Advances in Astronomy

Volume 2019, Article ID 8138067, 7 pages

https://doi.org/10.1155/2019/8138067

## Tsallis Holographic Dark Energy with Granda-Oliveros Scale in ()-Dimensional FRW Universe

Physics Department, Faculty of Science, Damanhour University, Damanhour, Egypt

Correspondence should be addressed to Ayman A. Aly; hc.nrec@yla.namya

Received 22 October 2018; Accepted 27 December 2018; Published 10 January 2019

Guest Editor: Pedro H. R. S. Moraes

Copyright © 2019 Ayman A. Aly. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Based on Tsallis holographic dark energy model recently proposed by using the general model of the Tsallis entropy expression, we reconstruct cosmographic parameters, , and we study their evolution in spatially flat ()-dimensional Friedmann-Robertson-Walker universe using Granda-Oliveros scale. Our results show that the universe is in an accelerating expansion mode described by phantom-like behavior. We go further and study the state finder operators and the diagnostic to understand the behavior of our model. The stability of the system is also studied by using the square of speed of sound showing that our model is stable over the low range of red-shift considered. The results indicate that the entropy formalism will play an important role in understanding the dynamics of our universe.

#### 1. Introduction

The study of accelerated expansion of the universe becomes one of the main hot topics in cosmology in the last few years [1–3]. Many theories are considered to explain this behavior. Actually, scientists believe that this accelerating behavior is mainly due to some repulsive gravity at large scale which is caused by a nonstandard component with negative pressure representing around two-thirds of universe known as dark energy . This accelerating behavior is characterized by equation of state (EoS) parameter which lies in a narrow region around the value EoS = -1. The region corresponding to EoS parameter -1 belongs to what is known as phantom dark energy (PDE) with an energy having positive energy density, but negative pressure. The PDE is theoretically proposed by some models like braneworlds models or scalar-tensor gravity model [4–11].

Recent observations by Planck 2015, Baryon Acoustic Oscillations (BAO), Supernova Type Ia (SNIa), Large Scale Galaxy Surveys (LSS), and Weak Lensing (WL), show that is the best candidate to explain the present acceleration of the universe [1–3, 12].

An interesting work by M. Li [13] assumed a new dark energy model called holographic dark energy model (HDE) based on what is called holographic principle. In this model the future event horizon of the universe can be used as infrared cutoff. Actually, This model gives a good explanation for accelerated expansion nature of the universe and that fits the current observations. Tsallis and Cirto assumed some quantum modification for HDE by assuming that the black hole horizon entropy could be given by [14]where is an unknown constant and denotes the nonadditivity parameter chosen to have a positive value. By choosing and the Bekenstein entropy is easily recovered [14].

The holographic principle, which states that the number of degrees of freedom of a physical system should be scaled with its bounding area rather than with its volume [13], should be constrained by an infrared cutoff. Cohen et al. [15] proposed a relation between the system entropy and the IR and UV () cutoffs as [16]After combining (1) with (3), one findswhere stands for the energy density of vacuum which is denoted as , the energy density of dark energy. Using this inequality, a new Tsallis holographic dark energy density (THDE) is established as [14]

where is an unknown parameter [14]. In this work we will use the IR cutoff as Granda-Oliveros () scale which has the following form [17]:where and are two positive constant parameters and H is the Hubble parameter. By combining (4) with (23) one can write THDE in scale as

For flat case, which we are going to consider in this study, the best estimated values for and are and as mentioned in [18]. Actually, dark energy theories use Granda-Oliveros scale as IR scale depending only on local quantities; that way it is possible to avoid the causality problem; moreover it is also possible to obtain the accelerated expansion mode of the universe [17].

In this work we will study the Tsallis holographic together with (n+1)-dimensional universe to explore the nature of our universe.

This paper is organized as follows: In the next section the cosmological model is considered. In Section 2 cosmography is discussed. In Section 3, we study statefinder operators of our model. In Section 4 reconstructed cosmological parameters are deduced and then the stability of our model is assumed through the study of the square of speed of sound. Finally, the conclusion is presented.

#### 2. The Cosmological Model

In this section, we are going to give a brief review about -dimensional FRW universe with Granda-Oliveros cutoff. The metric of -dimensional Lorentzian isotropic and homogeneous space-time is described by [19]where , is the cosmic time and is the metric of an -dimensional manifold with curvature for open, flat, and closed universe. Since is an -hyperboloid, then the flat space has metric on the form [19]where is the radial variable and is assumed to have a positive value and is the metric of -dimensional sphere. Inserting (7) and (8) into the Einstein field equations: where is the Einstein tensor, the universal gravitational constant, the cosmological constant, and the energy momentum tensor of an ideal fluid, we deducedandwhich is the Friedmann equation in (n+1)-dimensional universe. Here, the Hubble parameter is represented by , the dot represents the first derivative with respect to cosmic time , and the total energy density of the universe , and are the energy densities of and , respectively. Assuming a flat FRW universe , for DE-dominated universe filled with THDE, one can write after some algebraic stepsand this is the mathematical expression for the Hubble parameter for our model.

In all the analyses below we chose (red, blue, and black lines, respectively), also we consider the approximation and , present value of fractional is [14], and present value of Hubble is [20].

#### 3. Cosmography

To get more information about the universe evolution we study the higher order time derivative of the scale factor called cosmographic parameters in framework of -dimensional FRW universe. In a cosmological terms the first five time derivatives of the scale factor are used to define Hubble, deceleration , jerk , snap , and lerk parameters, respectively, as [21] Hubble parameter has a dimension of the inverse of time, while the rest of the cosmographic parameters are dimensionless parameters. By integrating (12) with respect to the cosmic time, one can write the cosmic scale factor as

From observational and theoretical views the deceleration parameter is main indicator about the expansion or contraction of universe according to its sign. The third order derivative, the jerk parameter , represents the time variation of deceleration parameter. Since is observed, is used to predict the future of the universe. Actually the higher order derivatives like jerk and snap factors are mainly used to study the dynamics of the universe, because they could be related to the emergence of the sudden future singularities [21].

By using the relation between the cosmic time and the red-shift [22],Figure 1 represents the evolution of , , and against the red-shift. In Figure 1(a), we notice that and that agrees with the observation that constrains the necessary condition for continuous expansion as . In Figure 1(b), we notice that the jerk parameter at shows a decreasing behavior until approximately and then tends to have a constant value indicating that the rate of change of deceleration is constant, which is away from that is corresponding to ; this may indicates that our model is not a . In Figures 1(c) and 1(d) the snap and lerk factors fluctuate around the zero showing some discontinuity in behavior around and , respectively.