Thermodynamics of Ricci-Gauss-Bonnet Dark Energy

We investigate the validity of generalized second law of thermodynamics of a physical system comprising of 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, logarithmic 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.


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 as 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 are exhibited by considering Einstein field equations as alternate of first law of thermodynamics [6,7]. Gibbons and Hawking developed the Bekenstein'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 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 of some objections [16,17]. The choice of the length scale L appearing in the holographic dark energy density ρ de = 3M pl L −2 gives rise to different dark energy models. One of the its crucial model is holographic Ricci dark energy model which is developed by assuming IR length scale as the average radius of Ricci scalar curvature, R −1/2 [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 non-flat 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 dynamical models such as cosmological constants, quintessence, k-essence, Chaplygin gas and holographic dark energy (HDE) models [31]- [37]. Moreover, several modified theories of gravity are f (R), f (T ) [38], f (R, T ) [39], f (G) [40], f (T, T G ) [41], f (T, T) [42,43] (where R is the curvature scalar, T denotes the torsion scalar, T is the trace of the energy momentum tensor and G is the invariant of Gauss-Bonnet defined as G = R 2 − 4R µν R µν + R µνλσ R µνλσ ). For clear review of DE models and modified theories of gravity, see the reference [37]. Some authors [44]- [47] have also discussed various DE models in different frameworks and found interesting results.
Recently, Saridakis [48] Ricci-Gauss-Bonnet holographic dark energy in which Infrared cutoff is determined by both Ricci scalar and the Gauss-Bonnet invariant. Such a construction has the significant advantage that the Infrared cutoff, and consequently the HDE density, does 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 1 L 2 = −αR + β |G| where α and β are model parameters. In flat FRW geometry, the Ricci scalar (R) and the Gauss-Bonnet invariant (G) are given as R = −6(2H 2 +Ḣ) and G = 24H 2 (H 2 +Ḣ), respectively [48].
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 that wether each entropy attain maximum (thermodynamic equilibrium) by satisfying the conditionS tot < 0. 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.

Generalized Second Law of Thermodynamics at Apparent Horizon
According to GSLT, the entropy of horizon and entropy of matter resources inside horizen does not decrease with respect to time. Following Eq.(1), we can writeṠ HereṠ h gives entropy of horizon and entropy of matter inside horizon is represented byṠ in . Now considering spatially flat FRW universe, the first Friedmann equation is Here ρ ef f and P ef f 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, (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 as Here For flat FRW universe, the Hubble horizon can be defined as By utilizing above horizon, P ef f = p de (for cold dark matter p m = 0) and ρ ef f = ρ d + ρ m in Eq.(5), we can geṫ From conservation equation, one can obtaiṅ Substituting the value of ω de in Eq.(7), we geṫ Moreover, Ricci-Gauss Bonnet dark energy can be defined as follows [48] Here α and β are the model parameters. Standard Ricci dark energy can be obtained by substituting β = 0 and α = 0 yields a pure Gauss-Bonnet HDE. The density parameters can be introduced as According to first Friedman equation, we can obtain Also, ρ m can be evaluated by using conservation equation as follows with ρ m 0 = 3H 2 0 Ω m 0 . By using this value of ρ m , Ω m takes the following form Using Eqs. (12) and (14), we can find H as Differentiating H, we obtaiṅ where prime denotes the differentiation with respect to x = lna. Also, differentiation of R A with respect to t leads tȯ We get the following value ofΏ de by differentiating Eq.(11) Now,Ṙ h takes the forṁ Also, Friedman first equation gives ρ m = 3H 2 − ρ d and hence we can writė By inserting Eq.(6) in above equation, we havė By using value ofṘ h from Eq. (19), we geṫ Next, we will discuss the various expressions of entropy-area relations in order analyze the validity of GSLT on Hubble horizon.

Bekenstein Entropy
The Bekenstein entropy is given by By using G = 1, c = 1 and A = 4πR 2 h being the area of horizon, we get By using the expressions of R h andṘ h , we havė Equations (22) and (25) join to forṁ whereṠ tot represents the total entropy, i.e.,Ṡ tot =Ṡ in +Ṡ h . Now, we assume the power law form of scale factor, i.e., a = a 0 t n , where n and a 0 appear as constant parameters. Under this assumption, the values of H and R h turns out to be n t , t n respectively. In this way,Ṡ tot reduces tȯ In order to analyze the clear picture of validity of GSLT for this entropy on the Hubble horizon, we plotṠ tot against cosmic time (t) by fixing constant parameters as α = 0.2, β = 0.001 and n = 4 as shown in Figure 1. This shows thatṠ tot remains positive with increasing value of t which confirms the validity of GSLT at apparent horizon with Bekenstein entropy.
To examine the thermodynamic equilibrium, we differentiate Eq. (27) to getS tot given beloẅ We plot X =S tot versus n in Figure 2 which shows thatS tot < 0 for the selected range of n. Hence, thermal equilibrium condition is satisfied for Bekenstein entropy at apparent horizon.

