Advances in High Energy Physics

Volume 2019, Article ID 1287932, 9 pages

https://doi.org/10.1155/2019/1287932

## Cosmological Implications of the Generalized Entropy Based Holographic Dark Energy Models in Dynamical Chern-Simons Modified Gravity

^{1}Department of Mathematics, COMSATS University Islamabad, Lahore Campus-54000, Pakistan^{2}Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), P.O. Box 55134-441, Maragha, Iran

Correspondence should be addressed to Abdul Jawad; kp.ude.erohaliuc@dawajludba

Received 28 August 2018; Revised 8 November 2018; Accepted 5 December 2018; Published 8 January 2019

Academic Editor: Chao-Qiang Geng

Copyright © 2019 M. Younas et al. 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. The publication of this article was funded by SCOAP^{3}.

#### Abstract

Recently, Tsallis, Rényi, and Sharma-Mittal entropies have widely been used to study the gravitational and cosmological setups. We consider a flat FRW universe with linear interaction between dark energy and dark matter. We discuss the dark energy models using Tsallis, Rényi, and Sharma-Mittal entropies in the framework of Chern-Simons modified gravity. We explore various cosmological parameters (equation of state parameter, squared sound of speed ) and cosmological plane (, where is the evolutionary equation of state parameter). It is observed that the equation of state parameter gives quintessence-like nature of the universe in most of the cases. Also, the squared speed of sound shows stability of Tsallis and Rényi dark energy model but unstable behavior for Sharma-Mittal dark energy model. The plane represents the thawing region for all dark energy models.

#### 1. Introduction

In the last few years, remarkable progress has been achieved in the understanding of the universe expansion. It has been approved by current observational data that the universe undergoes an accelerated expansion. The observations of type Ia Supernovae (SNeIa)[1–4], large scale structure (LSS) [5–8], and Cosmic Microwave Background Radiation (CMBR) [9, 10] determined that the expansion of the universe is currently accelerating. There is also consensus that this acceleration is generally believed to be caused by a mysterious form of energy or exotic matter with negative pressure so called dark energy (DE) [11–21].

The discovery of accelerating expansion of the universe is a milestone for cosmology. It is considered that of our universe is composed of two components, that is DE and dark matter [16]. The dark matter constitutes about of the total energy density of the universe. The existence of the universe is proved by astrophysical observation but the nature of dark matter is still unknown. Mainly the DE is also a curious component of our universe. It is responsible for current accelerating universe and DE is entirely different from baryonic matter. DE constitutes almost of the total energy density of our universe.

In order to describe the accelerated expansion phenomenon, two different approaches have been adopted. One is the proposal of various dynamical DE models such as family of Chaplygin gas, holographic dark energy, quintessence, K-essence, and ghost [16]. A second approach for understanding this strange component of the universe is modifying the standard theories of gravity, namely, general relativity (GR). Several modified theories of gravity are [17], [18], and [19], where is the curvature scalar, denotes the torsion scalar, is the trace of the energy-momentum tensor, and is the invariant of Gauss-Bonnet.

Holographic DE (HDE) model is favorable technique to solve DE mystery which has attracted much attention and is based upon the holographic principle that states the number of degrees of freedom of a system scales with its area instead of its volume. In fact, HDE relates the energy density of quantum fields in vacuum (as the DE candidate) to the infrared and ultraviolet cut-offs. In addition, HDE is an interesting effort in exploring the nature of DE in the framework of quantum gravity. Cohen et al. [22] reported that the construction of HDE density is based on the relation with the vacuum energy of the system whose maximum amount should not exceed the black hole mass. Cosmological consequences of some HDE models in the dynamical Chern-Simons framework, as a modified gravity theory, can be found in [23].

By considering the long term gravity with the nature of spacetime, different entropy formalism has been used to observe the gravitational and cosmological effects [24–29]. The HDE models such as Tsallis HDE (THDE) [27], Rényi HDE model (RHDE) [28], and Sharma-Mittal HDE (SMHDE) [29] have been recently proposed. In the standard cosmology framework and from the classical stability view of point, while THDE is not stable [27], RHDE is stable during the cosmic evolution [28] and SMHDE is stable only whenever it becomes dominant in the world [29]. In the present work, we use the Tsallis, Sharma-Mittal, and Rényi entropies in the framework of dynamical Chern-Simons modified gravity and consider an interaction term. We investigate the different cosmological parameters such as equation of state parameter and the cosmological plane where shows the evaluation with respect to . We also investigate the squared sound speed of the HDE model to check the stability and the graphical approach.

