# Rényi Holographic Dark Energy

Ratchaphat Nakarachinda 111Email<EMAIL_ADDRESS>The Institute for
Fundamental Study (IF), Naresuan University,
99 Moo 9, Tah Poe, Mueang Phitsanulok, Phitsanulok, 65000, Thailand
Department of Physics, School of Science, King Mongkut’s Institute of
Technology Ladkrabang, 1 Chalong Krung 1 Alley, Lat Krabang, Bangkok 10520,
Thailand  Chakrit Pongkitivanichkul 222Email<EMAIL_ADDRESS>Khon Kaen
Particle Physics and Cosmology Theory Group (KKPaCT), Department of Physics,
Faculty of Science, Khon Kaen University, 123 Mitraphap Rd., Khon Kaen, 40002,
Thailand
Daris Samart 333Email<EMAIL_ADDRESS>Khon Kaen Particle Physics and
Cosmology Theory Group (KKPaCT), Department of Physics, Faculty of Science,
Khon Kaen University, 123 Mitraphap Rd., Khon Kaen, 40002, Thailand
Lunchakorn Tannukij 444Email<EMAIL_ADDRESS>Department of Physics,
School of Science, King Mongkut’s Institute of Technology Ladkrabang, 1
Chalong Krung 1 Alley, Lat Krabang, Bangkok 10520, Thailand  Pitayuth Wongjun
555Email<EMAIL_ADDRESS>The Institute for Fundamental Study (IF), Naresuan
University,
99 Moo 9, Tah Poe, Mueang Phitsanulok, Phitsanulok, 65000, Thailand

###### Abstract

In this work, the holographic dark energy model is constructed by using the
non-extensive nature of the Schwarzschild black hole via the Rényi entropy.
Due to the non-extensivity, the black hole can be stable under the process of
fixing the non-extensive parameter. A change undergoing such a process would
then motivate us to define the energy density of the Rényi holographic dark
energy (RHDE). As a result, the RHDE with choosing the characteristic length
scale as the Hubble radius provides the late-time expansion without the issue
of causality. Remarkably, the proposed dark energy model contains the non-
extensive length scale parameter additional to the standard $\Lambda$CDM
model. The cosmic evolution can be characterized by comparing the size of the
Universe to this length scale. Moreover, the preferable value of the non-
extensive length scale is determined by fitting the model to recent
observations. The results of this work would shed light on the interplay
between the thermodynamic description of the black hole with non-extensivity
and the classical gravity description of the evolution of the Universe.

##  I Introduction

One of the significant theoretical challenges in modern cosmology is an
attempt to explain the accelerated expansion of the Universe nowadays [1, 2].
It is possible to categorize such theoretical models into two classes,
modified gravity theories and dark energy, adding exotic contributions to the
Einstein field equation. The simple and promising candidate for dark energy is
cosmological constant which is well-known as $\Lambda$CDM. Since its energy
density of the cosmological constant is constant throughout the whole
evolution of the Universe, it is consequently interpreted as the vacuum energy
[3, 4]. However, the value of the energy density calculated from standard
quantum field theory has a discrepancy of about 120 orders in magnitude with
the observed value [5]. This leads to the fine-tuning problem in which the
value of the energy density in the past must be tremendously fine-tuned.
Moreover, this leads to the puzzle of why the contributions from dark energy
and non-relativistic matter are comparable nowadays, namely coincidence
problem [6]. As a result, numerous dynamical models of dark energy have been
proposed [7, 8]. In this article, we are interested in the dark energy model
in which the root idea is inspired by a fundamental principle called the
holographic principle. Therefore, such a dark energy model is commonly known
as “holographic dark energy” (HDE).

The HDE is rooted from the holographic principle which generally refers to the
duality between theories of the bulk and its boundary [9, 10, 11]. By using
this notion with dimensional reduction, it is possible to prevent one from
over-counting the degrees of freedom for the black hole entropy [12]. This
provides a hint to solve the cosmological constant since it suggests that one
may over-count the degrees of freedom by using local quantum field theory.
Based on this notion, it may be a link between the effective field theory
which connects the energy density and length scale through a saturation
entropy. In other words, the ordinary matter with saturated energy should not
form a black hole, i.e., $E_{\text{HDE}}\sim E_{\text{BH}}=L/(2G)$. According
to such a definition, there exists a relationship between the IR scale
corresponding to the characteristic length scale of the dark energy $L$ and
the UV scale corresponding to the energy scale of the black hole or the
Newtonian gravitational constant $G$. Note that such a relation is also
obtained when we consider the Bekenstein bound [13, 14, 15], but it is
expressed by considering the entropy [16]. As a result, the energy density of
HDE is constructed as follows

$\displaystyle\rho_{\text{HDE}}=\frac{3b^{2}}{8\pi GL^{2}},$ (1)

where $b^{2}$ is a dimensionless constant. It is seen that the characteristic
length scale is arbitrary.

In describing the dynamics of the Universe, the IR length scale $L$ should be
interpreted as some cosmological length. One of the possibilities is the
Hubble radius. Unfortunately, the model of HDE with Hubble radius as the IR
length scale cannot drive the accelerated expansion [17]. In fact, the
evolution of this HDE is the same as the dominant species in the Universe. For
example, in the matter-dominated epoch, the energy density is proportional to
$a^{-3}$ where $a$ is the scale factor. This HDE is therefore diluted as time
grows. After that the successful model is proposed by treating the IR length
scale as the future particle horizon [18]. The cosmological implications of
this model are investigated in various aspects [19]. For example, many types
of interaction between HDE and dark matter and their observational constraints
are studied [20, 21, 22]. Even though this model can drive the accelerated
expansion of the Universe, it suffers from a problem of causality [23]. The
information in the future is required to describe the dynamics at the present
time.

Other models with different IR length scales are proposed [24, 25, 26, 27,
28]. In addition, HDE in modified gravity theories are also numerously active
[29, 30, 31, 32, 33, 34]. Based on the holographic principle, it is feasible
to construct more types of dark energy models. Using time scale as the IR
cutoff, the agegraphic dark energy concept has been put out [35, 36]. Since
the Ricci scalar $R$ does in fact have a dimension of $L^{-2}$, one can define
the dark energy based on the corresponding length scale, referred to as the
Ricci dark energy [37, 38]. The perturbation analysis of the Ricci model is
also investigated [39]. Since the notion of HDE roots from the black hole
entropy and there are various investigations on the generalized entropies, the
HDE model with generalized entropies have been investigated [40, 41, 42, 43,
44].

According to the thermodynamic properties of the cosmological horizon [45, 46,
47], there is a worthwhile proposal to extend the form of the energy density
of HDE by involving the first law of thermodynamics at the horizon [48].
Specifically, the energy density of the HDE can be written in derivative form
as $\rho_{\text{HDE}}=T\text{d}S/\text{d}V$. In this form, it provides a
consistent way to deal with various kinds of generalized entropy. Therefore,
there have been intensive investigations using this form of the energy density
[49, 50, 51, 52, 53]. Note that there is a suggestion that the Friedmann
equations are also modified when the generalized entropy is taken into account
[54]. Recently, rigorous investigation on this issue has been clarified by
using the continuity equation [55]. It was shown that this form of the energy
density is consistent with the standard HDE when we consider the Friedmann
equations or entropies that follow exponent stretched area law such as Barrow
entropy, Tsallis entropy, and Tsallis-Cirto entropy. For the Rényi entropy,
the energy density will be explored and it will be examined whether cosmic
accelerated expansion is achieved.

