# Slow-roll inflation and growth of perturbations in Kaniadakis Cosmology

G. Lambiase<EMAIL_ADDRESS>Dipartimento di Fisica, Università degli
Studi di Salerno, Via Giovanni Paolo II, 132 I-84084 Fisciano (SA), Italy
INFN Gruppo Collegato di Salerno - Sezione di Napoli c/o Dipartimento di
Fisica, Università di Salerno, Italy  G. G. Luciano
<EMAIL_ADDRESS>Applied Physics Section of Environmental
Science Department, Escola Politècnica Superior, Universitat de Lleida, Av.
Jaume II, 69, 25001 Lleida, Spain  A. Sheykhi<EMAIL_ADDRESS>Department of Physics, College of Sciences, Shiraz University, Shiraz 71454,
Iran  Biruni Observatory, College of Sciences, Shiraz University, Shiraz
71454, Iran

###### Abstract

Kaniadakis entropy is a one-parameter deformation of the classical Boltzmann-
Gibbs-Shannon entropy, arising from a self-consistent relativistic statistical
theory. Assuming a Kaniadakis-type generalization of the entropy associated
with the apparent horizon of Friedmann-Robertson-Walker (FRW) Universe and
using the gravity-thermodynamics conjecture, a new cosmological scenario is
obtained based on the modified Friedmann equations. By employing such modified
equations, we analyze the slow-roll inflation, driven by a scalar field with
power-law potential, at the early stages of the Universe. We explore the
phenomenological consistency of this model by computation of the scalar
spectral index and tensor-to-scalar ratio. Comparison with the latest Planck
data allows us to constrain Kaniadakis parameter to
$\kappa\lesssim\mathcal{O}(10^{-13}\div 10^{-12})$, which is discussed in
relation to other observational bounds in the past literature. We also
disclose the effects of Kaniadakis correction term on the growth of
perturbations at the early stages of the Universe by employing the spherically
symmetric collapse formalism in the linear regime of density perturbations. We
find out that the profile of density contrast is non-trivially affected in
this scenario. Interestingly enough, we observe that increasing Kaniadakis
parameter $\kappa$ corresponds to a faster growth of perturbations in a
Universe governed by the corrected Friedmann equations.

##  I Introduction

It is a general belief that our Universe has experienced two phases of
accelerated expansion. The first, named inflation, which was proposed to
address some internal problems (flatness, horizon, structure formation) of
standard modern Cosmology Guth ; Linde ; Staro . According to inflationary
model, our Universe expanded exponentially soon after the Big Bang, which
leads to produce a huge amount of energy. The second is thought to have
started roughly five billion years ago and has by now been confirmed by
measurements of the luminosity distances of type Ia Supernovae meas1 ; meas2 .
While being largely understood in most of their facets, concerns remain about
the origin of these phenomena. Among the candidate mechanisms, two
explanations are the most credited so far: on one hand, it is possible to
maintain the classical Einstein’s description of gravity and introduce extra
energy degrees of freedom that drive the Universe acceleration, such as an
inflaton field Bartolo:2004if  and the dark sectors of the cosmos
Peebles:2002gy ; Cai:2009zp ; Capolup . Alternatively, one can resort to a
class of modified theories of gravity and leave the energy content of the
Universe unaffected (see Capozziello:2011et  for a review). From the latter
perspective, potential developments have been achieved in the gravity-
thermodynamics picture, which conjectures that the gravitational field
equations in the cosmological context can be extracted from the first law of
thermodynamics on the Universe apparent horizon, and vice-versa
Jacobson:1995ab ; Padmanabhan:2003gd ; Padmanabhan:2009vy ; Cai:2005ra ;
Akbar:2006kj ; Cai:2006rs  (see Paranjape:2006ca ; Sheykhi:2007zp ;
Akbar:2006er ; Sheykhi:2007gi ; Jamil:2009eb ; Sheykhi:2010zz ; Fan:2014ala
for further applications).

In its traditional formulation, the gravity-thermodynamics conjecture
identifies the thermodynamic entropy of the Universe with the Boltzmann-Gibbs-
Shannon (BGS) entropy. In recent years, much efforts have been devoted to
study generalized scenarios based on extended entropies, such as Tsallis
Tsallis1 ; Tsallis2 ; Tsallis3 , Barrow Barrow:2020tzx  and Kaniadakis Kania0
; Kania1 ; Kania2 ; Scarf1 ; Scarf2  entropies, which all possess the BGS
framework as a limit. In particular, Kaniadakis entropy is a one-parameter
modification of the BGS entropy that arises from a coherent and self-
consistent relativistic statistics, while still retaining the basic features
of the classical BGS theory. Motivated by the relativistic essence of this new
formalism, implications have been examined in several areas Hasegawa ; OSC ;
NC  and the problem of how to equip the Standard Model of Cosmology (SMC) with
a fully relativistic description of the entropic degrees of freedom in the
early Universe has been investigated from multiple perspectives Myrev .

