theta$ takes.

With the help of Eqs. (76) and (77), we can plot the $r-n_{s}$ diagram and
compare the prediction of our model with the Planck 2018 CMB data Akrami  _et
al._ (2020). Fig. 1 shows the prediction of our model in the $r-n_{s}$ plane
for two typical values of $N_{*}$ and varying $\beta$ in the range of
$0<\beta<2$. The dashed and solid black curves illustrate the results of the
model for $N_{*}=50$ and $N_{*}=60$, respectively. Besides, The red solid line
between the dashed and solid black curves shows the prediction of the
potential in the standard inflation which corresponds to $\beta=1$ in our
model, in the range of $50\leq N_{*}\leq 60$. Moreover, the prediction of the
Starobinsky $R^{2}$ inflationary model Starobinsky (1980) has been shown by
the orange solid line with $50\leq N_{*}\leq 60$.

Figure 1: The $r-n_{s}$ diagram for the quadratic potential (56) in the slow-
roll inflation based on the Tsallis entropy for two different values of
$N_{*}$ with varying $\beta$ in the range of $0<\beta<2$. The results for
$N_{*}=50$ and $N_{*}=60$ are shown by the dashed and solid black curves,
respectively. The red solid line between the dashed and solid black curves
shows the results of the potential in the standard inflation, in the range of
$50\leq N_{*}\leq 60$. Moreover, the result of the Starobinsky $R^{2}$
inflation Starobinsky (1980) is specified by the orange solid line with
$50\leq N_{*}\leq 60$. The marginalized joint 68% and 95% CL regions of the
Planck 2018 TTTEEE+low$\ell$+lowE data Akrami  _et al._ (2020) are specified
by dark and light blue, respectively.

In Fig. 1, the parameter $\beta$ has been taken as a varying parameter in the
range $0<\beta<2$. It should be noted that each value of $\beta$ in this range
is related to a special case of our power-law $f(R)$ scenario, and in the case
with $\beta=1$ and $\gamma=1/(4G)$, our $f(R)$ model reduces to the Einstein
general relativity (GR). This does not mean that a transition from $f(R)$ to
GR has occurred during inflation in our scenario. In other words, to
investigate each case of our scenario, we should fix the value of $\beta$ at
the first step, and it is not the case that this parameter varies during
inflation and causes a transition from $f(R)$ to GR. Since we cannot determine
the parameter $\gamma$ in our investigation, therefore we cannot determine the
time at which the $f(R)$ gravity in Eq. (38) transits to GR.

Fig. 1 shows that the prediction of the potential $m^{2}\phi^{2}/2$ in the
standard setting is not in good consistency with the Planck 2018 observations
Akrami  _et al._ (2020), while in the framework of the Tsallis entropy-based
inflationary scenario, it can be in very good agreement with these data, and
its results can lie inside the 68% CL region of Planck 2018 data Akrami  _et
al._ (2020). From the figure, we see that for small $\beta$, the model shows
better consistency with the observations and the prediction of the model can
enter the 68% CL region of these data. As we have proved, in the limit
$\beta\ll 1$, the inflationary observables $n_{s}$ and $r$ approach to the
same values in the Starobinsky $R^{2}$ inflation Starobinsky (1980).

With the help of Eqs. (76) and (77), and also the Planck observational
constraints on the $r–n_{s}$ plane, we can estimate the ranges of the
parameter $\beta$ for which the results of the model in the $r-n_{s}$ plane
are consistent with the 68% CL region of the Planck 2018 data Akrami  _et al._
(2020). In the case $N_{*}=50$, if $0<\beta\lesssim 0.045$ the result of the
model is in agreement with the 68% CL constraint of these data, and for
$N_{*}=60$ the prediction of our model can enter the 68% CL region of the
Planck 2018 data Akrami  _et al._ (2020), provided that $0<\beta\lesssim
0.011$.