In the present work, we use a similar form to the one in Ref. [48]. However,
we adopt the first law of thermodynamics based on the properties of the black
hole instead of using one in thermodynamics at the apparent horizon as
commonly found in the literature. Therefore, in our approach, it can be
applied to other kinds of black holes. Actually, by using the first law of
thermodynamics of the AdS black hole, the accelerated expansion with proper
evolution of the Universe can be obtained [56]. As a result, the form of
energy density in Eq. (1) can be obtained from the first law of black hole
thermodynamics,

$\displaystyle\rho_{\text{HDE}}\sim\frac{\text{d}M_{\text{BH}}}{\text{d}V}=T_{\text{G}}\frac{\text{d}S_{\text{G}}}{\text{d}V},$
(2)

where $T_{\text{G}}=\partial M_{\text{BH}}/\partial S_{\text{G}}$ and
$S_{\text{G}}$ are the conjugate variable for the entropy which is interpreted
as temperature and the generalized entropy, respectively. $V=\partial
M_{\text{BH}}/\partial P$ is a conjugate variable for the pressure $P$ simply
interpreted as the volume of the Universe. Note that by keeping the
temperature defined in this way, the Legendre structure is preserved then
thermodynamic quantities are suitably defined.

From the black hole thermodynamics viewpoint, the thermal system associated
with the black hole is a non-extensive system since the entropy of the black
hole is adopted as the horizon area [57, 58]. Therefore, it is worthwhile to
formulate the entropy of the black hole as non-extensive entropy. Accordingly,
there are several investigations on the non-extensive nature of the black
holes [59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74]. One of
the useful choices of the non-extensive entropy is the Tsallis entropy [75,
76]. However, by using Tsallis statistics, it is not easy to define a proper
temperature which is the state variable based on the notion of maximum
entropy. In fact, the composition rule for the Tsallis entropy does not
provide the definition in such a way. Fortunately, the proper temperature can
be defined via the Rényi entropy which is the formal logarithm map of the
Tsallis entropy [77]. In this sense, other thermodynamic state variables and
thermodynamic laws can be defined consistently [66, 72]. This is a worth
result of the Rényi entropy as the black hole entropy. In this work, we
utilize the Rényi entropy along with the first law of thermodynamics and
examine how the accelerated expansion of the Universe can be achieved in terms
of HDE.

For the black hole thermodynamics with Rényi entropy, the non-extensive
parameter $\lambda$ can be promoted as the thermodynamic variable. This
variable can be adopted as a kind of effective pressure of the system while
its conjugate variable corresponds to the thermodynamics volume. To obtain a
consistent energy density in terms of HDE, we consider the thermodynamics
process at which the pressure is held fixed. Therefore, the non-extensive
parameter is a model parameter in the cosmological aspect. With this setup,
the model is similar to the HDE constructed from AdS black hole (AdS-HDE) [56]
while the non-extensive parameter can play the same role as the cosmological
constant. As a result, we found that it is possible to obtain the accelerated
expansion of the Universe similar to one in AdS-HDE. However, the
interpretation is significantly different since the expansion of the Universe
is due to the non-extensive nature of the black hole. By comparing the results
to the observation, we found that the non-extensive length scale is preferable
as $L_{\lambda}=\sqrt{G/(\pi\lambda)}\approx 0.07H^{-1}_{0}$ where $H_{0}$ is
the current observed value of the Hubble parameter. This may shed light on the
interplay between the non-extensive nature of the black hole in the
description of thermodynamics and the evolution of the Universe in the
description of classical gravity.

Following is a description of how this work is organized. Sec. II discusses
the effect of nonextensivity on black hole thermodynamics by analyzing it via
Rényi entropy. The form of the energy density constructed from the black hole
with Rényi statistics is defined in Sec. III. The dynamics of the Universe
with such dark energy are investigated in Sec. IV and the model parameter is
constrained by observational data in Sec. V. Finally, in Sec. VI, the key
results as well as the interesting remarks are discussed and summarized.

##  II Schwarzschild black hole with Rényi entropy

In this work, HDE motivated from Rényi thermodynamics of the Schwarzschild
black hole is proposed to drive the late-time acceleration of the universe. To
formulate the corresponding energy density for the proposed HDE, the Rényi
thermodynamics of Schwarzschild black hole is briefly reviewed in this
section.

A Schwarzschild black hole is the hole whose geometry is described, in terms
of the Schwarzschild coordinates $(r,\theta,\phi)$, by the following
Schwarzschild metric.

$\displaystyle\text{d}s^{2}=-\left(1-\frac{2GM}{r}\right)\text{d}t^{2}+\left(1-\frac{2GM}{r}\right)^{-1}\text{d}r^{2}+r^{2}\text{d}\Omega^{2},$
(3)

where $M$ is the mass of the black hole which can be expressed in terms of the
Schwarzschild radius, $r_{h}$, as

$\displaystyle M=\frac{r_{h}}{2G}.$ (4)

The Hawking temperature, $T_{\text{H}}$, of such black hole and its Gibbs-
Boltzmann entropy, $S_{\text{BH}}$, can be expressed in terms of $r_{h}$ as

$\displaystyle T_{\text{H}}=\frac{1}{4\pi r_{h}},\qquad
S_{\text{BH}}=\frac{A}{4G}=\frac{\pi r_{h}^{2}}{G},$ (5)

where $A$ is the area of the 2-sphere specified by the Schwarzschild radius.
However, $S_{\text{BH}}$ is non-extensive since it does not scale with the
size (3-volume) of the system. The non-extensivity can be incorporated into
the consideration by assuming that the entropy of black holes may obey the
following Tsallis composition rule [75]

$\displaystyle S_{\text{tot}}=S_{1}+S_{2}+\lambda S_{1}S_{2},$ (6)

where $S_{1}$ and $S_{2}$ represent entropies of subsystems 1 and 2,
$S_{\text{tot}}$ is the total entropy of the system whose components are the
previously mentioned subsystems, and $\lambda$ denotes the non-extensivity of
the system. Due to the ambiguity of realizing empirical temperature through
the zeroth law of thermodynamics, one may transform the entropies associated
with the Tsallis composition rule into the following Rényi entropy.

$\displaystyle S_{\text{R}}=\frac{1}{\lambda}\ln{\left(1+\lambda
S_{\text{BH}}\right)},$ (7)

which recovers the Gibbs-Boltzmann (GB) entropy when the non-extensive
parameter $\lambda$ approaches zero. The non-extensive parameter $\lambda$
plays both roles of the mapping parameter from non-additive Tsallis entropy to
the additive Rényi one and of characterization of how much the Rényi entropy
deviates from the standard GB entropy. In addition, this parameter is bounded
according to the statistical axiom of the maximized entropy as $\lambda<1$ in
the unit of $k_{\text{B}}=1$ [77]. It is clear from this bound that the
entropic function behaves as a concave function. According to the dS to AdS
phase transition in five-dimensional spacetime with the Gauss-Bonnet gravity,
however, there exists a negative value of $\lambda$ that produces a positive
Rényi temperature [78]. Note that the logarithmic function in Eq. (7) is
always well-behaved when $\lambda$ is positive. As a result, it is prudent to
stick with our consideration for the range of the non-extensive parameter of
$0\leq\lambda<1$. In addition, the aforementioned range satisfies the strong
condition of concavity in the context of quantum entanglement [79].