The modified Friedmann equations in Kaniadakis cosmology have been first
explored in Lymperis:2021qty , where the extra terms have been incorporated
into additional dark energy components that reflect the effects of the
corrected entropy. The ensuing theory has proved to exhibit a richer
phenomenology compared to the SMC, especially as for the ability to predict
the usual thermal history of the Universe Lymperis:2021qty , the evolution of
the Hubble rate Hernandez-Almada:2021aiw  and the early baryogenesis and
primordial Lithium abundance KanLuc . Recently, an alternative derivation has
been proposed in Sheykhi:2023aqa  by incorporating Kaniadakis effects as
geometrical (i.e. gravitational) corrections to the left-hand side of the
Friedmann equations and keeping the energy content of the Universe as standard
model. In this approach, the apparent horizon radius of the Universe varies
with time due to the cosmic expansion Sheykhi:2023aqa . In this context, it
has been argued that the generalized second law of thermodynamics still holds
for a Universe enclosed by the apparent horizon and endowed with Kaniadakis
entropy. From theoretical shores, the analysis of Kaniadakis cosmology has
landed on more experimental grounds in the last years. Observational attempts
to constrain Kaniadakis corrections appear in Hernandez-Almada:2021aiw ;
KanLuc ; CosmKan4 , leading to different upper bounds on Kaniadakis parameter.

Starting from the above premises, in this work we explore more in-depth the
cosmological implications of Kaniadakis entropy. Specifically, we study the
slow-roll inflationary model of the Universe driven by a scalar field with
power-law potential. By computing the scalar spectral index and tensor-to-
scalar ratio, we show that Kaniadakis cosmology is phenomenologically
consistent with the latest Planck data, provided that the entropic parameter
is constrained to $\kappa\lesssim\mathcal{O}(10^{-13}\div 10^{-12})$.
Comparison with previous bounds on $\kappa$ supports the idea that the
existing observational literature on Kaniadakis cosmology can be understood in
a unified picture if one allows $\kappa$ to have a running behavior, namely to
vary with the energy scale. We finally examine the influence of Kaniadakis
entropy corrections on the growth of cosmological perturbations in the early
stages of the Universe. We employ the Top-Hat Spherical Collapse (SC) model
SCM , which describes the evolution of uniform and spherical symmetric
perturbations in an expanding background by adopting the same Friedmann
equations for the underlying theory of gravity Gr1 ; Gr2 ; Gr3 . We find out
that the profile of density contrast differs from the standard cosmology, in
particular, the growth rate of the total perturbations becomes faster compared
to the standard cosmology. Throughout this work we use the natural units
$\hslash=c=k_{B}=1$.

This paper is organized as follows. The next section is devoted to a review on
the derivation of the modified Friedmann equations inspired by the Kaniadakis
entropy. In section III, we explore the slow-roll solution for the power-law
inflation in entropy-corrected Kaniadakis cosmology. In section IV we comment
on the growth of perturbations at the early stages of the Universe using the
spherically collapse formalism. We summarize our results in section V.

##  II Kaniadakis entropy and modified Friedmann equations

Motivated by the evidence that the spectrum of the relativistic cosmic rays
exhibits a power-law tailed behavior instead of the classical exponential one,
the problem of deriving the generalization of the Maxwell-Boltzmann
distribution in a special relativistic framework has been first addressed in
Kania1 . As a result, it has been shown that the symmetries of the Lorentz
transformations impose the following modification of the BGS entropy

$S_{\kappa}=-\sum_{i}n_{i}\ln_{\kappa}n_{i}\,,$ (1)

where

$\ln_{\kappa}x\equiv\frac{x^{\kappa}-x^{-\kappa}}{2\kappa}\,.$ (2)

Using the maximum entropy principle, the corresponding Boltzmann factor for
the $i$-th level of a given system becomes

$n_{i}=\alpha\exp_{\kappa}\left[-\beta\left(E_{i}-\mu\right)\right],$ (3)

where

$\alpha=\left[(1-\kappa)/(1+\kappa)\right]^{1/2\kappa}\,,\quad\,\,1/\beta=\sqrt{1-\kappa^{2}}\,\hskip
0.56905ptT\,,$ (4)

and the $\kappa$-deformed exponential is given by

$\exp_{\kappa}(x)\,\equiv\,\left(\sqrt{1+\kappa^{2}\,x^{2}}\,+\,\kappa\,x\right)^{1/\kappa}\,.$
(5)

From Eq. (1), it is evident that departure from the standard BGS entropy is
quantified by the (dimensionless) exponent $-1<\kappa<1$. The classical
framework is recovered in the limit of $\kappa\rightarrow 0$.

Kaniadakis entropy is formally adopted for black holes to explore relativistic
corrections to characteristic thermodynamic quantities, like the Hawking
temperature, mass and heat capacity. These studies are also useful for
holographic applications. In this context, it is convenient to express Eq. (1)
in the probabilistic language Jeans1 ; Jeans2

$S_{\kappa}=-\sum_{i=1}^{W}\frac{P_{i}^{1+\kappa}-P_{i}^{1-\kappa}}{2\kappa}\,,$
(6)

where $P_{i}$ is the probability for the system to be in the $i$-th microstate
and $W$ the total number of permitted configurations. For equiprobable states
(i.e. $P_{i}=1/W$) and recalling that the BGS entropy obeys $S\propto\log W$,
we have $P_{i}=e^{-S}$. For the case of black holes ($S=S_{BH}=A/(4G)$), this
takes the form $P_{i}=e^{-A/(4G)}$. Here, we have denoted the standard
Bekenstein-Hawking entropy by $S_{BH}$, which scales as the horizon surface
area $A$ of the black hole (entropy-area law). By plugging into Eq. (6), we
obtain

$S_{\kappa}=\frac{1}{\kappa}\sinh\left(\kappa\hskip 1.13809ptS_{BH}\right).$
(7)

Two comments are in order here: first, we notice that $S_{\kappa}$ as written
in Eq. (7) is apparently symmetric for $\kappa\rightarrow-\kappa$. Thus, one
can simply restrict to positive values of the entropic exponent. Furthermore,
considering that deviations from the classical entropy are expected to be
small (i.e. $\kappa\ll 1$), it seems reasonable to expand $S_{\kappa}$ to the
leading order as

$S_{\kappa}=S_{BH}+\frac{\kappa^{2}}{6}S_{BH}^{3}+\mathcal{O}(\kappa^{4})\,,$
(8)

where the zero-th order returns the entropy-area law, as expected. The above
approximation is useful to extract analytical solutions from Kaniadakis
entropy-based equations, especially in the cosmological framework
Lymperis:2021qty . We shall rely on Eq. (8) for our next considerations,
verifying a posteriori the validity of the condition $\kappa\ll 1$.