Here, we are interested in applying the recent constraint of Galloni  _et al._
(2023) on $r_{0.01}$ to present some observational constraint on the model
parameter $\beta$. For this purpose, we should determine the $e$-fold number
at which the comoving wavenumber $k=0.01\,\mathrm{Mpc}^{-1}$ exits the Hubble
horizon during inflation. To do so, we examine the behavior of the comoving
wavenumber $k$ as a function of the $e$-fold number $N$, at which the mode
with comoving wavenumber $k$ leaves the Hubble horizon, $k=aH$. With the help
of this relation, we can easily find

$k(N)=\left(\frac{a(N)H(N)}{a(N_{*})H(N_{*})}\right)k_{*},$ (83)

which $a(N_{*})$ and $H(N_{*})$ are the scale factor and the Hubble parameter
at the time of horizon exit of the mode $k_{*}=0.05\,\mathrm{Mpc}^{-1}$,
respectively. Using Eq. (50), we can obtain the scale factor $a$ in terms of
the $e$-fold number $N$ as

$a(N)\simeq\exp({N_{*}-N}),$ (84)

where we have normalized the scale factor to its value at the epoch of horizon
crossing of the mode $k_{*}=0.05\,\mathrm{Mpc}^{-1}$. Finally, applying Eqs.
(71) and (84) in Eq. (83), the comoving number $k$ can be found as a function
of $N$ as

$k(N)=\left(\sqrt{\frac{1+2N}{1+2N_{*}}}\exp({N_{*}-N})\right)0.05\,{\rm
Mpc}^{{\rm-1}}\,.$ (85)

Using Eq. (85) and setting $N_{*}=60$, we plot in Fig. 2 the variation of the
comoving wavenumber $k$ against the $e$-fold number $N$. In the figure, we
have also specified the comoving wavenumber $k=0.01\,{\rm Mpc}^{{\rm-1}}$ and
its corresponding $e$-fold number which is $N_{0.01}\simeq 61.62$.

Figure 2: Evolution of the comoving wavenumber $k$ versus the $e$-fold number
$N$ for the quadratic potential (56) in the Tsallis inflationary setting. The
pink horizontal and vertical dashed lines specify $k=0.01\,{\rm
Mpc}^{{\rm-1}}$ and its corresponding $e$-fold number $N_{0.01}\simeq 61.62$,
respectively.

In Fig. 3, with the help of Eq. (77), the variation of the tensor-to-scalar
ratio $r_{0.01}$ is plotted as a function of the parameter $\beta$ by taking
$N_{0.01}\simeq 61.62$. The gray shaded region is excluded by the constraint
reported by Galloni et al. Galloni  _et al._ (2023) at the CMB pivot scale
$k=0.01\,{\rm Mpc}^{{\rm-1}}$. Our results imply that the model satisfies the
constraint $r_{0.01}<0.028$, provided that $0<\beta\lesssim 0.202$.

Figure 3: Variation of the tensor-to-scalar ratio $r_{0.01}$ versus the
parameter $\beta$ by setting $N_{0.01}\simeq 61.62$, for the quadratic
potential (56) in the framework of the Tsallis inflation. The gray-shaded
region is excluded by the constraint on the upper bound on $r_{0.01}$,
reported by Galloni et al. Galloni  _et al._ (2023). The pink vertical dashed
line specifies $\beta\simeq 0.202$ which is the maximum value of the parameter
$\beta$ for which the model satisfies the constraint $r_{0.01}<0.028$.

Using Eqs. (72), (73), and (74), we plot the evolution of the slow-roll
parameters $\varepsilon_{1}$, $\varepsilon_{2}$, $\varepsilon_{3}$, and
$\varepsilon_{4}$ as a function of $N$ for $\beta=0.01$ in Fig. 8. From the
figure, it is obvious that the slow-roll approximation is valid until the end
of inflation. This point proves the viability of our examination which is
based on the slow-roll approximation. Besides, the figure shows that
$\varepsilon_{1}$ becomes unity at $N=0$ which corresponds to the end of
inflation.

Figure 4: Evolution of the slow-roll parameters $\varepsilon_{1}$,
$\varepsilon_{2}$, $\varepsilon_{3}$, and $\varepsilon_{4}$ versus the
$e$-fold number $N$ for the quadratic potential (56) in the setup of the
Tsallis inflation, with $\beta=0.01$.