###  II.1 Thermodynamic phase space

Since the non-extensivity $\lambda$ is introduced in the context of Rényi
entropy, its thermodynamic phase space can be generally extended from the one
in the context of GB entropy. To construct the thermodynamic phase space due
to Rényi entropy, the black hole’s geometrical quantities must be expressed in
the appropriate form [66]. It is found that the mass of the black hole in Eq.
(4) can be written in terms of a homogeneous function of degree $1/2$ of the
Rényi entropy and the non-extensive parameter as follows:

$\displaystyle M(S_{\text{R}},\lambda^{-1})$ $\displaystyle=$
$\displaystyle\sqrt{\frac{e^{\lambda S_{\text{R}}}-1}{4\pi G\lambda}}.$ (8)

Applying Euler’s theorem to this function, the Smarr formula and the first law
are given by

$\displaystyle M$ $\displaystyle=$ $\displaystyle
2T_{\text{R}}S_{\text{R}}-2\Phi\lambda,$ (9) $\displaystyle\text{d}M$
$\displaystyle=$ $\displaystyle
T_{\text{R}}\text{d}S_{\text{R}}+\Phi\text{d}\lambda.$ (10)

Here, $T_{\text{R}}$ and $\Phi$ are conjugate variables of $S_{\text{R}}$ and
$\lambda$, respectively. They are defined as

$\displaystyle T_{\text{R}}$ $\displaystyle=$
$\displaystyle\left(\frac{\partial M}{\partial
S_{\text{R}}}\right)_{\lambda}\,=\,\frac{1}{4\pi
r_{h}}\left(1+\lambda\frac{\pi r_{h}^{2}}{G}\right),$ (11) $\displaystyle\Phi$
$\displaystyle=$ $\displaystyle\left(\frac{\partial
M}{\partial\lambda}\right)_{S_{\text{R}}}\,=\,\frac{1}{4\pi
r_{h}\lambda^{2}}\left[\left(1+\lambda\frac{\pi
r_{h}^{2}}{G}\right)\ln\left(1+\lambda\frac{\pi
r_{h}^{2}}{G}\right)-\lambda\frac{\pi r_{h}^{2}}{G}\right].$ (12)

Interestingly, the conjugate variable $\Phi$ can be approximated for small
$\lambda$ as follows:

$\displaystyle\Phi$ $\displaystyle\approx$ $\displaystyle\frac{\pi
r_{h}^{3}}{8G^{2}}-\lambda\frac{\pi^{2}r_{h}^{5}}{24G^{3}}+\lambda^{2}\frac{\pi^{3}r_{h}^{7}}{48G^{4}}+\ldots$
(13)

It is obvious that the leading term is proportional to the cubic of the
horizon radius. This conjugate variable can be interpreted as a volume
including the non-extensive effect. Therefore, one, respectively, defines
thermodynamic volume and pressure as

$\displaystyle V_{\text{R}}=\frac{32G^{2}}{3}\Phi,\qquad
P_{\text{R}}=\frac{3}{32G^{2}}\lambda.$ (14)

Note that the factors in defining the volume and pressure are introduced in
order to obtain that the leading order of $V_{\text{R}}$ becomes $4\pi
r_{h}^{3}/3$. As a result, the thermodynamic relations (9) and (10) are
rewritten as

$\displaystyle M$ $\displaystyle=$ $\displaystyle
2T_{\text{R}}S_{\text{R}}-2V_{\text{R}}P_{\text{R}},$ (15)
$\displaystyle\text{d}M$ $\displaystyle=$ $\displaystyle
T_{\text{R}}\text{d}S_{\text{R}}+V_{\text{R}}\text{d}P_{\text{R}}.$ (16)

According to the first law (16), the mass of the black hole plays the role of
the chemical enthalpy.

###  II.2 Stabilities of black hole

The thermodynamic stabilities of the black hole can be categorized into two
cases. The first one is called local stability. Such stability is
characterized by the positiveness of the heat capacity,
$C=\text{d}E/\text{d}T$ where $E$ is the black hole’s energy. If the black
hole has a negative heat capacity, it is locally unstable. Imagine that the
black hole is surrounded by an environment with a lower temperature. Despite
trying to radiate its energy, it gets hotter. This is equivalent to the
evaporation of the black hole, and it eventually disappears. On the other
hand, if it is surrounded by an environment with a higher temperature, the
black hole will absorb energy from the outside. Its temperature, however, gets
progressively lower. In such a situation, the black hole is moving further and
further away from equilibrium with the surrounding environment. Hence, it
cannot be stable. For an isobaric process, the heat capacity is given by

$\displaystyle C_{P_{\text{R}}}$ $\displaystyle=$
$\displaystyle\left(\frac{\partial M}{\partial
T_{\text{R}}}\right)_{P_{\text{R}}}\,=\,T_{\text{R}}\left(\frac{\partial
S_{\text{R}}}{\partial T_{\text{R}}}\right)_{\lambda}\,=\,-\frac{2\pi
r_{h}^{2}}{G\left(1-\lambda\frac{\pi r_{h}^{2}}{G}\right)}.$ (17)

It is seen that, without the non-extensive effect (i.e., $\lambda\to 0$), the
heat capacity of the black hole is always negative. For $\lambda$ being
positive, the heat capacity can be positive. Moreover, the local stability of
the black hole requires only positive (not negative) values of $\lambda$.
These results are illustrated in the left panel of Fig. 1.

Figure 1: Heat capacity (left) and Gibbs free energy (right) of Schwarzschild
black hole with various values of non-extensive parameter $\lambda$

The black hole in the GB limit (black solid line) is always locally unstable
while a large black hole described by Rényi entropy (e.g., those with blue
dotted and red dashed lines) can be locally stable. From Eq. (17), the locally
stable black hole must have a sufficiently large radius satisfied a condition:

$\displaystyle r_{h}>\sqrt{\frac{G}{\pi\lambda}}.$ (18)

The second type of stability is called global stability. This stability
identifies which phase of all possibility preferably exists in nature. In
particular, the preferable phase must have lower free energy, e.g., Gibbs free
energy, than others. We are interested in whether the black hole is
thermodynamically formed compared to the no-black hole phase (i.e., the free
energy is set to be zero). Consequently, the globally stable black hole is
characterized as a phase with negative free energy. For our consideration, the
Gibbs free energy is defined by

$\displaystyle\mathcal{G}_{\text{R}}$ $\displaystyle=$ $\displaystyle
M-T_{\text{R}}S_{\text{R}}\,=\,\frac{r_{h}}{2G}-\frac{1}{4\pi
r_{h}\lambda}\left(1+\lambda\frac{\pi
r_{h}^{2}}{G}\right)\ln\left(1+\lambda\frac{\pi r_{h}^{2}}{G}\right).$ (19)

The effect of the non-extensivity appears only in the second term. It is found
that this term (i.e., non-zero $\lambda$) can make $\mathcal{G}_{\text{R}}$
negative for the sufficiently large black hole phase

$\displaystyle
r_{h}>\sqrt{\left[-1+\exp\left\\{2+\text{ProductLog}\left(-\frac{2}{e^{2}}\right)\right\\}\right]\frac{G}{\pi\lambda}}\approx
1.98\sqrt{\frac{G}{\pi\lambda}},$ (20)

where $\text{ProductLog}(y)$ is the solution for $z$ in $y=ze^{z}$. It can be
seen in the right panel of Fig. 1. Again, the black hole described by the GB
entropy is always globally unstable.

Since there exists a globally stable phase for the black hole with Rényi
entropy, the first-order Hawking-Page (HP) phase transition from no-black hole
to stable black hole phases occurs. It is represented as the blue crossed
point in Fig. 2.