###  II.1 Modified Friedmann equations

Modified cosmology through Kaniadakis entropy (7) has been explored in
Lymperis:2021qty . This study has revealed that new cosmological scenarios
emerge based on corrected Friedmann equations, which contain extra terms that
represent an effective dark energy sector depending on the model parameter
$\kappa$. It was argued that the effective dark energy equation of state
parameter deviates from standard $\Lambda$CDM cosmology at small redshifts,
and remains in the phantom regime during the history of the Universe
Lymperis:2021qty . While achieving the same formal result, a geometric re-
interpretation of these corrections has been provided in Sheykhi:2023aqa ,
motivated by the observation that entropy is a geometrical quantity and any
modification to it should change the geometry part of the field equations
Sheykhi:2023aqa . In what follows, we stick to the latter derivation and
consider a homogeneous and isotropic FRW flat geometry with metric111For the
general case of a curved geometry, see Lymperis:2021qty ; Sheykhi:2023aqa .

$ds^{2}=h_{\mu\nu}dx^{\mu}dx^{\nu}+\tilde{r}^{2}\left(d\theta^{2}+\sin^{2}\theta\,d\phi^{2}\right),$
(9)

where $\tilde{r}=a(t)\hskip 0.56905ptr$, $x^{0}=t$, $x^{1}=r$,
$h_{\mu\nu}=\mathrm{diag}\left(-1,a^{2}\right)$ and $a(t)$ is the time-
dependent scale factor.

Conceiving the Universe as a spherical thermodynamic system, the radius of the
apparent horizon is $\tilde{r}_{A}=1/H=a/\dot{a}$, where $H$ is the Hubble
rate and the overdot denotes ordinary derivative with respect to the cosmic
time $t$. In turn, the associated temperature follows from the definition of
the surface gravity Akbar:2006kj

$K=-\frac{1}{\tilde{r}_{A}}\left(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\right),$
(10)

which gives

$T_{h}=\frac{K}{2\pi}=-\frac{1}{2\pi\tilde{r}_{A}}\left(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\right).$
(11)

We now assume the Universe to be filled with matter and energy in the form of
perfect fluid. Denoting the equilibrium energy density and pressure by $\rho$
and $p$, respectively, the energy-momentum tensor is

$T_{\mu\nu}=\left(\rho+p\right)u_{\mu}u_{\nu}+p\,g_{\mu\nu}\,,$ (12)

where $u_{\mu}$ is the four-velocity of the fluid. The conservation equation
for the total matter and energy content reads $\nabla_{\mu}T^{\mu\nu}=0$,
which gives

$\dot{\rho}=-3H\left(\rho+p\right)$ (13)

for the background (9).

In order to derive the cosmological equations, let us employ the gravity-
thermodynamic conjecture. Toward this end, we apply the first law of
thermodynamics

$dE=T_{h}dS_{h}+WdV\,,$ (14)

on the apparent horizon of entropy $S_{h}$, where $E=\rho V$ is the total
energy in the spherical volume $V$ enclosed by the horizon. Due to the cosmic
expansion, the work density done by a change in the horizon radius is

$W=-\frac{1}{2}T^{\mu\nu}h_{\mu\nu}=\frac{1}{2}\left(\rho-p\right)\,.$ (15)

We omit standard textbook calculations. Assuming the entropy of the apparent
horizon is in the form of Kaniadakis entropy (8) and replacing the horizon
radius of the black hole with the radius of the apparent horizon, after some
algebra we get from Eq. (14) Sheykhi:2023aqa

$H^{2}-\kappa^{2}\hskip
0.56905pt\frac{\pi^{2}}{2\left(GH\right)^{2}}\simeq\frac{8\pi G}{3}\hskip
0.56905pt\rho\,,$ (16)

where we have neglected the irrelevant contribution from the cosmological
constant. This is the first Friedmann equation in Kaniadakis entropy-based
cosmology Sheykhi:2023aqa .

Similarly, one can derive the second Friedmann equation by taking the time
derivative of Eq. (16) and using the continuity equation (13). We get

$\dot{H}\left[1+\kappa^{2}\hskip
0.56905pt\frac{\pi^{2}}{2\left(GH^{2}\right)^{2}}\right]\simeq-4\pi
G\left(\rho+p\right)\,.$ (17)

It is worth noting that Eqs. (16) and (17) coincide with the leading order of
the exact equations found in Lymperis:2021qty . Clearly, the Friedmann
equations in standard cosmology are recovered for $\kappa\rightarrow 0$.

Before moving on to the study of the implications of Eqs. (16) and (17), we
remark that modified cosmic scenarios have also been investigated in Tsallis
TsallisCosm1 ; TsallisCosm2 ; TsallisCosm3 ; TsallisCosm4  and Barrow
BarrowCosm1 ; BarrowCosm2 ; BarrowCosm3 ; BarrowCosm4  entropies, inspired by
non-extensive and quantum gravitational considerations, respectively. Here, we
stress that Kaniadakis model has a relativistic foundation that makes it
conceptually different from the other modified cosmologies.