Here, it should be noted that throughout this paper, we have worked in the
framework of the $f(R)$ gravity given by Eq. (38), and we have treated GR as a
special case of our $f(R)$ gravity model that is realized by taking $\beta=1$
and $\gamma=1/(4G)$. To clarify this point, we note that for a given set of
model parameters $(\beta,\gamma)$, one can find $\Theta$ from Eq. (60) as a
function of the parameter $m$. Substituting $\Theta=\Theta(m)$ in Eq. (75) and
then fixing the amplitude of scalar perturbations ${\cal P}_{s}$ at the epoch
of horizon crossing with the $e$-fold number $N_{*}=60$ as ${\cal
P}_{s}(k_{*})\simeq 2.1\times 10^{-9}$ Akrami  _et al._ (2020), we can
determine the value of $m$. In Fig. 5, by applying Eq. (70), we plot the
evolution of the function $f(R)$ against the $e$-fold number $N$ by taking
$\beta=0.01$ for some typical values of $\gamma$. The prediction of general
relativity (GR) for the Ricci scalar $R$ is also shown in the figure by the
red color. The figure clearly shows the value of the function $f(R)$ depends
on the parameter $\gamma$. As we see, different values of $\gamma$ lead to the
different values of $f(R)$. It can be concluded that $\gamma$ manages the
behavior of the $f(R)$ function and causes the deviation from GR. Since
$\gamma$ is a free parameter in our model, we may take its value such that the
regime of the $f(R)$ gravity dominates over the GR regime throughout
inflation, and consequently any transition would not occur during inflation at
all. This means that in the effective action which may contain the $f(R)$ term
together with the Einstein-Hilbert term $R$, the contribution of the latter
will be negligible compared to the former contribution, and therefore the
effective action will be reduced to the action (16) which is used in the
present work. Besides, since our investigation is not able to determine the
precise value of $\gamma$, it is not possible to specify the time of the
transition from $f(R)$ to GR that may happen during inflation or in the post-
inflationary Universe, in the present work. However, if future studies provide
some understanding for us about the nature of the inflaton field, we may
determine its effective mass, and accordingly estimate value of the $\gamma$.
This enables us to estimate the time of such a transition. In addition, we may
provide some observational constraints of our model parameter $\beta$ in the
setup of the Tsallis inflation, like the analysis performed in Luciano and
Saridakis (2022) for the case of the Barrow cosmology.

Figure 5: Evolution of the function $f(R)$ versus the $e$-fold number $N$ for
the quadratic potential (56) in the Tsallis inflationary scenario with
$\beta=0.01$ and some typical values of $\gamma$. Also, the red curve
illustrates the Ricci scalar $R$ predicted by general relativity (GR).

###  VI.2 Natural inflation

The next model that we consider in our investigation is the natural inflation
model in which the inflaton field is presumed to be an axion or pseudo-Nambu-
Goldstone boson with a cosine-type periodic potential Freese  _et al._ (1990);
Adams  _et al._ (1993)

$V(\phi)=\Lambda^{4}\left[1+\cos\left(\frac{\phi}{\phi_{0}}\right)\right]\,,$
(86)

where $\Lambda$ is some non-perturbatively generated scale and $\phi_{0}$ is
the scale that determines the curvature of the potential. Both of these
constants have dimensions of mass. It seems that the super-Planckian value of
the scale $\phi_{0}$, i.e. $\phi_{0}\gtrsim M_{P}$, is impossible in the
context of string theory because all known controlled string theory
constructions are restricted to $\phi_{0}<M_{P}$ Svrcek and Witten (2006);
Banks  _et al._ (2003). In the framework of standard slow-roll inflation, the
prediction of the natural potential in the $r-n_{s}$ plane is in tension with
the latest observations Akrami  _et al._ (2020), in the sense that its
prediction can lie within the 95% CL region of these data only for $N_{*}=60$.
This issue motivates us to study natural inflation in the Tsallis entropy-
based scenario to see whether the novel framework can improve the consistency
of the model with the Planck 2018 observations Akrami  _et al._ (2020).

Using Eqs. (9), (22), (40), and (42), the slow-roll parameters in Eqs. (43),
(44), (45), and (46) for the Tsallis entropy-based model described by the
potential (86) can be obtained as