Figure 2: Diagram of the Gibbs free energy versus temperature for
Schwarzschild black hole

In this figure, the red solid and black dashed lines are represented as the
black hole with Rényi and GB entropies, respectively. It is seen that, at low
temperatures, the black hole cannot be formed as the phases lie on the green
dotted horizontal line. Until the temperature is sufficiently high (known as
the HP temperature), the black hole is preferred to form as phases lie on the
green dotted slanted line.

In addition, one can realize the effect of non-extensivity by interpreting it
as a length scale compared to the size of the system, i.e., the black hole’s
horizon radius. The non-extensivity translates to a quantity in the dimension
of length as follows:

$\displaystyle L_{\lambda}=\sqrt{\frac{G}{\pi\lambda}}.$ (21)

Note also that such a length scale is compatible only when the parameter
$\lambda$ is positive. By equipping all constants, the non-extensive length
scale is expressed as $L_{\lambda}=\sqrt{\hbar G/\left(\pi
k_{\text{B}}c^{3}\lambda\right)}\sim 10^{-35}/\sqrt{k_{\text{B}}\lambda}$ m.
This means that $L_{\lambda}$ can be small in the order of $10^{-35}$ m as
$k_{\text{B}}\lambda$ reaches its upper bound, i.e., $k_{\text{B}}\lambda\to
1$. On the other hand, as $k_{\text{B}}\lambda$ goes zero, or equivalently,
approaching the GB limit, the non-extensive length scale is infinite. Based on
GB statistics, this is an alternative viewpoint of the Schwarzschild black
hole’s instabilities. According to the stability conditions (18) and (20), it
is obviously seen that the globally stable black hole requires a larger radius
than the local one. In other words, the black hole radius-to-non-extensive
length ratio, i.e., $r_{h}/L_{\lambda}$, can identify the stability of the
black hole. One can conclude that the Schwarzschild black hole described by
Rényi statistics can be locally and globally stable when

$\displaystyle\frac{r_{h}}{L_{\lambda}}\gtrapprox 1.98.$ (22)

If the non-extensive length is not too large, the black hole is always stable.

It is very important to emphasize that, in the aspect of thermodynamic
stability, the effect of non-extensivity analyzed via the Rényi entropy on a
black hole in asymptotically flat spacetime is very closed to the effect of
existence of the cosmological constant analyzed via the GB entropy on a black
hole in asymptotically anti-de Sitter (AdS) spacetime [80]. In other words,
the stable phase can emerge due to non-extensivity or negative cosmological
constant. According to Ref. [64], the aforementioned similarity convinces the
authors to investigate a cosmological solution via HDE defined from the black
hole with Rényi entropy as will be discussed in the next sections.

##  III Holographic Dark Energy from Rényi black hole

As previously mentioned, the thermodynamic property of a black hole can play a
significant role in defining HDE. In this section, we consider the
Schwarzschild black hole with Rényi entropy. The behaviors of HDE defined from
such a black hole will be discussed in the context of dark energy in the
Universe.

The holographic dark energy is defined as matter with saturated energy
corresponding to the energy of a black hole. According to the thermodynamic
stabilities under the isobaric process, we are interested in defining the dark
energy from the black hole evolved under such a process. The change in the
black hole’s energy can be written in terms of the energy density and volume
as

$\displaystyle\text{d}E_{\text{saturate}}$ $\displaystyle\sim$
$\displaystyle\rho_{\text{HDE}}\text{d}V.$ (23)

In the mentioned scenario, the change of the black hole’s energy equals that
of the thermodynamic enthalpy, or equivalently, mass. The energy density of
the Rényi holographic dark energy (RHDE) is, therefore, given by

$\displaystyle\rho_{\text{RHDE}}$ $\displaystyle\sim$
$\displaystyle\left(\frac{\text{d}M}{\text{d}V_{\text{R}}}\right)_{P_{\text{R}}}\,=\,\left(\frac{\partial_{L}M}{\partial_{L}V_{\text{R}}}\right)_{\lambda}.$
(24)

Here, the volume $V$ is interpreted as the thermodynamic volume of the black
hole in Rényi description, i.e., $V_{\text{R}}$ in Eq. (14). The horizon
radius $r_{h}$ from the black hole side is replaced by the characteristic
length scale or the IR length scale $L$. It is very important to note that the
form of the energy density of the original model is modified due to the
realization of the thermodynamic “Rényi” volume, not the Rényi entropy. This
is one of the crucial notions in our approach for which the resulting energy
density is different from those in Refs. [54, 55]. It is because the change of
the black hole’s mass (under the isobaric process) is identical to that
described by the BH entropy,

$\displaystyle\text{d}M\big{|}_{P_{\text{R}}}$ $\displaystyle=$ $\displaystyle
T_{\text{R}}\text{d}S_{\text{R}}\,=\,T_{\text{H}}\text{d}S_{\text{BH}}.$ (25)

As a result, the energy density of RHDE can be expressed as

$\displaystyle\rho_{\text{RHDE}}$ $\displaystyle=$ $\displaystyle
3b^{2}\left[\frac{3}{16\pi
GL_{\lambda}^{2}\left\\{1+\left(1-\frac{L_{\lambda}^{2}}{L^{2}}\right)\ln\left(1+\frac{L^{2}}{L_{\lambda}^{2}}\right)\right\\}}\right],$
(26)

where the non-extensive length scale $L_{\lambda}$ was defined in Eq. (21).
The constant $3b^{2}$ is introduced to recover the original model in the GB
limit [17, 18], i.e.,
$\displaystyle{\lim_{L_{\lambda}\to\infty}\rho_{\text{RHDE}}=\frac{3b^{2}}{8\pi
GL^{2}}}$.

Let us analyze the behaviors of the energy density of RHDE in Eq. (26). Under
the assumption of fixing the pressure, the non-extensive parameter is
appropriately treated as a time-independent variable throughout the evolution
of the Universe. Together with the fact that the parameter $b^{2}$ is just a
constant, the dynamical variable of RHDE model is, thus, only the IR length
scale $L$. It is worthwhile to analyze the asymptotic behaviors of the energy
density $\rho_{\text{RHDE}}$ in different regimes of $L$. For a very small $L$
or, equivalently, $L\ll L_{\lambda}$, one can approximate

$\displaystyle\rho_{\text{RHDE}}\Big{|}_{L\ll L_{\lambda}}$
$\displaystyle\approx$ $\displaystyle\frac{3b^{2}}{8\pi
GL^{2}}+\frac{5b^{2}}{24\pi GL_{\lambda}^{2}}+\frac{13b^{2}L^{2}}{432\pi
GL_{\lambda}^{4}}+\ldots$ (27)

It is seen that the leading-order term is exactly the same as that in the
original model. Hence, the Universe filled with RHDE in this limit will mainly
evolve in a similar way to that of the original model. Interestingly, the
next-order term is a constant corresponding to the existence of the non-
extensive effect. Note also that such a constant term appears in the energy
density of HDE defined from the Schwarzschild-anti de Sitter black hole [56].
On the other hand, the energy density $\rho_{\text{RHDE}}$ in a very large $L$
regime is approximated as

$\displaystyle\rho_{\text{RHDE}}\Big{|}_{L\gg L_{\lambda}}$
$\displaystyle\approx$ $\displaystyle\frac{9b^{2}}{16\pi
GL_{\lambda}^{2}(1+x)}-\frac{9b^{2}(1-x)}{16\pi
GL^{2}(1+x)^{2}}+\frac{9b^{2}L_{\lambda}^{2}(5-x+2x^{2})}{32\pi
GL^{4}(1+x)^{3}}+\ldots$ (28)

where $x=\ln\left(L^{2}/L_{\lambda}^{2}\right)$. Even though $x$ is dynamical,
the rate of change is suppressed due to the logarithmic function. According to
this approximation, $x$ can be roughly thought of as a positive constant in
the order of unity. As a result, the first term is made constant and dominant
over the rest, hereafter the “approximated constant”. Consequently, the
Universe will be driven to expand with acceleration. If the aforementioned
scenario occurs at the late time of the cosmic evolution, the Universe will
behave similarly to that of the $\Lambda$CDM model or the AdS-HDE model at
late time [56].