##  III Slow-roll inflation in Kaniadakis Cosmology

We now study inflation in Kaniadakis cosmology. Following the analysis of
Keskin ; LucBarInf , we consider the high-energy era of the Universe from the
slow-roll condition perspective OdSR  and assume the evolution to be driven by
a scalar field $\phi$ with potential $V(\phi)$. In this setting, the
characteristic parameters measuring inflation are the tensor-to-scalar ratio
and the scalar spectral index of the primordial curvature perturbations, which
for a minimal coupling are defined by Inf1 ; Inf2

$\displaystyle r$ $\displaystyle=$ $\displaystyle 16\epsilon\,,$ (18)
$\displaystyle n_{s}$ $\displaystyle=$ $\displaystyle 1-6\epsilon+2\eta\,,$
(19)

respectively. Here, we have denoted the slow-roll parameters by

$\displaystyle\epsilon$ $\displaystyle=$
$\displaystyle-\frac{\dot{H}}{H^{2}}\,,$ (20) $\displaystyle\eta$
$\displaystyle=$ $\displaystyle-\frac{\ddot{H}}{2H\dot{H}}\,.$ (21)

Under the canonical scalar field assumption, we can write the model lagrangian
as

$\mathcal{L}=X-V(\phi)\,,$ (22)

where

$X=-\frac{1}{2}h^{\mu\nu}\partial_{\mu}\phi\hskip
0.56905pt\partial_{\nu}\phi\,,$ (23)

and $V(\phi)$ are the kinetic and (spatially homogenous) potential terms,
respectively. The energy density and pressure of the associated matter content
inside the early Universe are now

$\displaystyle\rho_{\phi}$ $\displaystyle=$
$\displaystyle\frac{\dot{\phi}^{2}}{2}+V(\phi)\,,$ (24) $\displaystyle
p_{\phi}$ $\displaystyle=$ $\displaystyle\frac{\dot{\phi}^{2}}{2}-V(\phi)\,.$
(25)

They obey the continuity equation (13), which can be recast in the Klein-
Gordon-like form

$\ddot{\phi}+3H\dot{\phi}+\partial_{\phi}V(\phi)=0\,,$ (26)

where we have used the shorthand notation
$\partial_{\phi}V\equiv\frac{dV}{d\phi}$.

Assuming the potential energy dominates all other forms of energy, the slow-
roll approximation is expressed as Inf1 ; Alha

$\ddot{\phi}\ll H\dot{\phi}\,,\qquad\,\frac{\dot{\phi}^{2}}{2}\ll V(\phi)\,.$
(27)

If we solve the first modified Friedmann equation (16) with respect to $H$ and
use the second of the above conditions, we obtain to the leading order in
$\kappa$

$H\simeq\sqrt{{\frac{8\pi G}{3}\hskip 0.56905ptV}}+\sqrt{\frac{27\pi}{2\hskip
0.56905ptG^{7}\hskip 0.56905ptV^{3}}}\frac{\kappa^{2}}{64}\,,$ (28)

where we have omitted the field dependence of $V$ to streamline the notation.

On the other hand, the second Friedmann equation (17) along with Eqs. (24) and
(25) give

$\dot{H}\simeq\left(-4\pi
G+\frac{9\pi\kappa^{2}}{32G^{3}V^{2}}\right)\dot{\phi}^{2}\,.$ (29)

Hence, the slow-roll parameters (20) and (21) take the form

$\displaystyle\epsilon$ $\displaystyle\simeq$
$\displaystyle\left(\frac{3}{2V}-\frac{27\kappa^{2}}{128}\frac{1}{G^{4}V^{3}}\right)\dot{\phi}^{2}\,,$
(30) $\displaystyle\eta$ $\displaystyle\simeq$
$\displaystyle\left[-\sqrt{\frac{3}{8\pi G\hskip
0.56905ptV}}\,\ddot{\phi}\right.$ (31)
$\displaystyle+\left.\sqrt{\frac{3}{2\pi\hskip
0.56905ptG^{9}\,V^{7}}}\frac{9\kappa^{2}}{512}\left(\ddot{\phi}\,V-2\dot{\phi}^{2}\,\partial_{\phi}V\right)\right]\frac{1}{\dot{\phi}}\,\,.$

As observed in Keskin , these two parameters are to be computed at the horizon
crossing, where the fluctuations of the inflaton freeze.

The amount of inflation that occurs is measured by the number $N$ of e-folds
Carrol

$N=\int_{t_{i}}^{t_{f}}H(t)dt\,,$ (32)

where we have denoted the initial (final) time of the inflation by $t_{i}$
$(t_{f})$. Since the primordial fluctuations of the inflaton are relevant
during the horizon crossing and the power spectrum of the inflation dynamics
is evaluated at this point, the beginning time of the inflation is set to the
horizon crossing time, i.e. $t_{i}\equiv t_{c}$. In terms of the field, this
amounts to saying $\phi_{i}\equiv\phi_{c}$, which allows us to write Eq. (32)
in the equivalent form

$N=\int_{\phi_{c}}^{\phi_{f}}\frac{H}{\dot{\phi}}\,d\phi\,,$ (33)

where we have defined $\phi_{c}\equiv\phi(t_{c})$ and
$\phi_{f}\equiv\phi(t_{f})$.