$\displaystyle\varepsilon_{1}\simeq\frac{3^{\frac{-1+\beta}{2-\beta}}\eta}{2(2-\beta)}\left(\frac{2^{3-2\beta}\pi^{2-\beta}(2-\beta)(1+\cos(\chi))}{\beta}\right)^{\frac{1}{\beta-2}}(1-\cos(\chi))\,,$
(87)
$\displaystyle\varepsilon_{2}\simeq\frac{3^{\frac{-1+\beta}{2-\beta}}\eta}{2(2-\beta)}\left(\frac{2^{3-2\beta}\pi^{2-\beta}(2-\beta)(1+\cos(\chi))}{\beta}\right)^{\frac{1}{\beta-2}}\Big{(}1-(2\beta-3)\cos(\chi)\Big{)}\,,$
(88)
$\displaystyle\varepsilon_{3}\simeq\frac{3^{\frac{-1+\beta}{2-\beta}}(-1+\beta)\eta}{2(2-\beta)}\left(\frac{2^{3-2\beta}\pi^{2-\beta}(2-\beta)(1+\cos(\chi))}{\beta}\right)^{\frac{1}{\beta-2}}(1-\cos(\chi))\,,$
(89)
$\displaystyle\varepsilon_{4}\simeq\frac{3^{\frac{-1+\beta}{2-\beta}}(-1+\beta)\eta}{(2-\beta)}\left(\frac{2^{3-2\beta}\pi^{2-\beta}(2-\beta)(1+\cos(\chi))}{\beta}\right)^{\frac{1}{\beta-2}}(1-\cos(\chi))$
$\displaystyle\times\Bigg{\\{}(4\pi)^{\beta}(-1+\beta)\big{(}1-(\beta-1)\cos(\chi)\big{)}\eta+8\pi^{2}(-2+\beta)^{2}(1+\cos(\chi))$
$\displaystyle\times
3^{\frac{1}{\beta-2}}\left(\frac{2^{3-2\beta}\pi^{2-\beta}(2-\beta)(1+\cos(\chi))}{\beta}\right)^{\frac{-\beta+1}{\beta-2}}\Bigg{\\}}\Big{/}\Bigg{\\{}(4\pi)^{\beta}(-1+\beta)^{2}(1-\cos(\chi))\eta$
$\displaystyle+(4\pi)^{2}3^{\frac{1}{\beta-2}}(-2+\beta)^{2}\left(\frac{2^{3-2\beta}\pi^{2-\beta}(2-\beta)(1+\cos(\chi))}{\beta}\right)^{\frac{-\beta+1}{\beta-2}}(1+\cos(\chi))\Bigg{\\}}.$
(90)

Here, we have defined

$\eta\equiv\frac{\Lambda^{4}}{{\phi_{0}}^{2}}\left(\frac{\Lambda^{4}}{\gamma}\right)^{\frac{1}{\beta-2}}\,,\hskip
28.45274pt\chi\equiv\frac{\phi}{\phi_{0}}.$ (91)

Applying Eqs. (9), (22), (40), (42), and (86), we can obtain the scalar power
spectrum (47) as the following form

$\displaystyle{\cal
P}_{s}\simeq\Bigg{\\{}2(-2+\beta)^{2}\eta^{-3}(\Xi)^{4}3^{\frac{2\beta}{-2+\beta}}\left(\frac{2^{3-2\beta}\pi^{2-\beta}(2-\beta)(1+\cos(\chi))}{\beta}\right)^{\frac{2+\beta}{2-\beta}}\csc^{2}(\frac{\chi}{2})\Bigg{\\}}\Big{/}$
$\displaystyle\Bigg{\\{}(4\pi)^{\beta}(-1+\beta)^{2}(1-\cos(\chi))\eta+16\pi^{2}(-2+\beta)^{2}\,3^{\frac{1}{-2+\beta}}\left(\frac{2^{3-2\beta}\pi^{2-\beta}(2-\beta)(1+\cos(\chi))}{\beta}\right)^{\frac{\beta-1}{2-\beta}}$
$\displaystyle\times(1+\cos(\chi))\Bigg{\\}}\,,$ (92)

where

$\Xi\equiv\Lambda/\phi_{0}\,.$ (93)

If we apply Eq. (9) and also the potential (86) in Eq. (52), we reach the
following differential equation

$\frac{d\chi}{dN}\simeq-3^{\frac{-1+\beta}{2-\beta}}\eta\left(\frac{2^{3-2\beta}\pi^{2-\beta}(2-\beta)(1+\cos(\chi))}{\beta}\right)^{\frac{1}{\beta-2}}\sin(\chi)\,.$
(94)

Solving the differential equation (94), we find the evolution of scalar field
$\chi$ with respect to the $e$-fold number $N$ in the slow-roll approximation.

