#  Accelerating universe in Kaniadakis cosmology without need of dark energy

A. Salehi111Email<EMAIL_ADDRESS>Department of Physics, Lorestan
University, Khoramabad, Iran

(August 28, 2024)

###### Abstract

Taking into consideration of Kaniadakis entropy associated with the apparent
horizon of Friedmann-Robertson-Walker (FRW) Universe and using the gravity-
thermodynamics conjecture, a new cosmological scenarios emerge based on
corrected Friedmann equations, which contains a correction term
$\alpha\left(H^{2}+\frac{k}{a^{2}}\right)^{-1}$ where
$\alpha\equiv\frac{K^{2}\pi^{2}}{2G^{2}}$ and $K$ is Kaniadakis parameter. We
show that it is possible to reconstruct the parameters of the model, in terms
of cosmographic parameters$\\{q,j,s\\}$ analytically. For the flat universe,
the parameters can be reconstructed in terms of only two cosmographic
parameters $\\{q,j\\}$. The advantage of this analytical reconstruction is
that not only it enables us to put constrain on parameters of the model in
terms of directly measurable cosmographic parameters, but also provides the
possibility to test observational measurements on Kaniadakis cosmology. We
examine the model for different sets of cosmographic parameters which
observationally have been constraint in Kakidrosis cosmology. As an
interesting result is that without any assumption about the value of
$\Lambda$, we found that the set $\\{q_{0}=-0.708,j_{0}=1.137\\}$
automatically gives $\Lambda\simeq 0$ and $\\{\Omega_{m0}\simeq
0.325,\Omega_{\alpha 0}=0.671\\}$. This result is in excellent agrement with
pervious observational studies. Reconstructing the evolution of deceleration
parameter against redshift $z$ for these values, shows that the correction
term could plays the role of dark energy without any dark energy component or
cosmological constant $\Lambda$. Finally, we formulate the deviation parameter
in terms of $\\{q,j\\}$ which reflects the deviation of model from $\Lambda
CDM$ model. We Show that the deviation factor is very sensitive to the jerk
parameter $j$, while the $\Omega_{m0}$ is semiactive to deceleration parameter
$q_{0}$. Hence the set ${j,q}$ can be regarded as useful parameters to test
the theoretical and observational studies in Kaniadakis cosmology.

###### pacs:

98.80.-k, 04.50.Kd, 04.25.Nx

The quantum phenomenon of Hawking radiation Hawking  indicates that black hole
has a temperature proportional to its surface gravity and an entropy
proportional to its horizon area Hawking ; Bekenstein ; Hooft ; Susskind .
This issue led to an interesting exploration that expresses there is a
profound connection between gravity and thermodynamics which was first
addressed by Jacobson Jac . This connection indicates that the field equations
of gravity can be written in the form of the first law of thermodynamics on
the horizon and vice versa (see Jac ; Pad1 ; Pad2 ; Pad3 ; CaiKim ; Cai2 ;
Shey1 ; Shey2 ; SheyCQ ; SheyLog ; SheyPL  and references therein). It also
indicates that the field equations of gravity can be extracted from
statistical mechanics. In this approach, the gravitational field equations can
be derived from the holographic principle and the equipartition law of energy
on the horizon degrees of freedom Ver . This approach describes gravity as an
entropic force emerges from the change in the information of the system. A
wide range of studies have focused on this scenario (see e.g. Cai5 ; sheyECFE
; ECEE ; Visser ; SRM ).

It is important to note that, in deriving the gravitational field equations
from the thermodynamic arguments, the entropy expression plays a crucial role.
Any modification to the entropy expression modifies the corresponding field
equations CaiLM ; SheT1 ; Odin1 ; SheT2 ; Emm2 ; SheB1 ; SheB2 ; Odin2 ; Odin3
; Odin4 ; Odin5 .

The effects of the generalized Kaniadakis entropy on the Friedmann equations
have been investigated in Luci ; Mor ; Her ; Dre ; Kum1 ; Kum2 . Modified
Friedmann equations based on the generalized Kaniadakis entropy was already
investigated by two approaches. In the first approach which explored by Lym ,
using the relation $-dE=TdS$ on the apparent horizon of FRW universe, it was
shown that the Friedmann equations are modified by new extra terms that
constitute an effective dark energy sector depending on the Kaniadakis
parameter $K$. Here $-dE$ is the energy flux crossing the horizon within an
infinitesimal period of time $dt$, while $T$ and $S$ are, respectively, the
temperature and entropy associated with the apparent horizon Lym . In the
second approach which recently presented by As , the geometry part of the
cosmological field equations is modified, and the energy content of the
universe is not affected by the generalized Kaniadakis entropy. While the
author of As  stated that since entropy is a geometrical quantity and any
modification to it should change the geometry part of the field equations,
hence the second approach is more reasonable and. In addition, since our
universe is expanding thus we consider the work term (due to the volume
change) in the first law of thermodynamics and write it as $dE=TdS+WdV$. In
this paper, we follow the second approach and use the modified Friedman
equations obtained byAs , however, we show that the two approaches for $\alpha
H^{-4}\equiv\frac{K^{2}\pi^{2}}{2G^{2}H^{4}}\ll 1$, are equivalent. We
reconstruct the parameters of the model
$\Omega_{m},\Omega_{\Lambda},\Omega_{\alpha}$ in terms of the cosmographic
parameter $(q,j,s)$ which express geometrical property of the model. This
approach provide us some advantages.

The structure of this paper is as follows. In section I, we review the origin
of the Kaniadakis entropy. In section II, we review the modified Friedmann
equations through the generalized Kaniadakis entropy from approaches explored
by As  and Lym  and show that this two approaches are equivalent. In section
IV, we reconstruct the parameter of the model in terms of cosmographic
parameters and constrain on parameters of the model . We finish with
conclusion in the last section.

##  I Kaniadakis horizon entropy

Kaniadakis entropy is a one-parameter entropy which generalizes the classical
Boltzmann-Gibbs-Shannon entropy. It originates from a coherent and self-
consistent relativistic statistical theory. The advantages of Kaniadakis
entropy is that it preserves the basic features of standard statistical
theory, and in the limiting case restore it Kan1 ; Kan2 . The general
expression of Kaniadakis entropy is given by Kan1 ; Kan2

$\displaystyle S_{K}=-k_{{}_{B}}\sum_{i}n_{i}\,\ln_{{}_{\\{{\scriptstyle
K}\\}}}\\!n_{i},$ (1)

with $k_{{}_{B}}$ is the Boltzmann constant, and

$\displaystyle\ln_{{}_{\\{{\scriptstyle K}\\}}}\\!x=\frac{x^{K}-x^{-K}}{2K}.$
(2)

Here $K$ is called the Kaniadakis parameter which is a dimensionless parameter
ranges as $-1<K<1$, and measures the deviation from standard statistical
mechanics. In the limiting case where $K\rightarrow 0$, the standard entropy
is restored.

