Kaniadakis Holographic Dark Energy in non-flat Universe
Umesh Kumar Sharmaa,1, Vipin Chandra Dubeyb,2, A. H. Ziaiec,3, H. Moradpourc,4

a Department of Mathematics, Institute of Applied Sciences and Humanities, GLA
University Mathura-281406, Uttar Pradesh, India

b Department of Mathematics, Jaypee Institute of Information Technology, A-10,
Sector-62, Noida-201309, Uttar Pradesh, India

c Research Institute for Astronomy and Astrophysics of Maragha (RIAAM),
University of Maragheh, P.O. Box 55136-553, Maragheh, Iran

1E-mail<EMAIL_ADDRESS>
2E-mail<EMAIL_ADDRESS>
3E-mail<EMAIL_ADDRESS>
4E-mail<EMAIL_ADDRESS>
###### Abstract

In present research, we construct Kaniadakis holographic dark energy (KHDE)
model within a non-flat Universe by considering the Friedmann-Robertson-Walker
(FRW) metric with open and closed spatial geometries. We therefore investigate
cosmic evolution by employing the density parameter of the dark energy (DE),
the equation of state (EoS) parameter and the deceleration parameter (DP). The
transition from decelerated to accelerated expanding phase for the KHDE
Universe is explained through dynamical behavior of DP. With the
classification of matter and DE dominated epochs, we find that the Universe
thermal history can be defined through the KHDE scenario, and moreover, a
phantom regime is experienceable. The model parameters are constrained by
applying the newest $30$ data cases of $H(z)$ measurements, over the redshift
span $0.07\leq z\leq 2.36$, and the distance modulus measurement of the recent
Union $2.1$ data set of type Ia supernovae. The classical stability of KHDE
model has also been addressed.

Keywords: KHDE, Non-Flat FLRW Universe, General Relativity, Generalized
Entropies
PACS: 98.80.Es, 95.36.+x, 98.80.Ck

##  1 Introduction

It was promulgated by Gibbs that the systems involving interactions (long-
range) are no longer extensive [1], a theory acts as a benchmark of different
entropy formalisms [2, 3, 4, 5]. These entropies provide considerable
cosmological models [6, 7, 8, 9, 10]. In this regard, new models of
holographic dark energy (HDE) [11, 12, 13, 14, 15] have recently been proposed
using these generalized entropies that have also been used to study black
holes [16, 17]. These entropies describe inflation without considering
inflaton [18], affects the Jeans mass [19], and in general, may be considered
as the sources of modified Newtonian dynamics [20]. They might also be
motivated through the quantum features of gravity [21, 22].

An attractive approach for understanding the origin of DE is holographic dark
energy hypothesis, based on the Bekenstein entropy [23] (which takes the kind
of degrees of freedom of horizon [24, 25]), called ordinary holographic dark
energy hypothesis (OHDE) [26]. By accepting apparent horizon as IR (Infrared)
cut-off, OHDE faces some drawbacks [26, 27], whereas apparent horizon is a
suitable decisive boundary from the thermodynamic, and causality viewpoints
[28, 29, 30, 31, 32, 33]. Contrarily, by taking apparent horizon as IR cut-
off, holographic dark energy (HDE) models based on three generalized entropy,
can provide substantial explanation for the Universe expansion [11, 12, 14,
34, 35, 36, 37]. Thus, practicing different entropies, numerous proper models
of HDE can be obtained. Many researchers have examined the compatibility of
these generalized entropies with the zeroth law of thermodynamics [38, 39, 40,
41, 42, 43, 44, 45].

Recently, Kaniadakis statistics [7, 46] as a generalized entropy measure [2,
3, 4] has been employed to study some gravitational and cosmological
consequences. The generalized $\kappa$-entropy (Kaniadakis), a single free
parameter entropy of a black hole is obtained as [47]

$\displaystyle S_{\kappa}=\frac{1}{\kappa}\sinh(\kappa S_{BH}),$ (1)

where $\kappa$ is an unknown parameter. Therefore, using this entropy, and
holographic dark energy hypothesis a new model of DE is introduced as
Kaniadakis holographic dark energy (KHDE) [47], which exposes considerable
properties [47, 48].