The dynamics of the inflaton field is governed by the potential $V(\phi)$. In
our model, we assume a power-law potential in the form

$V=V_{0}\phi^{n}\,,$ (34)

where $V_{0}>0$ is a constant with dimension $[E]^{4-n}$ associated with the
energy scale of inflation $E_{inf}\simeq 10^{15}\,\mathrm{GeV}$, and $n>0$ is
the power-term. From the latest data, we know that inflationary models with
$n\sim\mathcal{O}(10^{-1}\div 1)$ are observationally favored, while $n\geq 2$
tends to be excluded in the minimal coupling setting. In the following, we
assume $n\simeq 1$.

Analytical solutions of the inflationary observable indices can be extracted
by expressing $\dot{\phi}$ and $\ddot{\phi}$ as functions of the scalar field
$\phi$. Toward this end, we use the first of the two slow-roll conditions in
Eq. (27) to simplify the dynamics (26) as

$\dot{\phi}\simeq-\frac{1}{3H}\hskip 0.56905pt\partial_{\phi}V\,.$ (35)

By substitution of Eqs. (28) and (34), we are led to

$\dot{\phi}\simeq-\frac{n}{2}\sqrt{\frac{V_{0}}{6\pi
G}}\phi^{(n-2)/2}+\frac{3\hskip
0.56905ptn\kappa^{2}}{512}\sqrt{\frac{3}{2\pi\left(G^{9}\hskip
0.85358ptV_{0}^{3}\right)}}\phi^{-(2+3n)/2}$ (36)

Let us impose that inflation (and counting of e-folds $N$) ends when
$\epsilon(\phi_{f})\sim 1$. We first recast Eq. (30) as

$\epsilon(\phi_{f})\simeq\frac{1}{16\pi\hskip 0.56905ptG\hskip
0.56905pt\phi_{f}^{2}}-\frac{27\kappa^{2}}{2048\pi\hskip
0.56905ptG^{5}(V_{0}\phi_{f}^{2})^{2}}\,,$ (37)

where we have used the power-law potential (34) with $n\simeq 1$. Solving with
respect to $\phi_{f}$, we obtain

$\phi_{f}\simeq\frac{1}{4\sqrt{\pi
G}}-\frac{27\kappa^{2}}{64V_{0}^{2}}\sqrt{\frac{\pi}{G^{7}}}\,,$ (38)

where we have considered the only solution that recovers the correct limit for
$\kappa\rightarrow 0$.

The value $\phi_{c}$ of the inflaton at the horizon crossing can be then
estimated by Eq. (33), which gives

$N\simeq-\frac{1}{4}+4\pi\hskip
0.56905ptG\phi_{c}^{2}+\frac{9\pi\kappa^{2}\left[3+\log\left(16\pi\hskip
0.56905ptG\phi_{c}^{2}\right)\right]}{32\hskip 0.56905ptG^{3}V_{0}^{2}}\,.$
(39)

To solve this equation, we make the ansatz that the two terms in the square
brackets are of the same order. We check a posteriori the validity of this
assumption (see below Eq. (46)). Under the above condition, we obtain

$\phi_{c}\simeq\frac{1}{4}\sqrt{\frac{1+4N}{\pi\hskip
0.56905ptG}}-\frac{1.2\kappa^{2}}{V_{0}^{2}}\sqrt{\frac{\pi}{\left(1+4N\right)G^{7}}}\,.$
(40)

By use of this relation, we can cast the tensor-to-scalar ratio (18) and the
scalar spectral index (19) the in terms of the e-folding time $N$. To this
aim, we observe that

$\displaystyle\epsilon(\phi_{c})$ $\displaystyle\simeq$
$\displaystyle\frac{1}{1+4N}+\frac{6.2\pi\kappa^{2}}{G^{3}\left[V_{0}\left(1+4N\right)\right]^{2}}\,,$
(41) $\displaystyle\eta(\phi_{c})$ $\displaystyle\simeq$
$\displaystyle\frac{1}{-1-4N}-\frac{4\pi\kappa^{2}}{G^{3}\left[V_{0}\left(1+4N\right)\right]^{2}}\,.$
(42)

which can be inserted into Eqs. (18) and (19) to give

$\displaystyle r$ $\displaystyle\simeq$
$\displaystyle\frac{16}{1+4N}+\frac{10^{2}\pi\kappa^{2}}{G^{3}\left[V_{0}\left(1+4N\right)\right]^{2}}\,,$
(43) $\displaystyle n_{s}$ $\displaystyle\simeq$
$\displaystyle\frac{4N-7}{1+4N}-\frac{45.5\pi\kappa^{2}}{G^{3}\left[V_{0}\left(1+4N\right)\right]^{2}}\,.$
(44)

We now notice that the above description of inflation in Kaniadakis cosmology
is phenomenologically consistent, provided that Eqs. (43) and (44) match with
the latest Planck observations, which set the experimental bounds ConPlanck

$r<0.064\hskip 22.76228pt(95\%\,\,\mathrm{CL})\,,$ (45)

(from Planck TT,TE,EE + lensing + lowEB) and

$n_{s}=0.9649\pm 0.0042\,\,\,\,\,\,\,(68\%\,\,\mathrm{CL})\,,$ (46)

(from Planck TT, TE, EE + lowE + lensing).