Alternatively, Kaniadakis entropy can be expressed as Abreu:2016avj ;
Abreu:2017fhw ; Abreu:2017hiy ; Abreu:2018mti ; Yang:2020ria ; Abreu:2021avp

$S_{K}=-k_{{}_{B}}\sum^{W}_{i=1}\frac{P^{1+K}_{i}-P^{1-K}_{i}}{2K}.$ (3)

Here $P_{i}$ is the probability in which the system to be in a specific
microstate and $W$ represents the total number of the system configurations.
Throughout this paper we set $k_{{}_{B}}=c=\hbar=1$.

apply the Kaniadakis entropy in the black hole thermodynamics gives

$S_{K}=\frac{1}{K}\sinh{(KS_{BH})}.$ (4)

Where

$S_{BH}=\frac{A}{4G}$ (5)

Where $S_{BH}$ is the standard Bekenstein-Hawking entropy, with $A$ as the
horizon area and $G$ is is the gravitational constant. When $K\rightarrow 0$
one recovers the standard Bekenstein-Hawking entropy, $S_{K\rightarrow
0}=S_{BH}$.

##  II Modified Friedmann equations from different approaches

Considering an expanding Universe filled with a matter perfect fluid, with
energy density $\rho_{m}$ and pressure $p_{m}$, which is homogeneous and
isotropic Friedmann-Robertson-Walker (FRW) geometry metric with form

$ds^{2}=-dt^{2}+a^{2}(t)\left(\frac{dr^{2}}{1-kr^{2}}+r^{2}d\Omega^{2}\right),$
(6)

where $a(t)$ is the scale factor, and with $k=0,+1,-1$ corresponding to flat,
close and open spatial geometry respectively. In order to exploit the
gravitational thermodynamics conjecture in cosmology, the first law is
described in terms of the heat, considered as the energy that flows through
local Rindler horizons, applied on the horizon itself Jac ; Padmanabhan1 ;
Padmanabhan2 , and in particular on the apparent horizon Bak:1999hd ;
Frolov:2002va ; Cai:2005ra ; Cai:2008gw :

$r_{a}=\frac{1}{\sqrt{H^{2}+\frac{k}{a^{2}}}},$ (7)

where $H=\frac{\dot{a}}{a}$ the Hubble parameter and dots denoting derivatives
with respect to $t$. The authors of Lym  applied the first law of
thermodynamics as

$-dE=TdS$ (8)

Here dE is the energy flux crossing the apparent horizon within an
infinitesimal period of time dt which can be expressed as

$-dE=A(\rho_{m}+p_{m})Hr_{a}dt.$ (9)

Also they considered Temperature as

$T=\frac{1}{2\pi r_{a}}.$ (10)

Then use the generalized Kaniadakis entropy, namely equation (4). In
particular, differentiating (4) they derived

$dS_{K}=\frac{8\pi}{4G}\cosh{\left(K\frac{\pi
r_{a}^{2}}{G}\right)}r_{a}\dot{r_{a}}dt.$ (11)

Applying the first law of thermodynamics, they derived the modified Friedman
equations associated to Kaniadakis entropy as

$-4\pi
G(\rho_{m}+p_{m})=\cosh{\left[K\frac{\pi}{G(H^{2}+\frac{k}{a^{2}})}\right]}\left(\dot{H}-\frac{k}{a^{2}}\right).$
(12)

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\frac{8\pi
G}{3}\rho_{m}=\cosh{\left[K\frac{\pi}{G(H^{2}+\frac{k}{a^{2}})}\right]}\left(H^{2}+\frac{k}{a^{2}}\right)$
$\displaystyle\ \ \ \ \
-\frac{K\pi}{G}\text{shi}{\left[K\frac{\pi}{G(H^{2}+\frac{k}{a^{2}})}\right]}-\frac{\Lambda}{3},$
(13)

where $\Lambda$ is the integration constant and The function $\text{shi}{(x)}$
is defined in general as

$\displaystyle\text{shi}{(x)}=\int^{x}_{0}{\frac{\sinh(x^{\prime})}{x^{\prime}}dx^{\prime}}$
(14)

Which is an entire mathematical odd function of $x$ with no branch
discontinuities.

In the second approach which recently explored by As , He take the first law
of thermodynamics as

$dE=TdS+WdV$ (15)

But here, $dE$ is the change in the total energy inside the apparent horizon.
Also $W$ is work density associated with this volume change (due to the
universe expansion) and defined asHayward

$W=\frac{1}{2}(\rho-P)$ (16)

The total energy of the universe enclosed by the apparent horizon is $E=\rho
V$, where $V=\frac{4}{3}\pi r_{a}^{3}$ is the volume enclosed by a 3-
dimensional sphere with the area of apparent horizon. Hence using conservation
equation,

$dE=4\pi\tilde{r}_{A}^{2}\rho d\tilde{r}_{a}-4\pi
H\tilde{r}_{a}^{3}(\rho_{m}+p_{m})dt.$ (17)

Comparison the equations (17) and (9) shows that. The difference between $dE$
in two approaches is $4\pi\tilde{r}_{A}^{2}\rho d\tilde{r}_{a}$. This
difference due to this fact that the authors of Lym  assumed that within an
infinitesimal period of time $dt$ the apparent horizon radius $r_{a}$ is
fixed. As another difference between two approach is that the associated
temperature with the apparent horizon is given by

$T=\frac{\kappa}{2\pi}=-\frac{1}{2\pi\tilde{r}_{a}}\left(1-\frac{\dot{\tilde{r}}_{a}}{2H\tilde{r}_{a}}\right).$
(18)

Considering the fact that deviation from the standard Bekenstein-Hawking is
small, the author of As  assumed that that $K\ll 1$. Hence the expression (4)
can be expanded as

$S_{K}=S_{BH}+\frac{K^{2}}{6}S_{BH}^{3}+{\cal{O}}(K^{4}).$ (19)

apply the above expression to extract the modified friedmann equations, author
ofAs  find the first and second modified Friedman equations as (For more
details seeAs )

$H^{2}+\frac{k}{a^{2}}-\alpha\left(H^{2}+\frac{k}{a^{2}}\right)^{-1}=\frac{8\pi
G}{3}(\rho_{m}+\rho_{\Lambda}).$ (20)
$\displaystyle\left(\dot{H}-\frac{k}{a^{2}}\right)\left[1+\alpha\left(H^{2}+\frac{k}{a^{2}}\right)^{-2}\right]=-4\pi
G(\rho_{m}+p_{m}).$ (21)

Here,

$\displaystyle\alpha\equiv\frac{K^{2}\pi^{2}}{2G^{2}}$ (22)

When $\alpha\rightarrow 0$, the Friedmann equations in standard cosmology are
recovered.