By using Eqs. (9), (22), (40), (42), and (86) in Eq. (V), we find the scalar
spectral index $n_{s}$ as a function of $\chi$, $\beta$, and $\eta$ as

$\displaystyle n_{s}\simeq
1+\frac{3^{\frac{1-\beta}{-2+\beta}}}{2(-2+\beta)}\Bigg{\\{}5\,\pi^{\beta}\,2^{1+2\beta}\,3^{1+\beta}-19\,\beta\,3^{1+\beta}(4\pi)^{\beta}+2^{3+2\beta}\,3^{1+\beta}\,\pi^{\beta}\beta^{2}$
$\displaystyle+3^{1+\beta}(4\pi)^{\beta}\beta^{3}+\eta^{-1}\Big{(}448\,\pi^{2}\,3^{\frac{-1-\beta+\beta^{2}}{-2+\beta}}-64\,\pi^{2}\,3^{\frac{-5+\beta+\beta^{2}}{-2+\beta}}\beta+80\,\pi^{2}\,3^{\frac{-3+\beta^{2}}{-2+\beta}}\beta^{2}$
$\displaystyle-32\,\pi^{2}\,3^{\frac{-1-\beta+\beta^{2}}{-2+\beta}}\beta^{3}\Big{)}\left(\frac{2^{3-2\beta}\pi^{2-\beta}(2-\beta)(1+\cos(\chi))}{\beta}\right)^{\frac{\beta-1}{2-\beta}}$
$\displaystyle-\cos(\chi)\Big{[}12^{1+\beta}\pi^{\beta}(1-\beta)^{2}(2+\beta)+64\,\pi^{2}\,3^{\frac{-1-\beta+\beta^{2}}{-2+\beta}}(-2+\beta)^{3}\,\eta^{-1}$
$\displaystyle\times\left(\frac{2^{3-2\beta}\pi^{2-\beta}(2-\beta)(1+\cos(\chi))}{\beta}\right)^{\frac{\beta-1}{2-\beta}}\Big{]}-\cos(2\chi)\Big{[}-3^{1+\beta}(4\pi)^{\beta}(7+3\beta^{2})\beta$
$\displaystyle+2^{1+2\beta}\,3^{1+\beta}\pi^{\beta}(1+4\beta^{2})+16\,\pi^{2}\,3^{\frac{-1-\beta+\beta^{2}}{-2+\beta}}(2-\beta^{2})(-1+2\beta)\,\eta^{-1}$
$\displaystyle\times\left(\frac{2^{3-2\beta}\pi^{2-\beta}(2-\beta)(1+\cos(\chi))}{\beta}\right)^{\frac{\beta-1}{2-\beta}}\Big{]}\Bigg{\\}}\Big{/}\Bigg{\\{}3^{1+\beta}(1-\cos(\chi))$
$\displaystyle\times(4\pi)^{\beta}(1-\beta)^{2}\eta^{-1}\left(\frac{2^{3-2\beta}\pi^{2-\beta}(2-\beta)(1+\cos(\chi))}{\beta}\right)^{\frac{1}{2-\beta}}$
$\displaystyle+16\,\pi^{2}(1+\cos(\chi))\,3^{\frac{-1-\beta+\beta^{2}}{-2+\beta}}(2-\beta)^{2}\eta^{-2}\left(\frac{2^{3-2\beta}\pi^{2-\beta}(2-\beta)(1+\cos(\chi))}{\beta}\right)^{\frac{\beta}{2-\beta}}\Bigg{\\}}\,.$
(95)

We also can rewrite the tensor-to-scalar ratio $r$ in Eq. (49) as the
following form

$\displaystyle r\simeq$
$\displaystyle\frac{3^{\frac{2-\beta^{2}}{-2+\beta}}\,\beta\,\eta}{(\beta-2)^{3}\,\pi^{2}}\left(\frac{2^{3-2\beta}\,\pi^{2-\beta}(2-\beta)}{\beta}\right)^{\frac{1+\beta}{\beta-2}}\cos^{2}\big{(}\frac{\chi}{2}\big{)}\big{(}1+\cos(\chi)\big{)}^{\frac{2(\beta-3)}{2-\beta}}$
$\displaystyle\times\Big{[}3^{1+\beta}(4\pi)^{\beta}(1-\beta)^{2}\,\eta\left(\frac{2^{3-2\beta}\,\pi^{2-\beta}(2-\beta)}{\beta}\right)^{\frac{1}{2-\beta}}\big{(}\cos(\chi)-1\big{)}$
$\displaystyle-16\,\pi^{2}(2-\beta)^{2}\,3^{\frac{-1-\beta+\beta^{2}}{-2+\beta}}\big{(}1+\cos(\chi)\big{)}^{\frac{1}{2-\beta}}\left(\frac{2^{3-2\beta}\,\pi^{2-\beta}(2-\beta)}{\beta}\right)^{\frac{\beta}{2-\beta}}\Big{]}\sin^{2}(\chi)\,,$
(96)

where we have applied Eqs. (9), (22), (40), (42), and (86) in deriving Eq.
(VI.2).