On the other hand, one of the basic interests in cosmology is the question
concerning the shape or curvature of the Universe, or more precisely the
descriptive geometry of the visible Universe. Though recent observations have
provided evidences on the flat geometry, there exist some arguments that
support the idea of non-flat geometry due to the contribution of small
fraction of positive/negative spatial curvature. One can examine this topic by
defining the Universe spatial curvature or a quantity that describes the
variation of the historical geometry of the Universe from flat space geometry
[49]. Presently, the useful addition mode of spatial curvature to the Universe
energy density is quantified by the curvature parameter $\Omega_{K}$. In a
manner, this ordinary portion is to estimate the curvature parameter
$\Omega_{K}$, while $\Omega_{K}>0$ and $\Omega_{K}<0$ contributes to a
spatially open and closed Universe, respectively. Besides holding these ideas
in mind, polarization anisotropy data and Planck 2018 CMB temperature singly
introduced the constraint $\Omega_{K}=-0.044^{+0.018}_{-0.015}$ [50], here
$\Omega_{K}=0$ corresponds to a spatially flat Universe. Moreover, estimated
from the BOSS DR12 CMASS example of linking P18 data with the full-shape (FS)
universe power spectrum and including $\Omega_{K}=0.0023\pm 0.0028$ received
from this alliance [51], one can achieve less constraining outcomes. It is
also notable that in non-flat geometry, the closed Universe models exhibit
substantially higher lensing amplitudes in comparison to $\Lambda$CDM model
[50]. In either situation, from modern and future cosmic views, the influence
of spatial curvature on the evolution of the Universe is the cause why a large
number of researches have been attracted to find possible constraints on
$\Omega_{K}$ [52].

Keeping these studies in mind, our aim in the present study is to construct a
KHDE model in a non-flat universe. The work is organized as follows: In
Section 2 the fundamental equations of KHDE in a non-flat Friedmann-
Lema$\hat{i}$tre- Robertson- Walker (FLRW) Universe are presented. In section
3, the specific research on cosmic nature is continued, and we mainly focus on
the EoS and DE density parameters. Section 4 exposes the techniques used for
data analysis in this study. We examine the stability of the model in Section
5. Lastly, we recap our outcomes in Section 6.

##  2 KHDE in Non-Flat Universe

The cosmic principles and General Relativity (GR) are our basic hypotheses to
explain the Universe on large scales. The line element for a homogeneous and
isotropic non-flat Universe is presented through the FLRW metric, given by

$\displaystyle
ds^{2}=-dt^{2}+a^{2}(t)\Big{(}\frac{dr^{2}}{1-Kr^{2}}++r^{2}d\Omega^{2}\Big{)},$
(2)

where $a(t)$ is the scale factor which expresses the increase/decrease of the
isotropic and homogenous spatial parts. The constant parameter $K$ represents
the spatial curvature with $K=-1,0,1$ respectively corresponds to open, flat
and closed Universe models. The spatial curvature efficiently appears as an
extra matter-energy term within the Friedmann equations with sectional
contribution quantified through the curvature parameter
$\Omega_{K}\equiv-\frac{K}{(H_{0}a_{0})^{2}}$, where the Hubble parameter $H$
describes the Universe expansion rate and $0$ indicates the size estimated
presently. In general the estimated Hubble parameter $H_{0}$ is introduced as
the Hubble constant. Particularly, $K$ and $\Omega_{K}$ appear by opposing
signs, hence $\Omega_{K}<0$ represents a closed Universe and $\Omega_{K}>0$ an
open one.

