Advances in High Energy Physics

Volume 2018, Article ID 6139430, 12 pages

https://doi.org/10.1155/2018/6139430

## Thermodynamics of Ricci-Gauss-Bonnet Dark Energy

Correspondence should be addressed to Abdul Jawad; moc.oohay@181badawaj

Received 24 November 2017; Revised 10 January 2018; Accepted 21 January 2018; Published 4 March 2018

Academic Editor: Chao-Qiang Geng

Copyright © 2018 Ayesha Iqbal and Abdul Jawad. 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

We investigate the validity of generalized second law of thermodynamics of a physical system comprising newly proposed dark energy model called Ricci-Gauss-Bonnet and cold dark matter enveloped by apparent horizon and event horizon in flat Friedmann-Robertson-Walker (FRW) universe. For this purpose, Bekenstein entropy, Renyi entropy, logarithmic entropy, and power law entropic corrections are used. It is found that this law exhibits the validity on both apparent and event horizons except for the case of logarithmic entropic correction at apparent horizon. Also, we check the thermodynamical equilibrium condition for all cases of entropy and found its vitality in all cases of entropy.

#### 1. Introduction

The revelation of black holes thermodynamics motivated the physicist to examine the thermodynamics of cosmological models in accelerated expanding universe [1–3]. Bekenstein and Hawking determined that the entropy of black hole is proportional to its event horizon [4, 5] which leads to important law named generalized second law of thermodynamics (GSLT) for black hole physics. This law can be defined as the entropy of black hole and its exterior is always increasing. The primitive level of thermodynamics properties of horizons is exhibited by considering Einstein field equations as an alternate of first law of thermodynamics [6, 7]. Gibbons and Hawking developed the Beckenstein’s idea for cosmological system by exhibiting that the entropy of cosmological event horizon is proportional to horizon area [8]. They represented the equality of apparent horizon and event horizon for de Sitter universe. The validity of GSLT was deeply studied later [9–11]. GSLT in cosmological scenario implies that the rate of change of entropy of horizon along with that of fluid inside it will always be greater than or equal to zero. Its mathematical expression is

In addition, the holographic dark energy (HDE) is an interesting effort in exploring the nature of dark energy in the framework of quantum gravity. This model is motivated from the fundamental holographic principle that arises from black hole thermodynamics and string theory [12–15]. HDE fascinated a large amount of research despite some objections [16, 17]. The choice of the length scale appearing in the holographic dark energy density gives rise to different dark energy models. One of the crucial models is holographic Ricci dark energy model which is developed by assuming IR length scale as the average radius of Ricci scalar curvature, [18–20]. Moreover, its modified form is also presented and discussed widely [21–23].

Further, Wang et al. [24] observed that GSLT is verified at apparent horizon but not at event horizon for a specific model of dark energy. In case of new holographic dark energy, GSLT is valid fully on apparent horizon but partially on event horizon of universe [25]. The breakdown of GSLT was argued in case of event horizon enveloping the universe as compared to apparent horizon [26]. Setare [27] has derived the constraints on deceleration parameter in order to fulfill GSLT in case of nonflat universe enveloped by event horizon. The GSLT of thermodynamics has also been analyzed in case of Braneworld [28, 29] and generally Levelock gravity [30].

Moreover, modified matter part of Einstein Hilbert action results in dynamical models such as cosmological constants, quintessence, -essence, Chaplygin gas, and holographic dark energy (HDE) models [31–39]. Moreover, several modified theories of gravity are , [40–42], [43, 44], [45–47], [48–53], and [54, 55] (where is the curvature scalar, denotes the torsion scalar, is the trace of the energy momentum tensor, and is the invariant of Gauss-Bonnet defined as ). For clear review of DE models and modified theories of gravity, see [39]. Some authors [56–66] have also discussed various DE models in different frameworks and found interesting results.

Recently, Saridakis [67] proposed Ricci-Gauss-Bonnet holographic dark energy in which Infrared cutoff consists of both Ricci scalar and the Gauss-Bonnet invariant. Such a construction has the significant advantage that the Infrared cutoff and consequently the HDE density do not depend on the future or the past evolution of the universe, but only on its current features, and moreover it is determined by invariants, whose role is fundamental in gravitational theories. This model has IR cutoff form as where and are model parameters. In flat FRW geometry, the Ricci scalar and the Gauss-Bonnet invariant () are given as and , respectively [67].

In the present work, we examine the validity of GSLT by assuming various forms of entropy on apparent and event horizons. We have also examined whether each entropy attain maximum (thermodynamic equilibrium) by satisfying the condition . The plan of the paper is as follows. In Sections 2 and 3, we have examined the validity of GSLT as well as thermal equilibrium condition at apparent and event horizons, respectively. The results are summarized in the last section.

#### 2. Generalized Second Law of Thermodynamics at Apparent Horizon

According to GSLT, the entropy of horizon and entropy of matter resources inside horizon does not decrease with respect to time. Following (1), we can writeHere gives entropy of horizon and entropy of matter inside horizon is represented by . Now considering spatially flat FRW universe, the first Friedmann equation isHere and are effective density and pressure, respectively. We have made the following two assumptions: (i) an entropy is associated with the horizon in addition to the entropy of the universe inside the horizon and (ii) according to the local equilibrium hypothesis, there is no spontaneous exchange of energy between the horizon and fluid inside. Moreover, Gibb’s equation can be written asHere , , and which modified the above equation as follows:For flat FRW universe, the Hubble horizon can be defined asBy utilizing the above horizon, (for cold dark matter ) and in (5), we can getFrom conservation equation, one can obtainSubstituting the value of in (7), we getMoreover, Ricci-Gauss Bonnet dark energy can be defined as follows [67]:Here and are the model parameters. Standard Ricci dark energy can be obtained by substituting and yields a pure Gauss-Bonnet HDE. The density parameters can be introduced asAccording to first Friedman equation, we can obtainAlso, can be evaluated by using conservation equation as follows:with . By using this value of , takes the following form:Using (12) and (14), we can find asDifferentiating , we obtainwhere prime denotes the differentiation with respect to . Also, differentiation of with respect to leads toWe get the following value of by differentiating (11):Now, takes the formAlso, Friedman first equation gives and hence we can writeBy inserting (6) in above equation, we haveBy using value of from (19), we get Next, we will discuss the various expressions of entropy-area relations in order analyze the validity of GSLT on Hubble horizon.

##### 2.1. Bekenstein Entropy

The Bekenstein entropy is given byBy using , and being the area of horizon, we getBy using the expressions of and , we haveEquations (22) and (25) join to form where represents the total entropy; that is, .

Now, we assume the power law form of scale factor; that is, , where and appear as constant parameters. Under this assumption, the values of and turn out to be , respectively. In this way, reduces to where . In order to analyze the clear picture of validity of GSLT for this entropy on the Hubble horizon, we plot against cosmic time by fixing constant parameters as , , and as shown in Figure 1. This shows that remains positive with increasing value of which confirms the validity of GSLT at apparent horizon with Bekenstein entropy.