##  IV Cosmological solution

In this section, the evolution of the Universe with our proposed dark energy
model is investigated. The domination of the RHDE at the late time is then
illustrated with suitable values of the parameters in the model.
Interestingly, this RHDE can drive the Universe to expand at an agreed rate
with observation.

###  IV.1 Dark Energy Domination

To study the evolution of the homogenous and isotropic Universe, it is
supposed that the dynamics of the Universe are governed by Einstein’s general
theory of relativity. The field equation is given by

$\displaystyle G^{\mu}_{\hskip 4.26773pt\nu}=8\pi GT^{\mu}_{\hskip
5.69046pt\nu},$ (29)

where $G^{\mu}_{\hskip 4.26773pt\nu}$ and $T^{\mu}_{\hskip 5.69046pt\nu}$ are
the Einstein tensor describing the curvature of the spacetime and the energy-
momentum tensor existing in the spacetime, respectively. For the curvature
sector, we have considered the flat Friedmann-Lemaître-Robertson-Walker (FLRW)
metric,

$\displaystyle\text{d}s^{2}=-\text{d}t^{2}+a(t)^{2}\left(\text{d}r^{2}+r^{2}\text{d}\theta^{2}+r^{2}\sin^{2}\theta\text{d}\phi^{2}\right),$
(30)

where $a(t)$ is a scale factor characterizing how the Universe evolves. For
the matter sector, the energy-momentum tensor is chosen to be taken in the
perfect fluid form as follows:

$\displaystyle T^{\mu}_{\hskip
5.69046pt\nu}=\text{diag}\Big{(}-\rho(t),p(t),p(t),p(t)\Big{)}.$ (31)

Here, $\rho(t)$ and $p(t)$ are the energy density and pressure of the fluid,
respectively. Moreover, these energy density and pressure are assumed to be
linearly related as

$\displaystyle p=w\rho,$ (32)

where $w$ is called the equation of state parameter. For instance, $w=1/3$ and
$w=0$ for the radiation and non-relativistic matter, respectively. As a
result, the $(0,0)$ and $(i,j)$ components of the Einstein equation (29) are
respectively expressed as

$\displaystyle 3H^{2}$ $\displaystyle=$ $\displaystyle 8\pi
G\left(\rho_{\text{r}}+\rho_{\text{m}}+\rho_{\text{de}}\right),$ (33)
$\displaystyle 2\dot{H}+3H^{2}$ $\displaystyle=$ $\displaystyle 8\pi
G\left(-w_{\text{r}}\rho_{\text{r}}-w_{\text{m}}\rho_{\text{m}}-w_{\text{de}}\rho_{\text{de}}\right),$
(34)

where $H=\dot{a}/a$ is the Hubble parameter. The dot denotes the derivative
with respect to the cosmic time $t$. The quantities with subscripts “r”, “m”
and “de” denote those for the radiation, matter, and dark energy,
respectively. By defining the density parameter as follows:

$\displaystyle\Omega_{i}$ $\displaystyle=$ $\displaystyle\frac{8\pi
G}{3H^{2}}\rho_{i},$ (35)

Eq. (33) is then rewritten as a constraint equation,

$\displaystyle 1$ $\displaystyle=$
$\displaystyle\Omega_{\text{r}}+\Omega_{\text{m}}+\Omega_{\text{de}}.$ (36)

In our consideration, we are simply interested in the case of no interaction
among species in the Universe. Therefore, each species conserves independently
and its energy-momentum tensor obeys the conservation equation,
$\nabla_{\mu}T^{\mu}_{\hskip 7.11317pt\nu}=0$ which can be further expressed
as

$\displaystyle\dot{\rho}_{i}+3H\big{(}\rho_{i}+p_{i}\big{)}$ $\displaystyle=$
$\displaystyle\dot{\rho}_{i}+3H\big{(}1+w_{i}\big{)}\rho_{i}\,=\,0,$ (37)

where $w_{i}=p_{i}/\rho_{i}$ is the equation of state parameter. Note that,
for a species with the constant equation of state parameter, the energy
density is straightforwardly obtained as a monomial function of the scale
factor, $\rho\propto a^{-3(1+w)}$. By substituting this to Eq. (35), Eq. (36)
is eventually written as

$\displaystyle 1$ $\displaystyle=$
$\displaystyle\Omega_{\text{r,0}}l_{H}^{2}e^{-3(1+w_{\text{r}})\ln
a}+\Omega_{\text{m,0}}l_{H}^{2}e^{-3(1+w_{\text{m}})\ln
a}+\Omega_{\text{de}}.$ (38)

The quantities with subscript “0” stand for those at present. In the above
expression, the present-time scale is set to be unity $a_{0}=1$. We have
introduced the dimensionless Hubble radius as $l_{H}=H_{0}/H$. It is also
noticed that the cosmic time $t$ is now scaled as $\ln a$.

In our cosmological model, the characteristic length scale of RHDE $L$ is
chosen as the Hubble radius, i.e., $L=H^{-1}$. It is very important to note
that this choice of the length scale does not contain the problem of
causality. From Eq. (26), the density parameter of RHDE can be defined as

$\displaystyle\Omega_{\text{RHDE}}$ $\displaystyle=$ $\displaystyle\frac{8\pi
G}{3H^{2}}\rho_{\text{RHDE}}\,=\,\frac{3b^{2}}{2\frac{l_{\lambda}^{2}}{l_{H}^{2}}\left[1+\left(1-\frac{l_{\lambda}^{2}}{l_{H}^{2}}\right)\ln\left(1+\frac{l_{H}^{2}}{l_{\lambda}^{2}}\right)\right]}.$
(39)

For convenience in analyzing, we define the dimensionless non-extensive
variable $l_{\lambda}=L_{\lambda}H_{0}$. Let us emphasize that the RHDE model
obviously contains two non-dynamical parameters ($b^{2}$ and $l_{\lambda}$)
and one dynamical parameter ($l_{H}$).