Figure 1: Plot of $r$ versus the (rescaled) Kaniadakis parameter. The grey-
shaded region is excluded by the observational constraint (45), which is
represented by the horizontal red-dashed line. The ensuing upper bound
$\kappa\lesssim 6.05\times 10^{-13}$ on Kaniadakis parameter is indicated by
the vertical black-solid line. We have set $N\gtrsim 60$. Figure 2: Plot of
$n_{s}$ versus the (rescaled) Kaniadakis parameter. The light grey-shaded
regions are excluded by the observational constraint (46), which is
represented (within the margin of the experimental error) by the horizontal
red-dashed lines. The vertical black-solid line delimitates the dark-grey
shaded region excluded by the upper bound on $\kappa$ fixed by Eq. (45). We
have set $N\gtrsim 60$.

The behavior of $r$ and $n_{s}$ in Eqs. (43) and (44) is plotted in Fig. 1 and
2, respectively, versus the (rescaled) Kaniadakis parameter Following the
standard literature Carrol ; N1 ; N2 ; Lamb1 ; Lamb2 , we have fixed the
number of e-folds to be $N\gtrsim 60$. From Fig. 1 we see that the
$\kappa$-corrected tensor-to-scalar ratio is observationally consistent as far
as $\kappa\lesssim 6\times 10^{-13}$. Moreover, with this constraint we also
fit the prediction of the scalar spectral index within the margin of the
experimental error, as shown in Fig. 2.

It remains to be seen whether the ansatz below Eq. (39) is valid. By using Eq.
(40) with $N\gtrsim 60$, we then get $\log\left(16\pi\hskip
0.56905ptG\phi_{c}^{2}\right)/3\sim\mathcal{O}(1)$, which confirms our
assumption.

Therefore, we conclude that Kaniadakis cosmology allows for the slow-roll
inflationary era of a Universe driven by a scalar field with power-law
potential ($n\sim\mathcal{O}(1)$), setting the constraint

$\kappa\lesssim\mathcal{O}(10^{-13}\div 10^{-12})$ (47)

on the entropic parameter. More discussion on this result compared to the past
literature can be found in the concluding section.

##  IV Growth of Cosmological Perturbations

In this section we explore the influence of the modified Kaniadakis entropy on
the growth of cosmological perturbations at the early stages of the Universe.
Following Gr3 , we now consider, for simplicity, a universe filled with
pressure-less matter. This assumption does not affect the generality of our
next considerations. For later convenience, we recast the modified Friedmann
equations (17) as

$\frac{\ddot{a}}{a}\simeq-\frac{4\pi G}{3}\hskip
0.56905pt\rho+\frac{15\pi\kappa^{2}}{32G^{3}\rho}\,,$ (48)

where we have used

$\frac{\ddot{a}}{a}=\dot{H}+H^{2}\,,$ (49)

along with the first Friedmann equation (16). The conservation equation (13)
for the dust matter takes the form

$\dot{\rho}+3H\rho=0\,.$ (50)

To study the evolution of the cosmological perturbations, we use the the Top-
Hat Spherical Collapse (SC) model SCM . In this formalism, one considers a
uniform and spherical symmetric perturbation in an expanding background and
analyzes the growth of perturbations in a spherical region of radius $a_{p}$
and density $\rho^{c}$ by resorting to the same Friedmann equations as for the
underlying theory of gravity Gr1 ; Gr2 ; Gr3 . At time $t$, the density of the
fluid in this region can be written as Gr3

$\rho^{c}=\rho(t)+\delta\rho\,,$ (51)

where $\delta\rho$ denotes the density fluctuation. The conservation equation
reads

$\dot{\rho}^{c}+3h\rho^{c}=0\,,$ (52)

where $h=\dot{a}_{p}/a_{p}$ is the local Hubble rate of the perturbed region.
Additionally, since Eq. (48) is valid in the whole spacetime, it can be
specifically written for the perturbed region to give

$\frac{\ddot{a}_{p}}{a_{p}}\simeq-\frac{4\pi G}{3}\hskip
0.56905pt\rho^{c}+\frac{15\pi\kappa^{2}}{32G^{3}\rho^{c}}\,.$ (53)

Let us now define the density contrast $\delta$ of the fluid in the Universe
by

$\delta=\frac{\rho^{c}}{\rho}-1=\frac{\delta\rho}{\rho}\ll 1\,,$ (54)

where we have reasonably assumed the fluctuation to be much smaller than the
density itself (linear regime). Deriving the above relation respect to $t$ and
using Eqs. (50) and (52), we obtain

$\dot{\delta}=3\left(1+\delta\right)(H-h)\,,$ (55)

which by further derivation gives

$\ddot{\delta}=3\left(1+\delta\right)(\dot{H}-\dot{h})+\frac{\dot{\delta}^{2}}{1+\delta}\,.$
(56)

This is the differential equation for the evolution of the matter
perturbations.

We can observe that the simultaneous usage of Eqs. (48), (53) and (54) leads
to

$\dot{H}-\dot{h}\simeq h^{2}-H^{2}+\left(\frac{4\pi G}{3}\hskip
0.56905pt\rho+\frac{15\pi\kappa^{2}}{32G^{3}\rho}\right)\delta\,,$ (57)

where we have expanded to the leading order in $\delta$ due to the linear
regime we are working in. Hence, we are allowed to recast the differential
equation (56) as

$\ddot{\delta}+2H\dot{\delta}-\left(4\pi
G\rho+\frac{45\pi\kappa^{2}}{32G^{3}\rho}\right)\delta=0\,.$ (58)

In order to analyze the impact of Kaniadakis entropy on the evolution of the
density contrast, it is convenient to express Eq. (58) in terms of the
redshift parameter

$z=\frac{1-a}{a}\,,$ (59)

where we have set the present value of the scale factor to unity. First, we
recall that the matter energy density is given by