The HDE states that if DE is supposed to control the present accelerated
expansion of the Universe, then, considering the Kaniadakis black hole entropy
Eq. (1), the amount of vacuum energy accumulated within a box of size $L^{3}$
must not exceed the energy of its same size black hole [26]. One then gets

$\Lambda^{4}\equiv\rho_{D}\propto\frac{S_{\kappa}}{L^{4}},$ (3)

for the vacuum energy $\rho_{D}$. Now, as a general limit on the system under
study the apparent horizon is considered as the IR cutoff for the FLRW
Universe. It then can be located as [53, 54, 55, 30]

$\displaystyle\tilde{r}_{A}=\frac{1}{\sqrt{\frac{K}{a^{2}}+H^{2}}},$ (4)

whence the vacuum energy reads

$\rho_{D}=\frac{3C^{2}H^{4}}{8\pi\kappa}\left(\Omega_{K}+1\right)^{2}\sinh\left(\frac{\pi\kappa}{H^{2}\left(\Omega_{K}+1\right)}\right),$
(5)

where we have set $L=\tilde{r}_{A}$ and $C^{2}$ is an undefined standard
constant [26]. Clearly, for the case $\Omega_{K}=0$, Eq. (5) coincides with
KHDE in a flat Universe [47]. Also, this is apparent that whenever
$\kappa\rightarrow 0$ and $\Omega_{K}=0$ , we have
$\rho_{D}=\frac{3C^{2}H^{2}}{8\pi}$, the famous Bekenstein entropy-based HDE
(i.e. OHDE) [26]. In a non-flat Universe, the first and second Friedmann
equations including KHDE and dark matter (DM), are given by

$\displaystyle
H^{2}+\frac{1}{a^{2}}K=\frac{1}{\tilde{r}^{2}_{A}}=\frac{8\pi}{3}G\left(\rho_{D}+\rho_{m}\right),$
(6)

$\displaystyle
H^{2}+\frac{2}{3}\dot{H}+\frac{K}{3a^{2}}=-\frac{8\pi}{3}Gp_{D},$ (7)

where the energy density of KHDE, its pressure and the matter energy density
are presented as $\rho_{D}$, $p_{D}$ and $\rho_{m}$, respectively. Working on
the fractional densities, the corresponding density parameters for pressure-
less fluid, KHDE and spatial curvature are defined as

$\displaystyle\Omega_{m}=\frac{8\pi G}{3H^{2}}\rho_{m},\hskip
28.45274pt\Omega_{D}=\frac{8\pi G}{3H^{2}}\rho_{D},\hskip
28.45274pt\Omega_{K}=\frac{1}{a^{2}}\frac{K}{H^{2}},$ (8)

by the virtue of which Eq. (6) can be recast into the following form

$\displaystyle\Omega_{K}+1=\Omega_{m}+\Omega_{D}.$ (9)

The conservation law for KHDE and matter leads individually to the following
equations

$\displaystyle\dot{\rho}_{D}+3(1+\omega_{D})\rho_{D}H=0,$ (10)
$\displaystyle\dot{\rho}_{m}+3\rho_{m}H=0,$ (11)

where $H=\frac{\dot{a}}{a}$ and KHDE EoS prameter is shown as
$\omega_{D}=p_{D}/\rho_{D}$. In order to obtain the behavior of DP we proceed
with differentiating Eq. (5) with respect to cosmic time $t$. One then finds

$\displaystyle{\dot{\rho}_{D}}=2H^{3}\left(\frac{\dot{H}}{H^{2}}-\Omega_{K}\right)\left(\frac{2\rho_{D}}{H^{2}\left(\Omega_{K}+1\right)}-\frac{3}{8}C^{2}\cosh\left(\frac{\pi\kappa}{H^{2}\left(\Omega_{K}+1\right)}\right)\right).$
(12)

Now, taking the derivative of Eq. (6) along with using Eqs. (9)-(12) gives

$\displaystyle\frac{\dot{H}}{H^{2}}=\frac{-3}{2}\left(\omega_{D}\Omega_{D}+\frac{\Omega_{K}}{3}+1\right),$
(13)

whereby one finds the following expression for DP

$\displaystyle
q=-\frac{\dot{H}}{H^{2}}-1=\frac{1}{2}\left(3\omega_{D}\Omega_{D}+\Omega_{K}+1\right).$
(14)

We note that this result can be also obtained from Eq. (7). This is due to the
fact that Eqs. (6) and (7) along with conservation equations are not
independent from each other. We can also get the behavior of EoS parameter by
solving Eq. (10) with respect to $\omega_{D}$ and substituting for $\rho_{D}$
and $\dot{\rho}_{D}$ from Eqs. (5) and (12), respectively. By doing so, one
arrives at the following expression for EoS parameter