Logarithmic corrections to entropy
Logarithmic corrections arises from loop quantum gravity due to thermal equilibrium and quantum fluctuations [49]- [55]. The entropy on apparent horizon can be defined as follows here η, ξ and γ are dimensionless constants. Differentiating with respect to t, we getṠ which takes the following form by inserting value ofṘ h from Eq.(19) In the presence of logarithmic entropy,Ṡ tot can be obtained by using Eq. (22) and Eq.(31) By substituting value of scale factor, the above equation reduces tȯ Differentiating above equation, we geẗ Figure 3 presents the plot ofṠ tot at apparent horizon by taking logarithmic  entropy at apparent horizon. Here we have taken η = 3.8 and ξ = 3 along with same values of α, β and n as in above mentioned case. HereṠ tot remains negative for t < 1.5 while it moves in positive direction t ≥ 1.5. Hence, validity of GSLT is verified for t ≥ 1.5 at apparent horizon with logarithmic entropy. Figure 4 shows that X =S tot < 0 with increasing value of t and n = 1.5. Hence, for logarithmic entropy at apparent horizon, the condition of thermal equilibrium is satisfied.

Renyi Entropy
it behaves as Bekenstein entropy for λ = 0. Differentiating with respect to t, we obtainṠ Using Eq. (19) in above equation, we geṫ Combining Eqs. (22)and (37) to geṫ For power law scale factor, we obtaiṅ The plot ofṠ tot by taking Renyi entropy at apparent horizon is presented by Figure 5. Here α, β and n has same values like previous case and λ = 1.5. In this case,Ṡ tot behaves positively with the passage of time which verifies the validity of GSLT for the present case. Further, differentiating above equation, we geẗ The plot of this expression is shown in Figure 6 which shows thatS tot < 0 for n = 1.5 with the passage of time. Hence, the condition for thermal equilibrium is satisfied in case of Renyi entropy at apparent horizon.

Power Law Entropic correction
The power law corrections to entropy appear in dealing with entanglement of quantum fields in and out of the horizon [58]. The corrected entropy takes the form [59] with k µ = µ(4π) , r c is crossover length and µ appears as a constant.

Utilization of Eq.(19) in above equation leads tȯ
Joining Eqs. (22)and (43) to obtaiṅ In the presence of scale factor, the above expression turns out to bė By taking power Law entropy at apparent horizon,Ṡ tot is plotted at apparent horizon as shown in Figure 7. With same values for α, β and n, we have taken µ = 5 and r c = 2. Here the effectiveness of GSLT at apparent horizon is certified by positive moves ofṠ tot with increasing t. Differentiating above equation, we geẗ Just like above mentioned three cases, in case of power law entropy at apparent horizon, the condition for thermal equilibrium is satisfied with the passage of cosmic time as shown in Figure 8.  R h = HR h − 1. The temperature we used in this section is T = bH 2π , where b is a constant. For the present case, rewriting Eq.(4) by using value of T anḋ R h , we have following equation for entropy inside horizoṅ

Bekenstein Entropy
Under this scenario, Eq.(24) can be written aṡ The equation forṠ tot can be obtained by using Eqs. (43) and (48) as followṡ By putting values of scale factor and R h in above equation, we havė Differentiating above equation with respect to t, we geṫ   Figure 10 shows thatS tot < 0 for increasing values of n. Hence, at event horizon, the Bekenstein entropy fulfilled the condition of thermodynamic equilibrium.

Logarithmic Entropy
For this entropy at event horizon, Eq.(29) leads tȯ By using Eqs. (47) and (52), the expression ofṠ tot can be written aṡ The following equation is obtained by using values of scale factor and R ḣ Differentiating above equation with respect to t, we obtain   Figure 12: Plot of X =S tot by taking Logarithmic entropy as entropy at event horizon, where time is measured in second. Figure 11 presents the plot ofṠ tot by taking logarithmic entropy at event horizon. Here we have taken η = 4 and ξ = 6 along with same values of α, β and n as in above mentioned case. Clearly,Ṡ tot moves in positive direction as value of t increases. The validity of GSLT is verified at event horizon in the presence of logarithmic entropy. From Figure 12, we can see thatS tot < 0 for n = 1.5. Hence, for this case, thermodynamic equilibrium condition holds.