$\rho=\rho_{0}\left(1+z\right)^{3}\,.$ (60)

Then, we replace the time derivatives with derivatives respect to $a$ (denoted
by the prime). Toward this end, we note that

$\displaystyle\dot{\delta}$ $\displaystyle=$ $\displaystyle
aH\delta^{\prime}\,,$ (61) $\displaystyle\ddot{\delta}$ $\displaystyle=$
$\displaystyle a^{2}H^{2}\delta^{\prime\prime}+a\left(H^{2}-4\pi
G\rho+\frac{2\pi^{3}\kappa^{2}\rho}{GH^{4}}\right)\delta^{\prime}\,.$ (62)

Inserting into Eq. (58), we are finally led to

$\delta^{\prime\prime}+\frac{3}{2a}\left(1+\frac{9\kappa^{2}}{64G^{4}\rho^{2}}\right)\delta^{\prime}-\frac{3}{2a^{2}}\left(1+\frac{9\kappa^{2}}{32G^{4}\rho^{2}}\right)\delta=0\,.$
(63)

It can be easily checked that, for $\kappa\rightarrow 0$, this equation
reduces to

$\delta^{\prime\prime}+\frac{3}{2a}\delta^{\prime}-\frac{3}{2a^{2}}\delta=0\,,$
(64)

which correctly reproduces the result of the SMC in the absence of the
cosmological constant SCM .

Figure 3: Plot of the density contrast $\delta$ versus $z$. The values of
$\kappa$ have been overestimated to graphically appreciate Kaniadakis
corrections. We have assumed the same initial conditions as in Gr3  and worked
in units of $G$.

The behavior of the matter density contrast $\delta$ versus redshift is
plotted in Fig. 3 for different values of $\kappa$ and in comparison with the
prediction of the SMC ($\kappa=0$). We can see that Kaniadakis corrections
influence the growth of cosmological perturbations (in particular at lower
redshift), in such a way that the higher $\kappa$, the faster the growth rate.
Such a result can be understood by observing that, according to Kaniadakis’
prescription (7), the modified entropy (and, thus, the number of holographic
degrees of freedom) of the Universe gets increased compared to the standard
Boltzmann-Gibbs scenario, supporting a more rapid growth of fluctuations in
energy density.

It is interesting to note that a similar behavior has been found in Gr3  in
the framework of quantum gravity deformations of the entropy-area law,
motivated by the intricate fractal structure of the apparent horizon in Barrow
cosmology. On the other hand, corrections induced by non-extensive Tsallis
statistics may result in a faster or slower growth of perturbations, depending
on the value of the model parameter Gr3 .

##  V Conclusion and discussion

Kaniadakis entropy provides a coherent and self-consistent attempt to
generalize Boltzmann-Gibbs-Shannon statistics to the relativistic framework.
Furthermore, its holographic usage in cosmology underlies the derivation of
the modified Friedmann equations, which predict an interesting phenomenology.
In this work we have studied implications of Kaniadakis cosmology on the slow-
roll inflationary era. We have assumed the energy content of the Universe to
be represented by a scalar field with a perfect fluid form and a power-law
potential. Phenomenological consistency of our model has been explored by
computing the scalar spectral index and tensor-to-scalar ratio at the horizon
crossing. From the latest Planck data, we have obtained
$\kappa\lesssim\mathcal{O}(10^{-13}\div 10^{-12})$, to be compared with other
estimates from Baryon Acoustic Oscillation (BAO) Hernandez-Almada:2021aiw ,
Type Ia supernova222It is worth noting that the values of $\kappa$ given in
Hernandez-Almada:2021aiw ; CosmKan4  are in terms of the re-scaled parameter
$\beta=\kappa\frac{M_{p}^{2}}{H_{0}^{2}}$, where $H_{0}$ is the present Hubble
rate. Hernandez-Almada:2021aiw  and baryogenesis KanLuc  measurements, among
others (see Tab 1).

$|\kappa|$ |  Physical framework |  Ref.
---|---|---
$10^{-125}$ | Baryon Acoustic Oscillations (BAO) | Hernandez-Almada:2021aiw
$10^{-125}$ | CC+SNIa+BAO | Hernandez-Almada:2021aiw
$10^{-124}$ | Cosmological constant (CC) | Hernandez-Almada:2021aiw
$10^{-124}$ | Type Ia supernova (SNIa) | Hernandez-Almada:2021aiw
$10^{-123}$ | Hubble data | CosmKan4
$10^{-123}$ | Strong lensing systems | CosmKan4
$10^{-123}$ | HII galaxies | CosmKan4
$10^{-83}$ | Baryogenesis | KanLuc
Table 1: Cosmological constraints on Kaniadakis parameter.