##  III Equivalence of two approaches

Here we want to show that two approaches are not different and for,
$KS_{BH}\ll 1$, the condition which have been implemented in As ), the
equations (12) and (II) are reduced to (20) and (21). To prove this, we start
from expanding of $\cosh{x}$ and $\text{shi}{(x)}$

$\displaystyle\cosh{x}=1+\frac{x^{2}}{2!}+\frac{x^{4}}{4!}+..$ (23)
$\displaystyle\text{shi}{(x)}=\int^{x}_{0}{\frac{\sinh(x^{\prime})}{x^{\prime}}dx^{\prime}}=x+\frac{x^{3}}{3!3}+\frac{x^{5}}{5!5}+..$
(24)

Hence for $x\ll 1$, we can neglect the terms of orders more than 2, hence we
can use the following approximations

$\displaystyle\cosh{x}\simeq 1+\frac{x^{2}}{2!}$ (25)
$\displaystyle\text{shi}{(x)}\simeq=x$ (26)

Since $K\ll 1$ and consequently $K\frac{\pi}{G(H^{2}+\frac{k}{a^{2}})}\ll 1$,
we can use the following approximations

$\displaystyle\cosh{\left[K\frac{\pi}{G(H^{2}+\frac{k}{a^{2}})}\right]}\simeq
1+(\frac{K^{2}\pi^{2}}{2G^{2}(H^{2}+\frac{k}{a^{2}})^{2}})$
$\displaystyle=1+\alpha\left(H^{2}+\frac{k}{a^{2}}\right)^{-2}$ (27)
$\displaystyle\text{shi}{\left[K\frac{\pi}{G(H^{2}+\frac{k}{a^{2}})}\right]}\simeq\frac{K\pi}{G(H^{2}+\frac{k}{a^{2}})}=\sqrt{2\alpha}\left(H^{2}+\frac{k}{a^{2}}\right)^{-1}$
(28)

Inserting equations (III) and (28) in equations (12) and (II), we arrive the
equations (21) and (20) respectively. Hence, the apparent difference in the
form of the equations is not related to the difference in the way the first
law of thermodynamics is applied. Rather, in the second approach,
approximation (19) is used, and with this approximation, the first method is
equivalent to second method

##  IV Connection between parameters of the model and cosmographic parameters

In this section, we aim to reconstruct the parameters of the model in terms of
cosmographic parameters. Here, in order to simplify the equations of the
model, we introduce following dimensionless variables

$\displaystyle\Omega_{m}=\frac{8\pi
G\rho_{m}}{3H^{2}},\Omega_{\Lambda}=\frac{8\pi G\rho_{\Lambda}}{3H^{2}},$
$\displaystyle\Omega_{k}=\frac{k}{a^{2}H^{2}},\Omega_{\alpha}=\alpha\frac{\left(H^{2}+\frac{k}{a^{2}}\right)^{-1}}{H^{2}}$
(29)

Hence using (IV), we can derive the following equations which arranged in a
matrix differential equation
$\frac{d}{dx}\left(\begin{array}[]{c}\Omega_{m}\\\ \Omega_{k}\\\
\Omega_{\Lambda}\\\
H\end{array}\right)=\left(\begin{array}[]{cccc}-1+2q&0&0&0\\\ 0&2q&0&0\\\
0&0&2+2q&0\\\
0&0&0&-1-q\end{array}\right)\left(\begin{array}[]{c}\Omega_{m}\\\
\Omega_{k}\\\ \Omega_{\Lambda}\\\ H\end{array}\right)$

. (30)

Where, $x=\ln a$. By dividing both sides of the equations (20) and(21) by
$H^{2}$ and using relations (IV), the equations (20) and(21) can be rewritten
as,

$\displaystyle 1-\alpha
H^{-4}\left(1+\Omega_{k}\right)^{-2}=\frac{\Omega_{m}+\Omega_{\Lambda}}{1+\Omega_{k}}$
(31) $\displaystyle\left(1+q+\Omega_{k}\right)\left[1+\alpha
H^{-4}\left(1+\Omega_{k}\right)^{-2}\right]=\frac{3}{2}\Omega_{m}$ (32)

Cosmography is the study of scale factor by expanding it through the Taylor
series with respect to the cosmic time. It is a powerful approach provides
valuable information about the evolution of the scale factor and its
derivatives This mathematical framework is inherently kinematic in the sense
that it relies only on geometrical assumptions for the metric and is
independent of cosmological model.Chiba ; Caldwell ; Visser ; Dabrowski1 ;
Dabrowski ; Capozziello0 ; Cattoen ; Maciej . To study cosmography, it is
worthy of introducing the cosmographic parameters as follows define $n$th
cosmographic parameter with $C^{(n)}$ as

$\displaystyle C^{(n)}=\frac{1}{aH^{n}}\frac{d^{n}a}{dt^{n}}$ (33)

Hence, it is easy to show that

$\displaystyle\frac{dC^{(n)}}{dx}=C^{(n+1)}+\Big{(}(n-1)+nq\Big{)}C^{(n)},\ \
n=0,1,2,....$ (34)

Where,

$\displaystyle C^{(0)}=H,\ \ C^{(1)}=-q,\ \ C^{(2)}=j,C^{(3)}=s,\ \
C^{(4)}=l,..$ (35)

which are usually referred to as the Hubble, deceleration, jerk, snap and lerk
parameters, respectively. So from equation (34), we can extract the following
relations