The present-time ratio of RHDE is straightforwardly obtained by replacing
$l_{H}=1$, i.e.,
$\Omega_{\text{RHDE}}\big{|}_{l_{H}=1}=\Omega_{\text{RHDE,0}}$. The quantity
$\Omega_{\text{RHDE,0}}$ is tightly constrained by recent observational data,
e.g., Planck 2018 [81]. This implies that one of the non-dynamical parameters
can be eliminated and then be written in terms of the known observable
$\Omega_{\text{RHDE,0}}$ and another non-dynamical parameter. Conveniently,
one chooses to eliminate the parameter $b^{2}$ with the expression

$\displaystyle b^{2}$ $\displaystyle=$
$\displaystyle\frac{2}{3}\Omega_{\text{RHDE,0}}l_{\lambda}^{2}\left[1+\left(1-l_{\lambda}^{2}\right)\ln\left(1+\frac{1}{l_{\lambda}^{2}}\right)\right].$
(40)

As a result, in the recent model, there is only one extra free parameter
(i.e., $l_{\lambda}$) from those in the $\Lambda$CDM model. In addition, the
density parameter of RHDE can recover to that of the $\Lambda$CDM model in a
small $l_{\lambda}$ limit, i.e., $\displaystyle{\lim_{l_{\lambda}\to
0}\Omega_{\text{RHDE}}=\Omega_{\text{RHDE,0}}=\text{constant}}$. The larger
the non-extensive length $l_{\lambda}$, the greater the deviation of RHDE from
the $\Lambda$CDM model. It is important to note that the aforementioned limit
cannot be reached since there exists the bounded from the maximization of the
entropic function $k_{\text{B}}\lambda<1$ as mentioned in the previous
section. This leads to the fact that the magnitude of $l_{\lambda}$ is
constrained to be small as $l_{\lambda}>\sqrt{H_{0}^{2}\hbar G/(\pi
c^{5})}\sim 10^{-62}$. Therefore, RHDE can mathematically describe the
identical cosmic evolution of the Universe as the standard $\Lambda$CDM model
does when $l_{\lambda}$ approaches zero. However, this situation is forbidden
due to consistency in Rényi statistics.

For a given non-extensive length scale $l_{\lambda}$, the evolution of the
parameter $l_{H}$ is evaluated by solving the constraint equation (38)
together with Eqs. (39) and (40). The numerical result is illustrated in the
left panel of Fig. 3.

Figure 3: Evolutions of ratio $l_{H}/l_{\lambda}$ (left) and RHDE density
parameter (right) with respect to $\ln a$ for various values of $l_{\lambda}$.
In these plots, we set $\Omega_{\text{RHDE,0}}=0.69$.

According to the evolution, the magnitude of $l_{H}$ (or $H^{-1}$) compared to
$l_{\lambda}$ (or $L_{\lambda}$) is very small in the range $\ln a<-5$. In
this regime, the approximation in Eq. (27) appropriately explains the behavior
of RHDE. Hence, RHDE acts as the scaling solution in the early time. The
standard cosmic evolution corresponding to those of the radiation- and matter-
domination epoch is obtained for $\ln a<-5$. As time evolves, the left panel
of Fig. 3 also shows that the magnitude of $l_{H}$ gets larger and larger
until it equals the magnitude of $l_{\lambda}$ at $-2\lessapprox\ln
a\lessapprox 0.5$. Note that the time that $l_{H}=l_{\lambda}$ depends on the
non-extensivity. For a sufficiently small $l_{\lambda}$, the magnitude of
$l_{H}$ can be much greater than that of $l_{\lambda}$ in a regime $\ln a>0$.
The approximation in Eq. (28) is applicable to describe RHDE at the late time.
RHDE with a suitable $l_{\lambda}$ can provide the dark energy-dominated epoch
due to its approximated constant term in energy density, i.e., the first term
in the right-hand side of Eq. (28), being dominant. The results are confirmed
in the right panel of Fig. 3. RHDE will be dominant over other species at the
late time.

Another feature of RHDE is that the non-extensive length scale controls the
ratio of the dark energy in the early times as shown in the right panel of
Fig. 3 or Fig. 4.

Figure 4: Evolution of density parameters for all species with respect to $\ln
a$ for $l_{\lambda}=0.1$ (left), $l_{\lambda}=1$ (middle) and $l_{\lambda}=10$
(right). Note that we have set the value of the present-time density
parameters for radiation and matter as $\Omega_{\text{r,0}}=10^{-4}$ and
$\Omega_{\text{m,0}}=1-\Omega_{\text{r,0}}-\Omega_{\text{RHDE,0}}$,
respectively.

It can be realized by considering the density parameter in a limit of small
$l_{H}$ as follows:

$\displaystyle\lim_{l_{H}\to
0}\Omega_{\text{RHDE}}=\frac{2}{3}l_{\lambda}^{2}\Omega_{\text{RHDE,0}}\left[1+(1-l_{\lambda}^{2})\ln\left(1+\frac{1}{l_{\lambda}^{2}}\right)\right].$
(41)

If the non-extensive length scale is too large, the ratio of radiation will be
too low. Thus, there is not enough production of fundamental particles’
nuclei. Such a strong constraint is known as the Big Bang Nucleosynthesis
(BBN) constraint which should be satisfied for the early dark energy. This
constraint provides that $\Omega_{\text{de}}<0.045$ at the radiation-dominated
epoch [82]. It leads to a maximum bound in $l_{\lambda}$ as
$l_{\lambda}<0.142$ when $\Omega_{\text{RHDE,0}}=0.69$. In fitting the model
with low-redshift observational data sets (in Sec. V), the range of
$l_{\lambda}$ would appear to be limited as $0<l_{\lambda}<0.2$.

As discussed in this subsection, we have shown that the RHDE can be dominant
over other species in the Universe. The next aim of this work is to
investigate how the Universe fulfilled with the RHDE evolves. In other words,
we are interested in determining whether the proposed dark energy can drive
the accelerated expansion of the Universe at a suitable rate.

###  IV.2 Accelerated Expansion

According to the conservation equation (37), the equation of state parameter
for RHDE is given by

$\displaystyle w_{\text{RHDE}}$ $\displaystyle=$
$\displaystyle-1-\frac{\dot{\rho}_{\text{RHDE}}}{3H\rho_{\text{RHDE}}}\,=\,-1+\frac{\frac{\text{d}}{\text{d}(l_{H}^{2})}\rho_{\text{RHDE}}}{\rho_{\text{RHDE}}}l_{H}^{4}\left(\frac{2\dot{H}}{3H_{0}^{2}}\right).$
(42)

This quantity is essential for the dark energy model because it characterizes
the rate of the expansion of the Universe. For the effective evolution of the
Universe for all epochs, one can determine the effective equation of state
parameter using Eqs. (33) and (34),

$\displaystyle w_{\text{eff}}$ $\displaystyle=$
$\displaystyle\frac{\sum_{i}p_{i}}{\sum_{i}\rho_{i}}\,=\,-1-\frac{2\dot{H}}{3H^{2}}\,=\,-1-l_{H}^{2}\left(\frac{2\dot{H}}{3H_{0}^{2}}\right).$
(43)

Obviously, there is still an unknown quantity $2\dot{H}/(3H_{0}^{2})$. It can
be evaluated by subtracting Eq. (33) by (34) and using Eqs. (42). One finds,

$\displaystyle\frac{2\dot{H}}{3H_{0}^{2}}$ $\displaystyle=$
$\displaystyle\frac{(1+w_{\text{r}})\Omega_{\text{r}}+(1+w_{\text{m}})\Omega_{\text{m}}}{l_{H}^{2}\left[1+l_{H}^{2}\frac{\frac{\text{d}}{\text{d}(l_{H}^{2})}\rho_{\text{RHDE}}}{\rho_{\text{RHDE}}}\left(1-\Omega_{\text{r}}-\Omega_{\text{m}}\right)\right]}.$
(44)

From the fact that the energy density $\Omega_{\text{RHDE}}$ is a function of
$l_{H}$, both equation of state parameters $w_{\text{RHDE}}$ and
$w_{\text{eff}}$ are, therefore, described by the parameter $l_{H}$ for each
$\ln a$. The numerical results are shown in Figs. 5 and 6.