Although all these constraints on $\kappa$ lie within the domain fixed by the
upper bound (47), they span a relatively wide range. Despite not being
contemplated in the original $\kappa$-statistics, this gap can be understood
by allowing Kaniadakis parameter to have a running behavior, namely to vary
with the energy (or time) scale. Such a behavior is actually what one expects
from the holographic application of the relativistic Kaniadakis’ prescription.
In fact, entropy quantifies the physical degrees of freedom of a system – the
Universe in our cosmological setup. In the standard model of cosmology, it is
typically assumed that the dynamics of the early Universe was initially set by
relativistic energy content (radiation-dominated era), which was later
exceeded by the energy density of the non-relativistic matter (matter-
dominated era) as the Universe cooled down. It is then natural to apply the
same paradigm to the entropic degrees of freedom. In the ensuing scenario, the
relativistic nature of entropy would be quantified by a decreasing function of
time, $\kappa\equiv\kappa(t)$ (or, equivalently, by an increasing function of
the energy scale, $\kappa\equiv\kappa(E)$), in a way that it is maximal
($\kappa\sim\mathcal{O}(1)$) in the earliest stages of the Universe’s
existence, while it recovers the standard Boltzmann-Gibbs-Shannon behavior
($\kappa\simeq 0$) at present time. This conjecture would explain in a unified
picture the predictions $\kappa\lesssim\mathcal{O}(10^{-13}\div 10^{-12})$
inferred from inflation ($E_{inf}\simeq(10^{15}\div 10^{16})\,\mathrm{GeV}$),
$\kappa\simeq\mathcal{O}(10^{-83})$ from Baryogenesis KanLuc
($100\,\mathrm{GeV}\lesssim E_{bar}\lesssim 10^{12}\,\mathrm{GeV}$ Baryo ) and
$\kappa\simeq\mathcal{O}(10^{-125})$ from Baryon Acoustic Oscillations
measurements Hernandez-Almada:2021aiw  (pre-recombination stage,
$E_{rec}\simeq(10^{-1}\div 1)\,\mathrm{eV}$). In passing, we mention that a
similar analysis has been recently conducted within the framework on non-
extensive Tsallis thermodynamics with varying exponent Varexp ; LucBlas1 ;
LucBlas2 . In that case, the running behavior is motivated by quantum field
theoretical considerations associated with renormalization group flow, which
is in principle unavoidable when one attempts to incorporate Tsallis entropy
into a general framework that would be consistent with quantum gravity.
Clearly, to substantiate this view in our case, one should consider an ab
initio investigation of Kaniadakis cosmology equipped with a running
parameter. This goes beyond the scope of the present analysis and will be
explored elsewhere.

We have then explored the effects of the modified Kaniadakis entropic
corrections on the growth of perturbations in the early stages of the
Universe. Employing the spherically symmetric collapse formalism and working
in the linear regime, we have extracted the differential equation for the
evolution of the matter perturbations. Furthermore, for the matter density
contrast we have observed that the modified profile of the growth of
perturbations differs from the standard cosmological model, with increasing
values of $\kappa$ corresponding to a faster growth of perturbations.
Interestingly enough, a similar outcome has been recently exhibited in Pert1 ;
Pert2 ; Pert3  in the context of both Tsallis and Barrow Cosmologies.

Further aspects are yet to be investigated. In Imp  a new description of
inflation has been proposed by merging the Starobinsky model, the Appleby-
Battye-Starobinsky parameterization of dark energy and a correction arising
from the modified $F(R)$ gravity. It would be interesting to compare our Eqs.
(43) and (44) with predictions of this model, also in light of the Kaniadakis
holographic description of dark energy proposed in Drepanou . Moreover, it is
suggestive to examine Kaniadakis influence on the growth of perturbations and
structure formation in relation to results from other deformed cosmologies
based on extended gravity Gr3 ; MCo1 ; MCo2 . Finally, to better test our
model of cosmic inflation, we are interested in studying Kaniadakis
corrections to primordial local non-Gaussianity. In Bravo  (and references
therein) it has been pointed out that Maldacena’s consistency relation
$f_{NL}=5(1-n_{s})/12$ between the amount of local squeezed non-Gaussianity
$f_{NL}$ and the spectral index $n_{s}$ Malda  cannot be directly observed. In
fact, the quantity that is actually observable is

$f_{NL}^{obs}=0+\mathcal{O}\left(\frac{k_{L}}{k_{S}}\right)^{2}\,,$ (65)

where $k_{L(S)}$ denote the long (short) wavelength modes of the inflaton in
Fourier space and the $\mathcal{O}\left(\frac{k_{L}}{k_{S}}\right)^{2}$ terms
arise from non-primordial phenomena, such as gravitational lensing and
redshift perturbations. The reason why Eq. (65) holds is that the primordial
value of $f_{NL}$ predicted by Maldacena’s consistency in the co-moving gauge
is canceled out by a correction
$-5(1-n_{s})/12+\mathcal{O}\left(\frac{k_{L}}{k_{S}}\right)^{2}$, caused by a
change of coordinates to render observables gauge invariant. Clearly, the
simplification of the two terms occurs, leading to Eq. (65), provided that the
prediction of primordial non-Gaussianity corresponds to Maldacena’s
consistency relation. The latter is known to be true for _attractor_ models of
single-field inflation, i.e. models in which every background quantity during
inflation is set by a single parameter (e.g. the Hubble rate), regardless of
the initial conditions. Therefore, any appreciable measurement of local non-
Gaussianity would rule out attractor single-field models of slow-roll
inflation. On the other hand, more exotic descriptions of inflation, such as
multi-field or non-attractor models, do not satisfy Maldacena’s condition
Nonat ; Nonat2  and would potentially be compatible with observations of
primordial local non-Gaussianity. In this scenario, it is desirable to
understand whether our Kaniadakis model of inflation falls within the latter
class and, in that case, how the constraint (47) reconciles with the current
bound on local non-Gaussianity from Planck ConPlanck . Work along these
directions is under active consideration and is left for future
investigations.

###### Acknowledgements.

GGL acknowledges the Spanish “Ministerio de Universidades” for the awarded
Maria Zambrano fellowship and funding received from the European Union -
NextGenerationEU. He is also grateful for participation to LISA cosmology
Working group. GGL and GL acknowledge networking support from the COST Action
CA18108 “Quantum Gravity Phenomenology in the Multimessenger Approach”.

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