Renyi Entropy
The following form is obtained from Eq.(34), by substituting value forṘ ḣ Joining Eqs. (47) and (56), we geṫ By using values of scale factor and R h , above equation reduces tȯ Differentiating above equation with respect to t, we geẗ The plot ofṠ tot for Renyi entropy at event horizon is presented in Figure 13. Here α, β and n has same values like previous case while λ = 1.5. In this case, S tot behaves positively with the passage of time which verifies the validity of GSLT. Figure 14 shows that the trajectories ofS tot remains negative for increasing of t with n = 1.5. This means that the present scenario obeys the condition for thermodynamic equilibrium.

Power law Entropy
Under conditions of present section, Eq.(41) reduces tȯ Joining Eqs. (47) and (60) to get the following equatioṅ Inserting conditions for scale factor and R h in above equation, we geṫ (62) The plot of this expression is displayed in Figure 15 with same values for α, β and n while µ = 5 and r c = 2. Here the effectiveness of GSLT at event horizon is certified by positive moves ofṠ tot with increasing t. Differentiating with respect to t, we obtain (63) Figure 16 shows that the present scenario fulfils the thermodynamic equilibrium condition for power law entropy at event horizon.

Conclusion
The concept of thermodynamics in cosmological system originates through black hole physics. It was suggested [60] that the temperature of Hawking radiations emitting from black holes is proportional to their corresponding surface gravity on the event horizon. Jacobson [61] found a relation between thermodynamics and the Einstein field equations. He derived this relation on the basis of entropy-horizon area proportionality relation along with first law of thermodynamics (also called Clausius relation) dQ = T dS, where dQ, T and dS indicate the exchange in energy, temperature and entropy change for a given system. It was shown that the field equations for any spherically symmetric spacetime can be expressed as T dS = dE+P dV (E, P and V represent the internal energy, pressure and volume of the spherical system) for any horizon [62]. By utilizing this relation, GSLT has been studied extensively in the scenario of expanding behavior of the universe. In order to discuss GSLT, horizon entropy of the universe can be taken as one quarter of its horizon area [63] or power law corrected [64] or logarithmic corrected [65] forms. Many people have explored the validity of GSLT of different systems including interaction of two fluid components like DE and dark matter [66], as well as interaction of three components of fluid [67] in the FRW universe by using simple horizon entropy of the universe. The thermodynamical analysis widely performed in modified theories of gravity [68]. Motivated by above mentioned works, we have considered a newly proposed DE model named as Ricci Gauss-Bonnet DE in flat FRW universe. We have developed thermodynamical quantities and analyzed the validity of GSLT and thermodynamic equilibrium. For dense elaboration of thermodynamics of present DE model, we have assumed various entropy corrections such as Bekenstein entropy, Logarithmic corrected entropy, Renyi entropy and power law entropy at apparent horizon as well as event horizon of the universe. We have found that GSLT holds for all cases of entropies as well as horizons. Also, thermal equilibrium condition satisfied under certain conditions on constant parameters. The detailed of results are as follows:

On Apparent Horizon
By utilizing usual entropy, GSLT on the apparent horizon has shown in Figure 1 which shows thatṠ tot remains positive with increasing value of t and confirms its validity. Figure 2 has also indicated that thermal equilibrium condition is satisfied for Bekenstein entropy at apparent horizon. For logarithmic corrected entropy, GSLT on apparent horizon has displayed in Figure The plot ofṠ tot by taking Renyi entropy at apparent horizon has displayed in Figure 5 which behaves positively with the passage of time and exhibits the validity of GSLT. Also, for this entropy, the condition for thermal equilibrium has been satisfied in case of Renyi entropy at apparent horizon ( Figure 6). By taking power Law entropy at apparent horizon,Ṡ tot is plotted at apparent horizon as shown in Figure 7. Here the effectiveness of GSLT at apparent horizon is certified by positive moves ofṠ tot with increasing t. Just like above mentioned three cases, in case of power law entropy at apparent horizon, the condition for thermal equilibrium is satisfied with the passage of cosmic time as shown in Figure 8.

On Event Horizon
It has been observed from Figure 9 that GSLT remains valid at event horizon with Bekenstein entropy. Also, at event horizon, the Bekenstein entropy fulfilled the condition of thermodynamic equilibrium ( Figure 10). The validity of GSLT is verified at event horizon in the presence of logarithmic entropy ( Figure 11). From Figure 12, we can see thatS tot < 0 for n = 1.5 which leads to the validity of thermal equilibrium condition.
The plot ofṠ tot for Renyi entropy at event horizon is presented in Figure  13. It is observed thatṠ tot behaves positively with the passage of time which verifies the validity of GSLT. Figure 14 shows that the trajectories ofS tot remains negative for increasing of t with n = 1.5. This means that the present scenario obeys the condition for thermodynamic equilibrium. The plot ofṠ tot for power law corrected entropy is displayed in Figure 15 and observe that GSLT holds in this case. Figure 16 shows that the present scenario fulfils the thermodynamic equilibrium condition for power law entropy at event horizon.