$\displaystyle\frac{dq}{dx}=-j+(1+2q)q,$ (36)
$\displaystyle\frac{dj}{dx}=s+(2+3q)j$ (37)
$\displaystyle\frac{ds}{dx}=l+(3+4q)s$ (38)
$\displaystyle\frac{dl}{dx}=m+(4+5q)l$ (39) $\displaystyle...$ (40)
$\displaystyle\frac{dC^{(n)}}{dx}=C^{(n+1)}+\Big{(}(n-1)+nq\Big{)}C^{(n)}$
(41)

From equations (31) and (32), one can obtain

$\displaystyle\frac{(\Omega_{m}+\Omega_{\Lambda})}{1+\Omega_{k}}+\frac{3\Omega_{m}}{2(1+q+\Omega_{k})}=2$
(42)

Also the Kaniadakis parameter can be reconstructed as

$\displaystyle\alpha=\Big{(}1+\Omega_{k}-\Omega_{m}-\Omega_{\Lambda}\Big{)}(1+\Omega_{k})H^{4}$
(43)

and

$\displaystyle\Omega_{\alpha}=\Big{(}1+\Omega_{k}-\Omega_{m}-\Omega_{\Lambda}\Big{)}$
(44)

Taking differentiating of both sides of equation (32) with respect to $x=\ln
a$, we arrive

$\displaystyle\left(\frac{dq}{dx}+\frac{d\Omega_{k}}{dx}\right)\left[1+\alpha
H^{-4}\left(1+\Omega_{k}\right)^{-2}\right]$ (45)
$\displaystyle+\left(1+q+\Omega_{k}\right)\Big{(}-4\alpha
H^{-5}\frac{dH}{dx}\left(1+\Omega_{k}\right)^{-2}$ $\displaystyle-2\alpha
H^{-4}\left(1+\Omega_{k}\right)^{-3}\frac{d\Omega_{k}}{dx}\Big{)}=\frac{3}{2}\frac{d\Omega_{m}}{dx}$

Hence, using equations (30),(32), equation (45) can be simplified as

$\displaystyle\frac{dq}{dx}=$
$\displaystyle\frac{1}{3\Omega_{m}(1+\Omega_{k})}\Big{(}8-21\Omega_{m}q-30\Omega_{m}\Omega_{k}+24q^{2}+24\Omega_{k}^{2}$
$\displaystyle+24q^{2}\Omega_{k}+24q\Omega_{k}^{2}-15\Omega_{m}\Omega_{k}^{2}+48q\Omega_{k}-21\Omega_{m}q\Omega_{k}$
$\displaystyle+6\Omega_{m}q^{2}\Omega_{k}+8q^{3}+8\Omega_{k}^{3}+24\Omega_{k}+24q-15\Omega_{m}$
$\displaystyle-6\Omega_{m}q^{2}\Big{)}$ (46)

Using equation (36 and (IV), we can drive

$\displaystyle j=$
$\displaystyle\frac{-1}{3\Omega_{m}(1+\Omega_{k})}\Big{(}8-21\Omega_{m}q-30\Omega_{m}\Omega_{k}+24q^{2}+24\Omega_{k}^{2}$
$\displaystyle+24q^{2}\Omega_{k}+24q\Omega_{k}^{2}-15\Omega_{m}\Omega_{k}^{2}+48q\Omega_{k}-21\Omega_{m}q\Omega_{k}$
$\displaystyle+6\Omega_{m}q^{2}\Omega_{k}+8q^{3}+8\Omega_{k}^{3}+24\Omega_{k}+24q-15\Omega_{m}$
$\displaystyle-6\Omega_{m}q^{2}\Big{)}+q+2q^{2}$ (47)

Solving equation (IV) in terms of $\Omega_{m}$ gives

$\displaystyle\Omega_{m}=$ (48)
$\displaystyle\frac{8(1+3q^{2}+3\Omega_{k}^{2}+3q^{2}\Omega_{k}+3q\Omega_{k}^{2}+6q\Omega_{k}+q^{3}+\Omega_{k}^{3}+3\Omega_{k}+3q)}{3(8q+10\Omega_{k}+4q^{2}+5\Omega_{k}^{2}-j-j\Omega_{k}+8q\Omega_{k}+5)}$

Neglecting $\Omega_{k}$, it can be simplified in terms of $(q,j)$ as

$\displaystyle\Omega_{m}=\frac{8}{3}\Big{(}\frac{1+3q+3q^{2}+q^{3}}{5-j+8q+4q^{2}}\Big{)}$
(49)

Inserting this value in equation (42), parameter $\Omega_{\Lambda}$ is also
obtained as

$\displaystyle\Omega_{\Lambda}=\frac{2}{3}\Big{(}\frac{4q^{3}+6q^{2}-5+3j}{-5+j-8q-4q^{2}}\Big{)}$
(50)

Also from equation (43), parameter $\alpha$ can be obtained as

$\displaystyle\alpha=\frac{j-1}{5-j+8q+4q^{2}}H^{4}$ (51)

Consequently,

$\displaystyle\Omega_{\alpha}=\frac{j-1}{5-j+8q+4q^{2}}$ (52)

Equations (49), (50) and (51) indicate that it is possible to reconstruct the
parameters of the model, $(\Omega_{m},\Omega_{\Lambda},\Omega_{\alpha})$ in
terms of directly measurable parameters $(q,j)$. Also parameter $\alpha$ can
be obtained in terms of $(H,q,j)$. Note that if $j\rightarrow 1$, then
$\Omega_{\alpha}\rightarrow 0$ and viceversa. This result was predictable,
because for a flat universe $k=0$, when $\Omega_{\alpha}\rightarrow 0$, the
model reduces to the standard $\Lambda CDM$ model. Also by neglecting
radiation density, the $\Lambda CDM$ model is described by $j=1$. In fact for
a flat $\Lambda CDM$ model according to the equation (43),
$\Omega_{m}+\Omega_{\Lambda}=1$, so from equation (42) it is easy to get

$\displaystyle q=-1+\frac{3}{2}\Omega_{m}$ (53)