Figure 5: Equation of state parameter of RHDE with respect to $\ln a$ for
various values of $l_{\lambda}$ Figure 6: Effective equation of state
parameter with respect to $\ln a$ for various values of $l_{\lambda}$

According to the approximation in Eq. (27), the energy densities at early time
and late are, respectively, given by

$\displaystyle\rho_{\text{RHDE}}\big{|}_{l_{H}\ll l_{\lambda}}$
$\displaystyle\propto$ $\displaystyle l_{H}^{-2}.$ (45)

Applying it to Eq. (44), the equation of state parameter in Eq. (42) can be
expressed as that of the scaling solution as previously mentioned,

$\displaystyle w_{\text{RHDE}}$ $\displaystyle\approx$
$\displaystyle\frac{w_{\text{r}}\Omega_{\text{r}}+w_{\text{m}}\Omega_{\text{m}}}{\Omega_{\text{r}}+\Omega_{\text{m}}}.$
(46)

This scaling behavior is valid in the radiation domination epoch and the
beginning phase of the matter domination epoch when the non-extensive length
scale is sufficiently large. For example, the dynamic of RHDE with
$l_{\lambda}=10^{-1}$ represents as the solid magenta line in Fig. 5. It is
also noticed that the period at which RHDE behaves as the scaling solution
decreases as $l_{\lambda}$ gets smaller. Instead, RHDE with smaller
$l_{\lambda}$ gets closer to the cosmological constant because the magnitude
of $l_{H}$ is not sufficiently small. It means the approximations in Eq. (27)
or Eq. 46 are not applicable in describing the RHDE at the early time. Even
though the RHDE with very small $l_{\lambda}$ can occur, it does not have a
significant effect on the standard cosmic evolution (as seen from
$w_{\text{eff}}$ in Fig. 6). This is because the ratio of RHDE (represented by
$\Omega_{\text{RHDE}}$) is also very small in such a case.

For another asymptotic limit corresponding to Eq. (28), the energy density is
approximately constant at the late time. The equation of state parameter of
RHDE approaches $w_{\text{RHDE}}\to-1$ as seen in Fig. 5. Therefore, this
model of dark energy can drive the Universe at the late time to expand with
the correct rate of acceleration.

##  V Observational Constraints

In this section, we will fit the RHDE model to the low-redshift observational
data sets. As discussed in the previous section, the dynamics of the Universe
can be affected due to the existence of non-extensivity. For example, the left
panel of Fig. 7 shows the Hubble parameter $H$ for low redshift $z=(1-a)/a$.
In the right panel, we plot the apparent magnitude $m_{B}$ which is defined as
$\displaystyle{m_{B}=5\log_{10}\left(\frac{d_{L}}{\text{Mpc}}\right)+25+M}$,
where
$\displaystyle{d_{L}=c(1+z)\int_{0}^{z}\frac{\text{d}z^{\prime}}{H(z^{\prime})}}$
is the luminosity distance and $M$ is the absolute magnitude introduced as a
fitting parameter. It is seen that the modification in cosmic evolution due to
varying $l_{\lambda}$ affects the values of the observables predicted from the
model at each redshift. Accordingly, these plots suggest that the model
parameter $l_{\lambda}$ must be fitted by the observations.

Figure 7: Comparison of observables predicted from RHDE model for various
values of $l_{\lambda}$ to cosmic chronometer data set [83] (left) and
Pantheon+ data set [84] (right)

In this work, the likelihood analysis is performed by using the Markov Chain
Monte Carlo (MCMC) code SimpleMC.666It is publicly available in
https://igomezv.github.io/SimpleMC/index.html To investigate the best-fit
model parameter, the considered observational data sets are listed as follows:

  * •

Cosmic chronometers (CC) with covariance matrix (2020) [83]

  * •

1550 distinct type Ia supernova (SNIa) containing 1701 light curves from
Pantheon+ compilation [84]

  * •

Recent baryon acoustic oscillation (BAO) data from SDSS-IV DR16 [85, 86, 87,
88, 89]

To create samples for MCMC analysis, the Metropolis-Hastings algorithm was
used. Additionally, we assess the quality of fits where an increasing number
of parameters is penalized by using the Akaike Information Criterion (AIC).
With the $k$ parameters and the highest Likelihood $\mathcal{L}_{\rm max}$,
AIC is defined as

$\displaystyle{\rm AIC}=-2\ln\mathcal{L}_{\rm max}+2k.$ (47)

Notice that the last term is introduced in order to penalize and prevent the
over-fitting issue. When comparing two models, one can utilize the disparity
in the estimators to support the models’ statistical compatibility. By
applying Jeffreys’ scale [90], statistical compatibility is indicated by a
value of $\Delta$AIC $\leq 2$, and mild tension between models is indicated by
a criterion of $2<\Delta$AIC $\leq 5$. When $\Delta$AIC $>5$, there is a large
statistically significant tension between the models. To verify convergence,
the Gelman-Rubin diagnostic (GRstop) has been employed.

Unfortunately, it is complicated to do the statistical analysis by keeping
$l_{\lambda}$ as a free parameter. This is because the Hubble parameter cannot
be written as an explicit function of $l_{\lambda}$. We instead choose to
consider the parameter space:
{$\Omega_{\text{m,0}},\Omega_{\text{b,0}},h_{0}=H_{0}/100$} for a given
$l_{\lambda}$, and then evaluate the value of $l_{\lambda}$ which gives the
lowest AIC. The results of AIC with respect to $l_{\lambda}$ are illustrated
in Fig. 8. It is found that the existence of the non-extensive effect is not
preferable by CC data. In other words, RHDE with $l_{\lambda}\to 0$, or
equivalently $\Lambda$CDM is the best-fit model for CC data as seen in the
left-bottom panel of Fig. 8. In contrast, from the result shown the middle-
bottom panel of Fig. 8, SNIa data seems to prefer a large non-extensive length
scale $l_{\lambda}\approx 0.164$. Such a case indeed contradicts the bound
from the BBN constraint. The two data sets, however, give less contribution in
constraining the model parameter. The most significant effect on AIC in
fitting the model comes from BAO data which prefers the intermediate value of
$l_{\lambda}\approx 0.102$ (see the right-bottom panel of Fig. 8). As a
result, in fitting RHDE with all data sets, AIC is minimized as
$\text{AIC}_{\text{min}}\approx 1424.754$ when $l_{\lambda}\approx 0.07015$.

Figure 8:  AIC versus values of $l_{\lambda}$ in fitting RHDE model with all
observational data sets (top), only CC (left-bottom), Pantheon+ (middle-
bottom), and BAO (right-bottom) data sets

The observational analysis for $l_{\lambda}$ giving $\text{AIC}_{\text{min}}$,
the probability posterior distribution of parameters and their central values
are reported in Fig. 9 and the second row of Tab. 1, respectively.