With the help of Eqs. (9) and (86), we can rewrite Eq. (42) as

$f=\frac{4\Lambda^{4}}{\beta}(1+\cos(\chi))\,.$ (97)

Substituting Eqs. (9) and (86) in Eq. (40), the Hubble parameter $H$ is found
as the following form

$H=3^{\frac{1}{2(\beta-2)}}\,\Big{(}\frac{\Lambda^{4}}{\phi_{0}^{2}\eta}\Big{)}^{\frac{1}{2}}\left(\frac{2^{3-2\beta}\,\pi^{2-\beta}(2-\beta)(1+\cos(\chi))}{\beta}\right)^{\frac{1}{2(2-\beta)}}\,.$
(98)

We can apply the above equation together with Eq. (84) in Eq. (83), to find
the comoving number $k$ as

$k(N)=\Big{(}(1+\cos(\chi))^{\frac{1}{2(2-\beta)}}(1+\cos(\chi_{*}))^{\frac{1}{2(-2+\beta)}}\,\exp({N_{*}-N})\Big{)}\,0.05\,{\rm
Mpc}^{{\rm-1}}\,.$ (99)

To find the scalar field $\chi$ in terms of the $e$-fold number $N$, we
perform the same steps as the previous subsection. In the first step, using
Eq. (87) for a given set of the model parameters $(\beta,\eta)$, we solve the
equation $\varepsilon_{1}(\chi_{e})=1$ numerically to find $\chi_{\rm e}$.
After that, with the help of the obtained value of $\chi_{\rm e}$, we solve
differential equation (94), numerically, and consequently we find
$\chi=\chi(N)$. Substituting $\chi(N)$ into Eqs. (87), (88), (89), (VI.2),
(95), (VI.2), and (99), the slow-roll parameters $\varepsilon_{1}$,
$\varepsilon_{2}$, $\varepsilon_{3}$, $\varepsilon_{4}$, the inflationary
observables $n_{s}$ and $r$, and also the comoving number $k$ are derived in
terms of $N$, respectively.

In Fig. 6, we plot the $r-n_{s}$ diagram for $\beta=1/2$ and $N_{*}=50,60$
with varying $\eta$ in the range of $\eta>0$. The black dashed and solid
curves illustrate the prediction of the model for $N_{*}=50$ and $N_{*}=60$,
respectively. The standard natural inflation which corresponds to $\beta=1$ in
our model, is also presented by the red dashed ($N_{*}=50$) and solid
($N_{*}=60$) curves. From the figure, it is obvious that as $\eta$ increases,
both $n_{s}$ and $r$ get smaller. Moreover, Fig. 6 clears that the result of
the potential (86) in the standard inflation for $N_{*}=50$ lies completely
outside the allowed regions of the Planck 2018 observations Akrami  _et al._
(2020) and only for $N_{*}=60$ enters the 95% CL region of these data. In the
Tsallis entropy-based inflationary setting, however, its prediction can lie
well within the 68% CL region allowed by the Planck 2018 data Akrami  _et al._
(2020).

Using the Planck observational constraints on $r–n_{s}$ plane and also Eqs.
(95) and (VI.2), we also can estimate the ranges of parameter $\eta$ for which
the prediction of our model for $\beta=1/2$ and different values of $N_{*}$ is
compatible with 68% CL constraint of the Planck 2018 data Akrami  _et al._
(2020). In the case $N_{*}=50$, the model is compatible with 68% CL constraint
of the Planck 2018 data for $0.172\lesssim\eta\lesssim 0.433$. We also find
that for $N_{*}=60$, the results of our model can lie inside the 68% CL region
of the Planck 2018 data, if $0.281\lesssim\eta\lesssim 0.613$.

Figure 6: The $r-n_{s}$ diagram of the natural potential (86) in the Tsallis
inflationary setting by taking $\beta=1/2$ for two different values of $N_{*}$
with varying $\eta$ in the range of $\eta>0$. The results of our model for
$N_{*}=50$ and $N_{*}=60$ are shown by the dashed and solid black curves,
respectively. The red dashed ($N_{*}=50$) and solid ($N_{*}=60$) curves show
the prediction of the standard natural inflation. The marginalized joint 68%
and 95% CL regions of the Planck 2018 TTTEEE+low$\ell$+lowE data Akrami  _et
al._ (2020) are specified by dark and light blue, respectively.

