# Thermodynamics of Ricci-Gauss-Bonnet Dark Energy.

1. IntroductionThe 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

d[S.sub.horizon]/dt + d[S.sub.inside]/dt [greater than or equal to] 0. (1)

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 L appearing in the holographic dark energy density [[rho].sub.de] = 3[M.sub.pl][L.sup.-2] 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, [R.sup.-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 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, fc-essence, Chaplygin gas, and holographic dark energy (HDE) models [31-39]. Moreover, several modified theories of gravity are f(R), f(T) [40-42], f(R, T) [43,44], f(G) [45-47], f(T, [T.sub.G]) [48-53], and f(T, T) [54,55] (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.sup.2] - 4[R.sub.[mu]v][R.sup.[mu]v] + [R.sub.[mu]v[lambda][sigma]], [R.sup.[mu]v[lambda][sigma]]). 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 1/[L.sup.2] = -[alpha]R + [beta][square root of ([absolute value of G])] where [alpha] and [beta] are model parameters. In flat FRW geometry, the Ricci scalar (R) and the Gauss-Bonnet invariant (G) are given as R = -6(2[H.sup.2] + H) and G = 24[H.sup.2]([H.sup.2] + H), 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 [[??].sub.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.

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 write

[[??].sub.stot] = [[??].sub.h] + [[??].sub.in] [greater than or equal to] 0. (2)

Here [[??].sub.h] gives entropy of horizon and entropy of matter inside horizon is represented by [[??].sub.in]. Now considering spatially flat FRW universe, the first Friedmann equation is

[H.sup.2] = [[kappa].sup.2]/3 ([[rho].sub.eff] + [P.sub.eff]). (3)

Here [[rho].sub.eff] and [P.sub.eff] 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 as

Td[S.sub.in] = [P.sub.eff] dV + d[E.sub.in]. (4)

Here [E.sub.in] = [[rho].sub.eff]V, V = (4[pi]/3)[R.sup.2.sub.h], and T = 1/2[pi][R.sub.h] which modified the above equation as follows:

[mathematical expression not reproducible]. (5)

For flat FRW universe, the Hubble horizon can be defined as

[R.sub.h] = 1/H. (6)

By utilizing the above horizon, [P.sub.eff] = [p.sub.de] (for cold dark matter [p.sub.m] = 0) and [[rho].sub.eff] = [[rho].sub.d] + [[rho].sub.m] in (5), we can get

[mathematical expression not reproducible] (7)

From conservation equation, one can obtain

[mathematical expression not reproducible]. (8)

Substituting the value of [[omega].sub.de] in (7), we get

[mathematical expression not reproducible]. (9)

Moreover, Ricci-Gauss Bonnet dark energy can be defined as follows [67]:

[[rho].sub.d] = 3(6[alpha](2[H.sup.2] + [??]) + 2[square root of (3)][beta]H [square root of ([absolute value of [H.sup.2] + [??]])]). (10)

Here [alpha] and [beta] are the model parameters. Standard Ricci dark energy can be obtained by substituting [beta] = 0 and [alpha] = 0 yields a pure Gauss-Bonnet HDE. The density parameters can be introduced as

[[OMEGA].sub.m] = [[rho].sub.m]/3[H.sup.2], [[OMEGA].sub.d] = [[rho].sub.d]/3[H.sup.2]. (11)

According to first Friedman equation, we can obtain

[[OMEGA].sup.d] + [[OMEGA].sub.m] = 1. (12)

Also, [[rho].sub.m] can be evaluated by using conservation equation as follows:

[mathematical expression not reproducible], (13)

with [mathematical expression not reproducible]. By using this value of [[rho].sub.m], [[OMEGA].sub.m] takes the following form:

[mathematical expression not reproducible]. (14)

Using (12) and (14), we can find H as

[mathematical expression not reproducible]. (15)

Differentiating H, we obtain

[mathematical expression not reproducible], (16)

where prime denotes the differentiation with respect to x = ln a. Also, differentiation of [R.sub.A] with respect to t leads to

[mathematical expression not reproducible]. (17)

We get the following value of [[??].sub.de] by differentiating (11):

[mathematical expression not reproducible]. (18)

Now, [[??].sub.h] takes the form

[mathematical expression not reproducible]. (19)

Also, Friedman first equation gives [[rho].sub.m] = 3[H.sup.2] - [[rho].sub.d] and hence we can write

[mathematical expression not reproducible] (20)

By inserting (6) in above equation, we have

[mathematical expression not reproducible]. (21)

By using value of [[??].sub.h] from (19), we get

[mathematical expression not reproducible]. (22)

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 by

[S.sub.h] = A/4G. (23)

By using G = 1, c = 1, and A = 4[pi][R.sup.2.sub.h] being the area of horizon, we get

[mathematical expression not reproducible]. (24)

By using the expressions of [R.sub.h] and [[??].sub.h], we have

[mathematical expression not reproducible]. (25)

Equations (22) and (25) join to form

[mathematical expression not reproducible], (26)

where [[??].sub.tot] represents the total entropy; that is, [[??].sub.tot] = [[??].sub.in] + [[??].sub.h].