Figure 9: Two-dimensional 68% and 95% CL allowed regions and the probability posterior distributions for model parameters from RHDE model with $l_{\lambda}\approx 0.07015$ (it gives $\text{AIC}_{\text{min}}$) for the parameter space $\\{\Omega_{m,0},\Omega_{b,0},h_{0}\\}$ Table 1: Observational constraints on the parameters of RHDE, AdS-HDE and $\Lambda$CDM using CC, Pantheon+, and BAO data sets Models | $\Omega_{\text{m,0}}$ | $\Omega_{\text{b,0}}h^{2}$ | $h_{0}$ | $l_{\lambda}$ | $\Omega_{R,0}$ | $b^{2}$
---|---|---|---|---|---|---
RHDE with $l_{\lambda,\text{AIC}_{\text{min}}}$ | $\,0.2856^{+0.0130}_{-0.0125}$ |  $0.0221^{+0.0004}_{-0.0004}$ |  $0.6643^{+0.0178}_{-0.0174}$ |  0.07015 | – | 0.0144
AdS-HDE: fix $b^{2}$ | $\,0.3128^{+0.0127}_{-0.0125}$ |  $0.0220^{+0.0004}_{-0.0005}$ |  $0.6978^{+0.0195}_{-0.0188}$ | – |  $0.0036^{+0.0044}_{-0.0029}$ | 1
AdS-HDE: | $\,0.3124^{+0.0136}_{-0.0131}$ |  $0.0220^{+0.0005}_{-0.0005}$ |  $0.6996^{+0.0194}_{-0.0191}$ | – |  $0.0043^{+0.0050}_{-0.0028}$ |  $0.8442^{+0.4069}_{-0.2151}$
$\Lambda$CDM | $0.3121^{+0.0134}_{-0.0130}$ | $0.0220^{+0.0005}_{-0.0005}$ | $0.6970^{+0.0194}_{-0.0188}$ | – | – | –

By comparing to other models, the values of AIC for each model are shown in
Tab. 2.

Table 2: Estimator of each model compared with $\Lambda$CDM Models | AIC |  $\Delta$AIC
---|---|---
RHDE with $l_{\lambda,\text{AIC}_{\text{min}}}$ |  1424.754 |  $-2.632$
AdS-HDE: fixed $b^{2}$ | 1429.912 | 2.526
AdS-HDE | 1431.910 | 4.524
$\Lambda$CDM | 1427.386 | 0

Note that AdS-HDE in Tabs. 1 and 2 represents the model with the
characteristic length scale as the particle horizon [56]. To compare RHDE and
AdS-HDE models, even though both can explain the late-time expansion equally
well, the RHDE is preferred by observations since it has a lower value of AIC.
To compare with $\Lambda$CDM in Tab. 2 the RHDE with fixed $l_{\lambda}$
performs fits with experimental data statistically better than $\Lambda$CDM
with the mild tension. However, the non-extensive length scale $l_{\lambda}$
should be treated as an additional parameter in the parameter space.
Therefore, the value of AIC for RHDE in Tab. 2 should be added by two due to
the increase in the number of parameters in Eq. 47. With this consideration
the difference of their AIC values falls instead in the range
$\Delta\text{AIC}\leq 2$ implying that RHDE and $\Lambda$CDM models are
statistically compatible.

##  VI Conclusion

In this work, the dark energy is defined from the thermodynamic properties of
the Schwarzschild black hole with a non-extensive effect. Such non-extensivity
is analyzed by treating the entropy as the logarithmic function of the area.
This consideration is associated with the Rényi description from a
thermodynamic point of view. The non-extensive effect contributed from Rényi
entropy can significantly stabilize the black hole similar to the effect of
the negative cosmological constant on the anti-de Sitter black hole. It also
provides an alternative aspect in identifying the stable phase of the black
hole by comparing the radius of the black hole to the non-extensive length
scale $L_{\lambda}$. The global stability requires larger $L_{\lambda}$ than
the local one does (see Eqs. (18) and (20)). The non-extensive thermodynamics
will be reduced to the standard GB one by taking this length scale to be
infinity. Hence, the black hole described by the GB statistics is always
unstable. As motivated from the stability of the black hole, the energy
density of the holographic dark energy can be obtained from the change of
enthalpy under the isobaric process.

It is interestingly found that the energy density of RHDE contains the
approximated constant term. This term can be dominant in the late-time
evolution of the Universe. Therefore, it can drive the expansion with the
corrected rate of acceleration in the same way as the cosmological constant in
the $\Lambda$CDM model does. One of the important remarks is that the non-
extensive length scale $l_{\lambda}$ is only one additional parameter to the
standard $\Lambda$CDM model. By fitting the proposed RHDE model to several
observations, we use the likelihood analysis. The lowest AIC is an estimator
in identifying which value of $l_{\lambda}$ is preferred. In the recent
procedure, the parameter space is kept the same as that of the $\Lambda$CDM
model. The result shows that the minimum AIC is obtained when the non-
extensive length scale is $l_{\lambda}\approx 0.07$ (or $L_{\lambda}\approx
0.07H_{0}^{-1}$). It can be concluded that the RHDE and $\Lambda$CDM models
are not different in the quality of fit of the observational data sets from a
statistical point of view. Moreover, both of them are more favorable than the
AdS-HDE model with mind tension.

It is worthwhile to note that there are two length scales $H^{-1}$ and
$L_{\lambda}$ which are dynamical and non-dynamical, respectively. One could
interpret that the behavior of the Universe is explained by comparing the
aforementioned length scales. The RHDE behaves similarly to the scaling
solution as found in the standard model of the holographic dark energy [17,
18] when $H^{-1}\ll L_{\lambda}$. When the Universe is sufficiently large
corresponding to the magnitudes of $H^{-1}$ and $L_{\lambda}$ are in the same
order, the non-extensivity due to the Rényi entropy is active, making the RHDE
evolves differently. Eventually, the RHDE behaves similarly to the dark energy
in $\Lambda$CDM model when $H^{-1}\gg L_{\lambda}$. In other words, the RHDE
cannot make the Universe expand with acceleration if the size of the Universe
is too small compared to the non-extensive length. After that, when the
Universe evolves until its size is significantly large, i.e., $H^{-1}\gg
L_{\lambda}$, the RHDE can drive the agreeing late-time evolution.

By interpreting that the RHDE behaves as a thermal system, there are
conditions in which this system is thermodynamic stable as
$H^{-1}/L_{\lambda}\gtrapprox 1.98$. According to the numerical results for
the evolution of the ratio $l_{H}/l_{\lambda}$ in the left panel of Fig. 3,
the Universe in early time ($\ln a\lessapprox-2$) corresponds to the unstable
phase while the Universe will undergo to the stable phase as time evolves.
From the numerical solving of the evolution of the ratio $l_{H}/l_{\lambda}$
with $l_{\lambda,\text{AIC}_{\text{min}}}$, the condition for the Universe
being in the stable phase is $\ln a\gtrapprox-1.69$. The lower bound of the
aforementioned period is around the end of the matter-dominated epoch. It is
possible to interpret that the Universe evolves in such a way that the thermal
system undergoes from unstable to stable phases. Moreover, from the fact that
the logarithmic function in Eq. (7) is the monotonically increasing function,
an examination of the evolution of the IR length scale (i.e. Hubble radius,
for our cosmological model) shown in the left panel of Fig. 3 remarkably
implies an increase in the Rényi entropy over time. The second law of Rényi
black hole thermodynamics is therefore satisfied in this aspect. As a result,
this work may shed light on the interplay between the non-extensive nature of
the black hole in the description of thermodynamics and the evolution of the
Universe in the description of classical gravity.

Throughout our consideration, we have focused on the HDE model constructed
only from the simple Schwarzschild black hole. It may be worthwhile to extend
this study to other types of black holes. With the complicated phase
structures of those black holes, quaint implications for cosmic evolution are
possibly obtained. These investigations are left as the future works.

## Acknowledgement

This research has received funding support from the NSRF via the Program
Management Unit for Human Resources & Institutional Development, Research and
Innovation [grant number B13F660066]. PW, LT, CP and DS are supported by
Thailand NSRF via PMU-B grant number B37G660013. CP is also supported by
Fundamental Fund 2566 of Khon Kaen University and Research Grant for New
Scholar, Office of the Permanent Secretary, Ministry of Higher Education,
Science, Research and Innovation under contract no. RGNS 64-043.

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