To present some constraint on the model parameter $\eta$ by using the recent
constraint of Galloni  _et al._ (2023) on $r_{0.01}$, we determine the
$e$-fold number at which the comoving wavenumber $k=0.01\,\mathrm{Mpc}^{-1}$
exits the Hubble horizon during inflation, with the help of Eq. (99) by taking
$\beta=1/2$ and $N_{*}=60$. It is found that $N_{0.01}\simeq 61.6$.

In Fig. 7, using Eq. (VI.2), we plot the variation of the tensor-to-scalar
ratio $r_{0.01}$ against the parameter $\eta$ by setting $\beta=1/2$ and
$N_{0.01}\simeq 61.6$. The gray-shaded region is excluded by the constraint on
the upper bound on $r_{0.01}$ reported by Galloni et al. Galloni  _et al._
(2023). The prediction of the model is in agreement with this constraint if
$\eta\gtrsim 0.593$.

Figure 7: Variation of the tensor-to-scalar ratio $r_{0.01}$ against the
parameter $\eta$ for the natural potential (86) in the Tsallis inflationary
scenario with $\beta=1/2$ and $N_{0.01}\simeq 61.6$. The gray-shaded region is
excluded by the constraint on the upper bound on $r_{0.01}$, reported by
Galloni et al. Galloni  _et al._ (2023). The pink vertical dashed line
specifies $\eta\simeq 0.593$ which is the minimum value of the parameter
$\eta$ for which the model agrees with the constraint $r_{0.01}<0.028$.

The evolution of the slow-roll parameters $\varepsilon_{1}$,
$\varepsilon_{2}$, $\varepsilon_{3}$, and $\varepsilon_{4}$ versus the
$e$-fold number $N$ for the natural potential (86) is plotted in Fig. 8, by
using Eqs. (87)-(VI.2). In this figure, we have considered $\beta=1/2$ and
$\eta=0.65$. The figure shows that the slow-roll approximation holds during
inflation. This point verifies the viability of our investigation which is
based on the slow-roll approximation. Moreover, from the figure, we see that
the first slow-roll parameter $\varepsilon_{1}$ reaches unity at the end of
inflation with $N=0$.

Figure 8: Evolution of the slow-roll parameters $\varepsilon_{1}$,
$\varepsilon_{2}$, $\varepsilon_{3}$, and $\varepsilon_{4}$ as a function of
the $e$-fold number $N$ for the natural potential (86) in the Tsallis entropy-
based model with $\beta=1/2$ and $\eta=0.65$.

In Fig. 9, with the help of Eq. (97), the evolution of the function $f(R)$ as
a function of the $e$-fold number $N$ is plotted for some typical values of
$\gamma$, and by setting $\beta=1/2$ and $\eta=0.65$. Note that to find the
parameter $\Lambda$, we follow this approach: in the first step, for given
values of the parameters $\beta$ and $\eta$, the value of $\Xi$ is fixed by
imposing the CMB normalization at the pivot scale $k_{*}=0.05\,{\rm
Mpc}^{{\rm-1}}$ corresponding to $N_{*}=60$, with the help of Eq. (VI.2).
Afterwards, for a specific value of the parameter $\gamma$ by using the first
definition in Eq. (91) and also Eq. (93), one can find the value of the
parameter $\Lambda$. In the figure, We also show the Ricci scalar $R$
predicted by GR by the red color. Fig. 9 clears the effect of the parameter
$\gamma$ on the function $f(R)$. Since in this model we are free to take any
values for the parameter $\gamma$, by choosing an appropriate value of this
parameter, we can guarantee that the regime of $f(R)$ gravity dominates over
the GR regime in the whole of inflation. Therefore, if the effective action
includes the Einstein-Hilbert term $R$ in addition to the $f(R)$ term, then
the former contribution can be dropped versus the latter contribution, and the
effective action turns into the action (16). To determine the epoch of the
transition from $f(R)$ to GR, we need the precise value of the $\gamma$
parameter that cannot be determined in the present work.

Figure 9: Evolution of the function $f(R)$ against of the $e$-fold number $N$
for the natural potential (86) in the Tsallis inflation with $\beta=1/2$ and
$\eta=0.65$, for some typical values of $\gamma$. The prediction of general
relativity (GR) for the Ricci scalar $R$ is also shown by the red color.