This paper is organized as follows. In Section 2, we provide the basics of Chern-Simons modified gravity. In Section 3, we observe the equation of state parameter (EoS), cosmological plane, and squared sound speed for THDE model. Sections 4 and 5 are devoted to finding the cosmological parameter, cosmological plane, and squared of sound speed for RHDE and SMHDE models, respectively. In the last section, we conclude the results.

#### 2. Dynamical Chern-Simons Modified Gravity

In this section, we give a review of dynamical Chern-Simons modified gravity. The action which describes the Chern-Simons modified gravity is given aswhere represents the Ricci scalar, is a topological invariant called the Pontryagin term, is a coupling constant, shows the dynamical variable, represents the action of matter, and is the potential term. In the case of string theory, we use . By varying the action equation with respect to and the scalar field , we get the following field equations:Here, and are Einstein tensor and Cotton tensor, respectively. The Cotton tensor is defined asThe energy-momentum tensor is given bywhere shows the matter contribution and represents the scalar field contribution, while and represent the pressure and energy density, respectively. Furthermore, is the four velocity. In the framework of Chern-Simons gravity, we get the following Friedmann equation:where is the Hubble parameter and the dot represents the derivative of with respect to and . For FRW spacetime, the pony trying term vanishes identically; therefore, the scalar field in (2) takes the following form: We set and get the following equation:which implies that , is a constant of integration. Using this result in (5), we have

We consider the interacting scenario between DE and dark matter and thus equation of continuity turns to the following equations:Here, is the energy density of the DE, is the energy density of the pressureless matter, and is the interaction term. Basically, represents the rate of energy exchange between DE and dark matter. If , it shows that energy is being transferred from DE to the dark matter. For , the energy is being transferred from dark matter to the DE. We consider a specific form of interaction which is defined as and is interacting parameter which shows the energy transfers between CDM and DE. If we take , then it shows that each component, that is, the nonrelativistic matter and DE, is self-conserved. Using the value of in (9) we havewhere is an integration constant. Hence, (10) finally leads to the expression for pressure as follows:

The ** EoS parameter** is used to categorize the decelerated and accelerated phases of the universe. This parameter is defined asIf we take , it corresponds to nonrelativistic matter and the decelerated phase of the universe involves radiation era . , and correspond to the cosmological constant, quintessence, and phantom eras respectively. To analyze the dynamical properties of the DE models, we use [30]. This plane describes the evolutionary universe with two different cases, freezing region and thawing region. In the freezing region the values of EoS parameter and evolutionary parameter are negative ( and ), while for the thawing region, the value of EoS parameter is negative and evolutionary parameter is positive ( and ). In order to check the stability of the DE models, we need to evaluate the

**which is given byThe sign of decides its stability of DE models, when , the model is stable; otherwise, it is unstable.**

__squared sound speed__#### 3. Tsallis Holographic Dark Energy

The definition and derivation of standard HDE density are given by , where represents reduced Plank mass and denotes the infrared cut-off. It depends upon the entropy area relationship of black holes, i.e., , where represents the area of the horizon. Tsallis and Cirto [31] showed that the horizon entropy of the black hole can be modified aswhere is the nonadditivity parameter and is an unknown constant [31]. Cohen at al. [22] proposed the mutual relationship between IR (L) cut-off, system entropy (S), and UV cut-off asAfter combining (15) and (16), we get the following relation:where is vacuum energy density and . So, the Tsallis HDE density [29] is given asHere, is an unknown parameter and IR cut-off is taken as Hubble radius which leads to , where is Hubble parameter. The density of Tsallis HDE model along with its derivative by using (18) becomesHere, is the derivative of Hubble parameter w.r.t. . The value of is calculated in terms of using which is given as follows.Inserting these values in (12) yields The EoS is obtained from (13):The plot of versus is shown in Figure 1. In this parameter and further results, the function is being utilized numerically. The other constant parameters are mentioned in Figure 1. The trajectory of EoS parameter remains in quintessence region at early, present, and latter epoch.