$\displaystyle\omega_{D}=\frac{1}{3}\left(\frac{2\pi\left(H^{2}+\dot{H}\right)\kappa\coth\left(\frac{\pi\kappa}{H^{2}\left(\Omega_{K}+1\right)}\right)}{H^{4}\left(\Omega_{K}+1\right){}^{2}}-\frac{2\left(\pi\kappa\coth\left(\frac{\pi\kappa}{H^{2}\left(\Omega_{K}+1\right)}\right)+2\left(H^{2}+\dot{H}\right)\right)}{H^{2}\left(\Omega_{K}+1\right)}+1\right).$
(15)

Additionally, from the second part of Eq. (8) and Eq. (5) the density
parameter of KHDE can be expressed as

$\displaystyle\Omega_{D}=\frac{C^{2}}{\kappa}H^{2}\left(\Omega_{K}+1\right){}^{2}\sinh\left(\frac{\pi\kappa}{H^{2}\left(\Omega_{K}+1\right)}\right).$
(16)

As expected for the flat case $\Omega_{K}=0$, Eq. (16) reduces to the
expression found in [47].

##  3 Cosmological evolution

In this section, we continue to study the cosmic evolution of KHDE for both
open and closed Universe models. As we mentioned above, Eq. (16) determines
the behavior of the density parameter of DE for the two spatially curved
cases. One can easily express the cosmic evolution in terms of the more
convenient redshift parameter, setting the current scale factor value to
$a_{0}=1$. We elaborate equations numerically, imposing the initial conditions
$\Omega_{D}=0.70$ and $H_{0}=67.9$ in agreement with recent observations [50].

Figure 1:  The evolution trajectories of deceleration parameter $q$ verses
redshift $z$ for $C=0.35$ and different values of model parameter $\kappa$
with initial conditions $\Omega_{D0}$= $0.70$ and $H_{0}=67.9$. The left panel
has been plotted for an open Universe with $\Omega_{K}$= $0.0026$ and the
right one has been plotted for a closed Universe with $\Omega_{K}$= $-0.0012$.

Figure 2: Behavior of EoS parameter $\omega_{D}$ versus redshift $z$ for
$C=0.35$ and different values of model parameter $\kappa$ with initial
conditions $\Omega_{D0}$= $0.70$ and $H_{0}=67.9$ for open Universe
$\Omega_{K}$= $0.0026$ (left panel) and closed Universe $\Omega_{K}$=
$-0.0012$ (right panel).

Figure 3:  Behavior of density parameter $\Omega_{D}$ versus redshift $z$ for
an open (left panel) and a closed (right panel) Universe for $C=0.35$ and
different values of model parameter $\kappa$ with initial conditions
$\Omega_{D0}$= $0.70$ and $H_{0}=67.9$.

The behavior of deceleration parameter $q$ for as a function of redshift $z$,
can be obtained by examining Eq. (14) which is solved numerically by taking
the initial conditions stated above. The numerical solution of $q$ is depicted
in Fig. (1). We have taken different observationally best-fitted values of
KHDE model parameter $\kappa=3100,\,3150$ and $3200$ and $C=0.35$ for an open
Universe $\Omega_{K0}$= $0.0026$ (left panel) and close Universe
$\Omega_{K0}$= $-0.012$ (right panel). We observe that the evolution of both
closed as well as open Universe models from early decelerating to late-time
accelerating phase of expansion occurs at transition redshift $z_{tr}$ where
$q(z_{tr})=0$. The transition redshift for the present model is obtained as,
$z_{tr}\approx 0.5$ as required from observations.