##  VII Conclusions

Inflation has occurred in the regime of high energy physics at which the
gravitational theory is expected to be modified. Therefore, the entropy-area
relation may undergo some modifications in those energy scales. This motivated
us to regard the entropy of the early Universe to be in the form of the
Tsallis entropy which is a generalization for the Bekenstein-Hawking entropy
Bekenstein (1973) and possesses the non-additivity and non-extensivity
property. This form of entropy has a relation with the horizon area as
$S_{h}=\gamma A^{\beta}$, in which $A$ is the area of the horizon and $\beta$
and $\gamma$ are unknown constants. We have studied the inflationary era in
the context of the Tsallis entropy-based cosmology. Since there is no definite
action for this setup, it was not possible to derive the power spectra of the
primordial scalar and tensor perturbations in our Tsallis inflation scenario.
To resolve this issue, we reconstructed an $f(R)$ model which is
thermodynamically equivalent to our setting in the slow-roll approximation.
This equivalence allows us to use the equations of the scalar and tensor power
spectra obtained in the $f(R)$ gravity for our Tsallis inflation model. We
have considered two different inflationary potentials in our scenario and
checked their viability with the Planck 2018 observations Akrami  _et al._
(2020).

First, we studied the observational consistency of the quadratic potential
(56) which provides a chaotic inflation model. In the standard inflationary
setting based on the Einstein gravity, the consistency of this potential is
not favored by the Planck 2018 observational data Akrami  _et al._ (2020).
This motivated us to investigate whether this potential can be resurrected in
light of the Planck 2018 results in the setting of the Tsallis entropy-based
inflation. We have derived analytic formulas for $n_{s}$ and $r$ in terms of
the parameter $\beta$ and the $e$-fold number $N$, and then plotted the
$r-n_{s}$ diagram for $N_{*}=50,60$, with varying the parameter $\beta$ in the
range of $0<\beta<2$. Our results imply that this potential can be in
excellent consistency with the Planck 2018 data in the Tsallis entropy-based
scenario, such that its results can lie inside the 68% CL region of the
observational data.

Moreover, we have proved that in the limits $\beta\ll 1$ and $N\gg 1$, the
behavior of the quadratic potential in the $r-n_{s}$ plane coincides exactly
with the prediction of the Starobinsky $R^{2}$ inflation Starobinsky (1980).
Furthermore, we have estimated that for $N_{*}=50$, the prediction of the
model is compatible with the 68% CL constraint of the Planck 2018
observations, if $0<\beta\lesssim 0.045$. In the case of $N_{*}=60$, the model
is consistent with the 68% CL region of the observational data provided that
$0<\beta\lesssim 0.011$. The recent constraint $r_{0.01}<0.028$ (95% CL)
Galloni  _et al._ (2023) on the tensor-to-scalar ratio at the scale
$k=0.01\,\mathrm{Mpc}^{-1}$ constrains this parameter to $0<\beta\lesssim
0.202$.

We further examined the viability of the natural potential (86) in the
inflationary setting based on the Tsallis entropy. In the framework of the
standard inflation, the prediction of this potential in the $r-n_{s}$ plane is
not very preferred according to the current CMB observations, regarding the
fact that its results can satisfy only the 95% CL constraints of the Planck
2018 data Akrami  _et al._ (2020). In this case, we have found the
inflationary observables $n_{s}$ and $r$ in terms of the model parameters
$\beta$ and $\eta$ numerically. We have focused on the cases $\beta=1/2$ and
drawn the $r-n_{s}$ diagram for $N_{*}=50,60$, with varying $\eta$ in the
range of $\eta>0$. We have demonstrated that in the framework of the Tsallis
entropy-based inflation, the natural potential provides a great fit to the
Planck 2018 data Akrami  _et al._ (2020). The prediction of the model lies
inside the 68% CL region of these data. Furthermore, we presented some
observational constraints on the model parameters by using the Planck 2018
data Akrami  _et al._ (2020). Our results imply that for $N_{*}=50,60$, the
results of our model can lie inside the 68% CL region of the Planck 2018 data
Akrami  _et al._ (2020), provided that $0.172\lesssim\eta\lesssim 0.433$ and
$0.281\lesssim\eta\lesssim 0.613$, respectively. The observational bound on
$r_{0.01}$ provided in Galloni  _et al._ (2023), also gives rise to the
condition $\eta\gtrsim 0.593$ for the model parameter.

###### Acknowledgements.

The authors thank the referees for their valuable comments.

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