Hence, $\frac{dq}{dx}=\frac{3}{2}\frac{d\Omega_{m}}{dx}$. So using relations
(36) and (30), it will be obtained that $j=1$. (For more details see also
Harrison -Hut ) In principal the case $j=1$ is a third order ODE which is hold
only for $\Lambda CDM$ model (for flat case $k=0$ and neglecting radiation
density energy $\Omega_{r}=0$). It indicates that based on the Friedman
equation, the evolution of the Universe in $\Lambda CDM$ can be represented by
the scale factor $a(t)$ in which $\frac{\dddot{a}}{aH^{3}}=1$. Note that
although the deceleration parameter changes in different epoches of the
universe in $\Lambda CDM$ model, however the parameter $j$ remain unchanged in
different apaches of the universe and it changes only when the model changes.
So this parameter can be regarded as factor which enable us to distinguish
between different cosmological models. In principle the parameter $j$ as one
of the pairs statefinder diagnostic $\\{r\equiv
j,s_{statefinder}\equiv\frac{j-1}{3(q-\frac{1}{2})}\\}$ can distinguish
$\Lambda CDM$ form other cosmological models(For more discussion one can see
(Sahni2003  and Alam ). Hence, it is expected that for those cosmological
models whose behavior is more similar to the $\Lambda CDM$, the $j$ value be
closer to one. Therefore, it is expected that models whose behavior is more
similar to a $\Lambda CDM$ such as (CPL) model, the $\omega CDM$ and the
$XCDM$ model will have their $j$ closer to 1. There are some previous studies
which have also reported $j\simeq 1$ (i.e. the parameter $j$ should be near
$1$). There are also some previous studies which have obtained jerk parameter
close to 1. For example see Maciej ; Alam ; Herrera-Zamorano ; Escamilla ;
Capozz ; Munoz ; Escamilla2 ; Nikodem . Hence, according to the equation (52)
the parameter $\Omega_{\alpha}$ can also be regarded as a deviation factor
which determine the amount of deviation from $\Lambda CDM$ model. We can
rewrite equation (52) as

$\displaystyle\Omega_{\alpha}=\frac{j-1}{4(1+q)^{2}-(j-1)}$ (54)

For $q_{0}\leq-0.5$, which is the result of many robust observations, then
$4(1+q_{0})^{2}<1$, hence

$\displaystyle\frac{j-1}{4(1+q_{0})^{2}-(j_{0}-1)}>\frac{j_{0}-1}{1-(j_{0}-1)}$
(55)

If we get $j_{d}=j_{0}-1$, then

$\displaystyle\frac{j_{d}}{1-j_{d}}<\Omega_{0\alpha}$ (56)

Which yields

$\displaystyle j_{d}<\frac{\Omega_{0\alpha}}{1+\Omega_{0\alpha}},\ \
j_{0}<1+\frac{\Omega_{0\alpha}}{1+\Omega_{0\alpha}}$ (57)

Since, $0<\Omega_{0\alpha}<1$, so $(0<j_{d}<\frac{1}{2})$ and $(1<j_{0}<1.5)$.
There are many independent studies that have investigated cosmographic
parameters and put observational constraints on them. Here, we use the results
of Her  which constrained on the cosmographic parameters $\\{q_{0},j_{0}\\}$
for two cases $\Lambda=0$ and $\Lambda\neq 0$ using robust observations. For
$\Lambda\neq 0$ and $\Lambda=0$ the values
$\\{q_{0}=-0.708^{+0.016}_{-0.016},j_{0}=1.137^{+0.014}_{-0.013}\\}$ and
$\\{q_{0}=-0.610^{+0.028}_{-0.035},j_{0}=1.041^{+0.051}_{-0.036}\\}$ was
reported respectively.

At first we examined the case
$\\{q_{0}=-0.708^{+0.016}_{-0.016},j_{0}=1.137^{+0.014}_{-0.013}\\}$ which
found for $\Lambda=0$. For this values, the equations (49), (50) and (52)
gives

$\displaystyle\Omega_{m0}\simeq 0.325,\Omega_{\Lambda 0}\simeq
0.003,\Omega_{\alpha 0}\simeq 0.671$ (58)

As an interesting result is that without making any assumption about the value
of $\Lambda$, our analysis shows that for the case
$\\{q_{0}=-0.708,j_{0}=1.137\\}$. $\Lambda\simeq 0$. In order to have a more
accurate comparison with the results of the Her , we check the results
obtained for the dimensionless parameter $\beta$, which has been defined as

$\displaystyle\beta\equiv\frac{K\pi}{GH_{0}^{2}}$ (59)

Using equation (IV), we can express the parameter $\Omega_{0\alpha}$ in terms
of this parameter as

$\displaystyle\Omega_{0\alpha}=\frac{\alpha}{H_{0}^{4}}=\frac{K^{2}\pi^{2}}{2G^{2}H_{0}^{2}}=\frac{\beta^{2}}{2}$
(60)

The author of Her  for $\Lambda=0$ using joint sample found
$\beta=1.161^{+0.013}_{-0.013}$, inserting this value in equation (60)
yields$\Omega_{0\alpha}=0.673$ which is in good agrement with that we have
found as $\Omega_{0\alpha}=0.671$. Also, for this case the author of Her
found $\Omega_{m0}\simeq 0.325$ which is in good agrement with that we have
found as$\Omega_{m0}\simeq 0.325$. This results indicate that our analysis are
consistence with that obtained by Her , also confirm the validation of two
studies, because from different approaches the same results have been
obtained. In the next section, we explain the reason for the slight
differences in the parameter values in our approach with Her . Now we aim to
check the results of Her  for case $\Lambda\neq 0$. The cosmographic values
which reported by Her  are as
$\\{q_{0}=-0.610^{+0.028}_{-0.035},j_{0}=1.041^{+0.051}_{-0.036}\\}$.
Inserting the above values in the equations (49),(50) and (51), the parameters
of the model are obtained as

$\displaystyle\Omega_{m0}=0.278,\ \ \Omega_{\Lambda 0}=0.651,\ \Omega_{\alpha
0}=\simeq 0.0703$ (61)

Although the results of $\Omega_{m0}=0.278$ and $\Omega_{\Lambda 0}=0.651$ are
comparable with many observation results, but the value of $\Omega_{\alpha 0}$
is expected to be much smaller than $0.0703$. In order to carefully examine
why the two models do not match very well in this case we recall the equation
(57 and express it in terms of $\beta$ as

$\displaystyle j_{d}<\frac{\frac{\beta^{2}}{2}}{1+\frac{\beta^{2}}{2}}$ (62)

Where for very small value of $\beta$, it can be simplified as
$\frac{\beta^{2}}{2}$. This equation indicates that for those values of
$\beta$ that are of order $10^{-2}$, the value of $j_{d}$ must be
approximately of order $10^{-4}$. For example if for the set
$\\{q_{0}=-0.610^{+0.028}_{-0.035},j_{0}=1.041^{+0.051}_{-0.036}\\}$, we
remain $q_{0}$ unchanged, but instead of $j_{0}=1.04$, we consider
$j_{0}=1.001$ and $j_{0}=1.0001$, according to the equations (49),(50) and
(51), The following results would be obtained