In order to study the nature of the EoS parameter of KHDE and specifically to
investigate whether it is influenced by $\kappa$ and $\Omega_{K}$ parameters,
we plot $\omega_{D}$ against redshift in Fig. (2) for different values of
$\kappa$ parameter and for open (left panel) and closed (right panel) Universe
models. As we can see, for different values of $\kappa$ parameter, the
evolution of $\omega_{D}$ and its prevailing value
$\omega_{D}(z=0)\equiv\omega_{D0}$ tend to assume more moderate values.
Interestingly, for $z>0$ the EoS parameter of DE lies completely within the
quintessence region. As the Universe evolves to redshift $z\leq 0$, the
phantom divide ($w_{DE}=-1$) is crossed at a certain value of redshift. This
value of the redshift depends on the model parameters of KHDE model.
Therefore, in the framework of present model, it is possible for the EoS of
KHDE to cross to the phantom regime, contrary to the case of standard HDE.

In Fig. (3) we depict the behavior of DE density parameter $\Omega_{D}$ during
the evolution of an open (left panel) and a closed (right panel) Universe for
different values of the Kaniadakis parameter $\kappa$. Since we witnessed the
general thermal history of the Universe through the classification of matter
and radiation dominated eras, the transition from matter to DE domination
complies with the required scenario of structure formation of the Universe. It
is remarkable that for more transparency we should extend the evolution of the
Universe to the distant futurity, from where we can observe the outcomes of
cosmic evolution in complete DE domination as expected. It is also noteworthy
that in comparison to the flat KHDE model, where the phantom regime was not
obtained, the incorporation of spatial curvature can drive the Universe into
the phantom region, which can be an interesting subject of research and
discussion in the future.

Figure 4: The evolution of $H(z)$ versus redshift $z$ with error bars for
$C=0.35$ and different values of model parameter $\kappa$ with initial
conditions $\Omega_{D0}$= $0.70$ and $H_{0}=67.9$ for open Universe
$\Omega_{K}$= $0.0026$ (left panel) and closed Universe $\Omega_{K}$=
$-0.0012$ (right panel). In this plot, the dots correspond to the 30 $H(z)$
data points

.

##  4  Observational data analysis

This section deals with the most advanced cosmological investigations of KHDE
model. We have explained the datasets considered in our analysis in-brief,
specifically, the current data of Hubble parameter has been achieved through
the CC (cosmic chronometer) mode and Type Ia Supernova (SNIa).

###  4.1 Observational Hubble Data (OHD)

We take Hubble datasets $H(z)$ from table 4 of Ref. [56], which contains 30
$H(z)$ observational dataset limits spanning the redshift interval of
$0.07\leq z\leq 1.965$. These uncorrelated data can be obtained by the cosmic
chronometric (CC) technique. The purpose of taking these set of data lies in
the case that the OHD dataset acquired of the CC method is individualistic.
The CC data of passively evolving Universes depends on the method of distinct
dating of galaxies. The evolution of $H(z)$ (Hubble parameter) for our KHDE
model has been shown in Fig. (4), for both closed and open Universes, and
compared with the newest 30 points of $H(z)$ data [56]. We observe that the
present KHDE model is fully compatible with the OHD dataset at variance with
the redshift parameter.

###  4.2 Distance modulus $\mu$

To examine the cosmological models, the datasets extracted from the SNIa are
very beneficial particularly as a primary evidence for accelerated Universe.
Therefore, we have further used the distance modulus dataset sample of 580
scores of Union 2.1 combined with SNIa [57], besides the CC data. To
investigate the evolution of the Universe, there is a prominent observational
tool namely the redshift-luminosity distance relation [58]. Due to the
expansion of the universe, the light reaching out towards a distant
luminescent body becomes redshifted and we can get the equation for the
luminosity distance (DL) in terms of $z$. With help of luminosity distance, we
can obtain the flux of a source, which is presented as

$\displaystyle D_{L}=a_{0}r(z+1),$ (17)

here $r$ denotes the radial coordinate of the source. Copeland et al.[59] have
formulated DL as given below

$\displaystyle D_{L}=\frac{c(1+z)}{H_{0}}\int_{0}^{z}\frac{dz}{h(z)},\quad
h(z)=\frac{H}{H_{0}}.$ (18)

The distance modulus $\mu$ is then obtained as [59]

$\displaystyle\mu=25+5\log_{10}\left(\frac{D_{L}}{M_{pc}}\right).$ (19)

From Eq. (18), one gets the following expression for distance modulus

$\displaystyle\mu=25+5\log_{10}\left[\frac{c(1+z)}{H_{0}}\int_{0}^{z}\frac{dz}{h(z)}\right].$
(20)

Figure 5: Evolution of Luminosity Distance versus redshift $z$ for $C=0.35$
and different values of model parameter $\kappa$ with initial conditions
$\Omega_{D0}$= $0.70$ and $H_{0}=67.9$ for open Universe $\Omega_{K}$=
$0.0026$ (left panel) and closed Universe $\Omega_{K}$=$-0.0012$ (right
panel).

Figure 6: The behavior of $v_{s}^{2}$ against redshift $z$ for $C=0.35$ and
different values of model parameter $\kappa$ with initial conditions
$\Omega_{D0}$= $0.70$ and $H_{0}=67.9$ for open Universe $\Omega_{K}$=
$0.0026$ (left panel) and closed Universe $\Omega_{K}$= $-0.0012$ (right
panel).

Figure (5) shows the most suitable period of distance modulus $\mu$(z) in
terms of $z$ for KHDE model combining the 580 scores of Union 2.1 [57] with
SNIa data. This figure displays the error graph for KHDE model presented in
solid line by applying 580 scores of Union 2.1 combined with SNIa datasets.
Also from this figure one could easily witness the moderate reflection of KHDE
model on the observed values of distance modulus $\mu$ for every data point,
which indeed justifies our model.

##  5 Stability

The present section deals with examining classical stability of KHDE model
through the squared speed of sound ($v_{s}^{2}$) for a closed as well as open
FRW Universe. The squared speed of sound is given as [60, 61]

$\displaystyle
v_{s}^{2}=\frac{dp_{D}}{d\rho_{D}}=\frac{\rho_{D}}{\dot{\rho}_{D}}\dot{\omega}_{D}+\omega_{D}.$
(21)

In figure (6), we plotted the behavior of squared speed of sound ($v_{s}^{2}$)
against redshift using different values of $\kappa$ parameter. As we see for
some values of Kaniadakis parameter namely, $\kappa=3150$, we have
$v_{s}^{2}<0$ and thus, KHDE model is not stable in both open and closed
Universe models, see the blue dotted curve. However, for other values of
$\kappa$ parameter $v_{s}^{2}>0$ and the KHDE model is stable for late-time,
see the black solid and red dashed curves. Hence the condition on classical
stability of the model puts limit on the allowed values of Kaniadakis
parameter.

##  6 Conclusions

In this research, we have examined KHDE model in the non-flat universe which
is dependent on two model parameters $C$ and $\kappa$. While estimating these
model parameters we have also used the dataset samples of distance modulus of
580 scores of Union 2.1 combining SNIa along with the $30$ CC data points of
$H(z)$ analyses. Besides, we have reconstructed the evolving behavior of
$q(z)$, $\omega_{D}$ and $\Omega_{D}$ for the existing model, while working on
the fittest values of $C$ and $\kappa$.

Proceeding to the detailed investigation, we showed that the classification of
DE and matter dominated eras through the supporting diagrams can explain the
Universe’s thermal history. Furthermore, we examined the effect of the
Kanadakis parameter $\kappa$, as well as of the curvature density parameter at
present, on the EoS parameter of DE. As we observed, the EoS parameter of DE
prevails to the quintessence region, crossing the phantom-divide has been
realized at present, namely KHDE favors the phantom region. However, the most
attractive characteristic is that comparing to the flat case, where the
phantom regime was not obtained, the incorporation of curvature leads the
Universe to the phantom regime. This is an advantage since one expects that
only small deviations from standard entropy could be one of the cases

It is however normal to develop the present study with the extension of other
data from BAO (Baryon Acoustic Oscillation), SNIa, and CMB (Cosmic microwave
background) probes in order to restrain the new parameters more accurately.
Work along this line is in progress and further results on this topic will be
reported in forthcoming researches.

## Acknowledgements

The authors are thankful to Ms. Gunjan Varshney, GLA University, Mathura,
India for her help in removing the glitches of the literature.

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