$\displaystyle For\ \ j_{0}=1.001,$ $\displaystyle\ \ \Omega_{m0}=0.260,\ \
\Omega_{\Lambda 0}=0.737,\ \Omega_{\alpha 0}=\simeq 0.0016,\ \beta\simeq
0.057$ (63) $\displaystyle For\ \ j_{0}=1.0001,$ $\displaystyle\ \
\Omega_{m0}=0.260,\ \ \Omega_{\Lambda 0}=0.739,\ \Omega_{\alpha 0}=\simeq
0.0001,\ \beta\simeq 0.018$ (64)

The authors of Her  for case $\Lambda\neq 0$ and for different observational
measurements (see Table 1) found that $\beta<0.01$, According to our analysis
(64) in order to $\beta\simeq 0.01$, ($j_{0}<1.0001$). In order to investigate
the cause of this discrepancy and investigate whether this discrepancy is a
serious problem or not we should note that the difference between what our
theoretical model predicts $j_{0}=1.0001$ and what is observationally measured
by Her  as $j_{0}=1.041^{+0.051}_{-0.036}$ is very small, also the value
predicted by our approach is within $(1-\sigma)$ of the observational
measurement, however this slight difference may be caused by the systematic
errors or other errors . However this discussion has at least two interesting
result as
The parameter $j_{0}$ in case$\Lambda\neq 0$ is very close to 1 and a small
change in the value of this parameter leads to a large change in the value of
the parameters $\beta$ and $\Omega_{\alpha 0}$. Hence parameter $j_{0}$ can be
regarded as deviation parameter which measures the deviation of the model from
$\Lambda CDM$ model.
2-Our theoretical approach could test the observational measurements in
Kaniadakis cosmology and estimate their difference with the theory model.

Our analysis showed that $\Omega_{\alpha 0}$ is very sensitive to changes of
$j_{0}$, the question which may arises is that the parameter $\Omega_{m0}$ is
sensitive to which of the parameters $q_{0}$ and $j_{0}$. In order to
investigate this issue, we call the equation (49). We can rewrite this
equation in terms of $q_{0}$ and $j_{0}$ as

$\displaystyle\Omega_{m0}=\frac{8}{3}\Big{(}\frac{1+3q_{0}+3q_{0}^{2}+q_{0}^{3}}{4(1+q_{0})^{2}-(j_{0}-1)}\Big{)}$
(65)

For$\Lambda\neq 0$, we showed that $j_{0}$ is very close to 1, hence
$j_{0}-1\simeq 0$ and we can neglect this term in comparison with
$4(1+q_{0})^{2}$. So the equation (66) can be simplified as

$\displaystyle\Omega_{m0}\simeq\frac{8}{3}\Big{(}\frac{1+3q_{0}+3q_{0}^{2}+q_{0}^{3}}{4(1+q_{0})^{2}}\Big{)}=\frac{2}{3}(1+q_{0})$
(66)

This is exactly the relation that hold for the $\Lambda CDM$. So we can
conclude that changes of $\Omega_{m0}$ is sensitive to changes of $q_{0}$.
Also, since $\Omega_{m0}+\Omega_{\alpha 0}+\Omega_{\Lambda 0}=1$, it can be
concluded that the parameter $\Omega_{\Lambda 0}$ is sensitive to both $q_{0}$
and $j_{0}$.

The previous analysis was hold for $\Omega_{k}=0$. Here we aim to consider
also $\Omega_{k}$ as well. for this case, we require the forth cosmographic
parameter $s$ to reconstruct
$(\Omega_{m},\Omega_{\Lambda},\Omega_{k},\alpha)$. Hence four unknown
parameter $(\Omega_{m},\Omega_{\Lambda},\Omega_{k},\alpha)$ can be
reconstructed in terms of the four known parameter $(H,q,j,s)$. Hence, taking
derivative in both sides of equation (IV) and using equation (37), we can find
$X$ as

$\displaystyle s=$
$\displaystyle-\frac{1}{6}(24q-24j+120q^{3}+96q^{2}+64q\Omega_{m}^{2}j+8\,{\Omega_{k}}^{2}$
$\displaystyle+120\Omega_{m}\Omega_{k}^{3}q^{2}+48q^{4}+15\Omega_{k}\Omega_{m}-24\Omega_{m}\Omega_{k}^{2}j$
$\displaystyle+96\Omega_{m}q^{3}\Omega_{k}+204\Omega_{m}q^{2}\Omega_{k}^{2}+54\Omega_{m}qj-80\Omega_{k}^{4}q^{2}$
$\displaystyle-12\Omega_{m}\Omega_{k}^{3}j+32q\Omega_{k}^{3}j-24\Omega_{m}q\Omega_{k}^{2}j+12\Omega_{m}q^{2}\Omega_{k}$
$\displaystyle+24q^{2}\Omega_{k}j+120\Omega_{m}q^{3}\Omega_{k}^{2}+15\Omega_{m}+16q\Omega_{k}^{2}-30\Omega_{m}q$
$\displaystyle+96q^{2}\Omega_{k}-160q^{2}\Omega_{k}^{2}-24q\Omega_{k}^{3}-108\Omega_{m}q^{2}-24q^{3}\Omega_{k}$
$\displaystyle-256q^{3}\Omega_{k}^{2}-224q^{2}\Omega_{k}^{3}-8q\Omega_{k}^{4}-24\Omega_{m}q\Omega_{k}j-24q^{2}j$
$\displaystyle+56q\Omega_{k}+30j\Omega_{m}-48qj-72\Omega_{m}q^{3}+24\Omega_{k}^{3}j$
$\displaystyle-24\Omega_{k}j-32q^{4}\Omega_{k}^{2}-160q^{3}\Omega_{k}^{3}+16\Omega_{k}^{2}j-60\Omega_{m}q\Omega_{k})$
$\displaystyle-64q^{4}\Omega_{k}$ (67)

Hence, by solving simultaneously the equations (IV), (IV), (42) and (43), we
can reconstruct $(\Omega_{m},\Omega_{\Lambda},\Omega_{k},\alpha)$ in terms of
the four known parameter $(H,q,j,s)$. For example if we use the previous
values $q=-0.53,j=1.01$ and result of Capozziello  for $s$ which reported
$s_{0}=-0.39$, we find

$\displaystyle\Omega_{m}=0.3172,\Omega_{\Lambda}=0.6735\ \
\Omega_{k}=0.00026,\ \ \alpha H_{0}^{-4}\simeq 0.010$ (68)

Note that there are many independent bounds on the cosmographic parameters.
Although they all obtained values for $H_{0}$ and $q_{0}$ that were not
significantly different, they obtained different values for higher-order
cosmographics $s_{0},l_{0},m_{0}$. Here, from the observational bounds
reported for $s_{0}$ in the previous studies, we have selected $s_{0}=-0.39$
fromCapozziello . The reason for this selection is that this value is closer
to the value that the $\Lambda CDM$ predicts theoretically. Since by the same
way that equation (53) was extracted, the cosmographic parameter $s$ in
$\Lambda CDM$ model, can be obtained as $s=1-\frac{9}{2}\Omega_{m}$ which
implies that in the range $-\frac{2}{9}\simeq
0.22<\Omega_{m}<\frac{4}{9}\simeq 0.44$, the cosmographic parameter $s$
changes in the range [-1,0]. Hence from previous studies, we consider a case
that bounded to this range.

### Analysis for case $\Lambda=0$

In this subsection, we want to investigate the results in absence of
cosmological constant $\Lambda$. For this case by considering a flat universe,
we can benefit the equation (50). Hence by considering $\Omega_{\Lambda}=0$,
the jerk parameter $j$ is simplified in terms of deceleration parameter $q$,
as

$\displaystyle j=-\frac{4}{3}q^{3}-2q^{2}+\frac{5}{3}$ (69)

We can immediately test the results of Her  for $\Lambda=0$. As we discussed
in the previous section the best fitted cosmographic parameters
$q_{0},j_{0}\\}$ in Her  was obtained as $\\{q_{0}=-0.708,j_{0}=1.137\\}$.
Inserting the value $\\{q_{0}=-0.708$ in equation (69) gives
$j_{0}=1.137331883$. This result not only confirms that both our analysis and
analysis of Her  are correct, but also shows the sensitivity of the model
parameters to changes in the $j$. As we discussed in previous section, for
$j_{0}=1.137$, we found $\Omega_{\Lambda}=0.003$. This means that a change of
order $10^{-4}$ in $j$ leads to a change of order $10^{-3}$ in
$\Omega_{\Lambda}$ and consequently the other parameters of the model. Also by
inserting $j$ from equation (69) in to the equation (49) and (51), the matter
energy density $\Omega_{m}$ and parameter $\alpha$ are obtained as

$\displaystyle\Omega_{m}=\frac{4(1+q)}{2q+5}$ (70)
$\displaystyle\Omega_{\alpha}=-\frac{2q-1}{2q+5}$ (71)

Hence, we can reconstruct all of the parameters of the model in terms of
deceleration parameter $q$. A wide range of observational studies on
deceleration parameter, found $-1<q<-0.5$. Equation (49) implies that for this
range the matter energy density $\Omega_{m}$ changes from 0 to 0.5. and jerk
parameter $j$ changes from 1 to 1.33.

This is an interesting point which indicates that Kaniadakis cosmology could
explain the current acceleration expansion of the universe without need of
dark energy or cosmological constant $\Lambda$. It is also interesting to note
that for $\Lambda=0$, we can reconstruct the evolution of the universe only by
knowing the current value of the deceleration parameter $q$. Since using the
equations (36) and (69),

$\displaystyle\frac{dq}{dx}=\frac{4}{3}q^{3}+4q^{2}+q-\frac{5}{3}$ (72)

Which leads to following integral

$\displaystyle\int dx=3\int\frac{dq}{4q^{3}+12q^{2}+3q-5}$ (73)

The solution of integral gives

$\displaystyle
x=\frac{1}{6}\ln\Big{(}4-\frac{9}{(q+1)^{2}}\Big{)}+\frac{1}{6}\ln C$ (74)

Where $C$ is constant of integration. Since, $x=\ln a=-\ln(1+z)$, the equation
(74)can be solved in terns of deceleration parameter $q$ as

$\displaystyle q=-1+\frac{3(1+z)^{3}}{\sqrt{4(1+z)^{6}-C}}$ (75)

By set $q_{0}$ for $z=0$ in equation(75), we can reconstructed the constant
$C$ in terms of$q_{0}$ as

$\displaystyle C=4-\frac{9}{(q_{0}+1)^{2}}$ (76)

By setting $q=0$ in equation (75), the transition redshift can be obtained

$\displaystyle z_{tr}=-1+\sqrt[6]{\frac{9}{5(q_{0}+1)^{2}}-\frac{4}{5}}$ (77)

For example for $q_{0}=-0.708$, the equation (77) yields $z_{tr}\simeq
0.6517$. This result is very close to results of (Her ) who found found
$z_{tr}\simeq 0.652^{+031}_{-0.031}$. The equation (75) also shows that in
absence of radiation energy density, for large value of $z$, the deceleration
parameter $q\rightarrow\frac{1}{2}$.( when $z\rightarrow\infty$ the second
terms of r.h.s of the equation $\rightarrow\frac{3}{2}$). Note that equation
(72) can also be solved numerically only by knowing $q_{0}$. For
$q_{0}=-0.708$, we have plotted the evolution of deceleration parameter
against redshift $z$. In top panel of Fig.1, the evolution of deceleration
parameter against redshift $z$ has been show. Also the evolution of
$\Omega_{m}$ and $\Omega_{\alpha}$ against redshift $z$ have been shown in low
panel of Fig.1. As can be seen the universe exhibits the usual thermal
history, with the sequence of matter and dark-energy eras with the transition
from deceleration to acceleration at $z_{tr}\simeq 0.65$. Additionally, in the
future the universe tends asymptotically to a complete dark-energy dominated,
de-Sitter state.

Fig. 1: Top panel: the evolution of deceleration parameter against redshift
$z$. Low panel: The evolution of $\Omega_{m}$ and $\Omega_{\alpha}$ against
redshift $z$

One can find easily from equations (31) and (32) that

$\displaystyle q=-\frac{4-5\Omega_{m}}{4-2\Omega_{m}}$ (78)

Also taking derivative respect to $x$ and using equation (36) and (30),

$\displaystyle
j=\frac{64-240\Omega_{m}+412\Omega_{m}^{2}-309\Omega_{m}^{3}+82\Omega_{m}^{4}}{(2-\Omega_{m})^{2}(4-\Omega_{m})^{2}}$
(79)

By the same way, for $\Lambda=0$, we can find all of the cosmographic
parameters in terms of $\Omega_{m}$. An interesting points that we found is
that the modified term $\alpha\left(H^{2}+\frac{k}{a^{2}}\right)^{-2}$ could
play a dual role, both the role of dark energy and the role of small
correction in dark energy. For example for cosmographic set
$q_{0}=-0.66,j_{0}=1.23,s_{0}=0.394$ which reported by Capozziello2 , our
analysis gives $\Omega_{m}=0.39,\Omega_{\Lambda}=0.01\ \ \Omega_{k}=0.024,\ \
\alpha H_{0}^{-4}\simeq 0.60$

##  V conclusion

Taking into account Kaniadakis entropy associated with the apparent horizon of
Friedmann-Robertson-Walker (FRW) Universe and using the gravity-thermodynamics
conjecture, a new cosmological scenarios emerge based on corrected Friedman
equations. This modified equations in Kaniadakis entropy was first presented
by Lym  then using different approach the author of As  find the modified
equations which are apparently different from the equations of the first
method. At first we showed that this two approaches are equivalent. In other
word, the modified Friedman equations derived by first approach for
$K\frac{\pi}{G(H^{2}+\frac{k}{a^{2}})}\ll 1$ leads to those obtained in the
second approach. In fact applying the first low of thermodynamic on Kaniadakis
entropy, the modified Friedman equations contains a correction term
$\alpha\left(H^{2}+\frac{k}{a^{2}}\right)^{-2}$ which could play different
roles for different cases $\Lambda=0$ and $\Lambda\neq 0$.

In this paper we showed that it is possible to reconstruct parameters of the
model $\\{\Omega_{m},\Omega_{\Lambda},\Omega_{k}\\}$ and $\Omega_{\alpha}$ or
its equivalent $\beta$ in terms of directly measurable cosmographic parameters
$\\{q,j,s\\}$. Establishing a relationship between the parameters that express
energy content of the universe and cosmographic parameters that express the
geometry of the universe indicates how this two set of parameters affect by
each other and is reminder of Willam’s interesting interpretation of general
relativity ”matter tells space-time how to curve, and curved space-time tells
matter how to move”.
Some of the interesting results of our analysis are as follows
$\bullet$ Solving Friedman’s equations analytically is only possible in very
simple cases. But in this article, we related two set of parameters
analytically, a set include energy component and another set include geometric
parameters of the universe. If each set of the parameters be known the another
set would be obtained obtained with high accuracy.

$\bullet$The cosmographic parameters are model independent parameters which
are directly measurable. Hence, if we have enough quality observational data
they can be constrained with high accuracy. Hence, in any cosmological model,
if we could make a relation between parameters of the model with these
parameters, we could make a good constrain on parameters of the model. Here we
have made this relation and found the interesting results.

$\bullet$The relations, we extracted are analytical relation, in which, the
model parameters are on one side and cosmographic parameters on the other side
of the equation. The efficiency of these equations is not limited to the
estimation of the model parameters based on the cosmographic parameters, but
these two way relations which enable us to test observational and numerical
studies on Kaniadakis entropy and estimate the deviation of the observational
measurement from that the analytical relation predicts. In order to test the
efficiency of our model, we test previous study Her  which numerically and
using robust observational have put constrain on both parameters of the model
and cosmographic parameters. For case $\Lambda=0$, the author of Her  found
$\\{q_{0}=-0.708^{+0.016}_{-0.016},j_{0}=1.137^{+0.014}_{-0.013}\\}$. In order
to test the validation of this results with our analytical approach, we insert
these results in analytical equation without assumption that $\Lambda=0$, we
found $\Omega_{\Lambda 0}\simeq 0.003,$ which is close to 0, also for this
case we found $\Omega_{m0}\simeq 0.325,\Omega_{\alpha 0}\simeq 0.671$ which
are very close to the results of Her , who using joint sample found
$\Omega_{m0}\simeq 0.325$ and $\beta=1.161^{+0.013}_{-0.013}$ or
$\Omega_{0\alpha}=0.673$. Interestingly we show that this slight difference is
related to sensitivity of the model with parameter $j_{0}$. In other word our
analytical model indicates that for this value
$\\{q_{0}=-0.708,j_{0}=1.137331883\\}$, the parameter $\Lambda$ will be
exactly equal to zero. Hence the difference between $j_{0}=1.137331883$ and
$j_{0}=1.137$ which is $\simeq 0.0003$ leads to deviation value $\simeq 0.003$
of parameter $\Omega_{\Lambda 0}$ from 0. Also for $q_{0}=-0.708$, the
equation (77) yields $z_{tr}\simeq 0.6517$. This result is very close to
results of (Her ) who found found $z_{tr}\simeq 0.652^{+031}_{-0.031}$. This
results show that according to our analytical approach, the numerical analyzes
in Her  have been done with very good accuracy so that they obtained a value
for $j_{0}$ which has a very small deviation from that theoretical approach
predicts.

$\bullet$We showed that for $\Lambda=0$, the correction term plays the role of
dark energy and exhibits the thermal history of the universe which is expected
evolution of the universe.

$\bullet$ As interesting result that theoretical approach predicts is that
$\Omega_{\alpha 0}$ or $\beta$ is very sensitive to the changes of $j_{0}$ and
$\Omega_{m0}$ is very sensitive to changes of $q_{0}$. We also defined the
deviation parameters $j_{d}$ which determine amount of deviation from
$j_{0}=1$ which hold for ($\Lambda CDM$. and showed that for $q_{0}<-0.5$ the
condition $j_{d}<\frac{\beta^{2}}{2}$ is hold. Note that it is expected that
the amount of deviation of the model from ($\Lambda CDM$ be more sensitive to
changes of $j$, because in all of the history of evolution of the universe in
$\Lambda CDM$ model $\Omega_{r}=0$, while the deceleration parameter changes
from matter dominated with $q=\frac{1}{2}$ to dark energy dominated ($\Lambda$
dominated with $q=-1$), the jerk parameter is unchanged with $j=1$, hence $j$
changes only when the model deviate from $\Lambda CDM$. Therefore, it is
expected that the greater the deviation of the model from the $\Lambda CDM$
model, the greater the difference of $j$ value with 1.

In summary, it can be concluded that as a useful approach, the theoretical
approach could test the future theocratical and observational studies on
kakidrosis entropy.

##  VI Data availability statement

The manuscript has no associated data or the data will not be deposited

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