Now, we assume the power law form of scale factor; that is, a = [a.sub.0][t.sup.n], where n and [a.sub.0] appear as constant parameters. Under this assumption, the values of H and [R.sub.h] turn out to be n/t, t/n respectively. In this way, [[??].sub.tot] reduces to

[mathematical expression not reproducible], (27)

where U = 3(6[alpha](2[n.sup.2] - n) + 2[square root of (3)][beta]n [square root of ([n.sup.2] - n)]). In order to analyze the clear picture of validity of GSLT for this entropy on the Hubble horizon, we plot [[??].sub.tot] against cosmic time (t) by fixing constant parameters as [alpha] = 0.2, [beta] = 0.00l, and n = 4 as shown in Figure 1. This shows that [[??].sub.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 (27) to get [[??].sub.tot] given below

[mathematical expression not reproducible]. (28)

We plot X = [[??].sub.tot] versus n in Figure 2 which shows that [[??].sub.tot] < 0 for the selected range of n. Hence, thermal equilibrium condition is satisfied for Bekenstein entropy at apparent horizon.

2.2. Logarithmic Corrections to Entropy. Logarithmic corrections arises from loop quantum gravity due to thermal equilibrium and quantum fluctuations [68-74]. The entropy on apparent horizon can be defined as follows:

[S.sub.h] = A/4G + [eta] ln [A/4G] - [xi] [4G/A] + [gamma], (29)

where [eta], [xi], and [gamma] are dimensionless constants. Differentiating with respect to t, we get

[mathematical expression not reproducible], (30)

which takes the following form by inserting value of [[??].sub.h] from (19):

[mathematical expression not reproducible]. (31)

In the presence of logarithmic entropy, [[??].sub.tot] can be obtained by using (22) and (31):

[mathematical expression not reproducible]. (32)

By substituting value of scale factor, the above equation reduces to

[mathematical expression not reproducible]. (33)

Differentiating the above equation, we get

[mathematical expression not reproducible]. (34)

Figure 3 presents the plot of [[??].sub.tot] at apparent horizon by taking logarithmic entropy at apparent horizon. Here we have taken [eta] = 3.8 and [xi] = 3 along with the same values of [alpha], [beta], and n as in the above-mentioned case. Here [[??].sub.tot] remains negative for t < 1.5 while it moves in positive direction t [greater than or equal to] 1.5.

Hence, validity of GSLT is verified for t [greater than or equal to] 1.5 at apparent horizon with logarithmic entropy. Figure 4 shows that X = [[??].sub.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.

2.3. Renyi Entropy. A novel type of Renyi entropy was recommended by Biro and Czinner [75] on black hole horizons by considering Bekenstein-Hawking entropy as nonextensive Tsalis entropy. The modified Renyi entropy can be defined as [76]

[S.sub.h] = [1/[lambda]] ln [l + [lambda] [A/4G]]. (35)

It behaves as Bekenstein entropy for [lambda] = 0. Differentiating with respect to t, we obtain

[mathematical expression not reproducible]. (36)

Using (19) in the above equation, we get

[mathematical expression not reproducible]. (37)

Combine (22) and (37) to get

[mathematical expression not reproducible]. (38)

For power law scale factor, we obtain

[mathematical expression not reproducible]. (39)

The plot of [[??].sub.tot] by taking Renyi entropy at apparent horizon is presented by Figure 5. Here [alpha], [beta], and n have the same values like previous case and [lambda = 1.5. In this case, [[??].sub.tot] behaves positively with the passage of time which verifies the validity of GSLT for the present case. Further, differentiating the above equation, we get

[mathematical expression not reproducible]. (40)

The plot of this expression is shown in Figure 6 which shows that [[??].sub.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.

2.4. Power Law Entropic Correction. The power law corrections to entropy appear in dealing with entanglement of quantum fields in and out of the horizon [77]. The corrected entropy takes the form [78]

[S.sub.h] = A/4G (l - [k.sub.[mu]] [A.sup.1-[mu]/2]), (41)

with [k.sub.[mu]] = [mu][(4[pi]).sup.[mu]/2-1]/(4 - [mu])[r.sup.(2-[mu])].sub.c]; [r.sub.c] is crossover length and [mu] appears as a constant.

[mathematical expression not reproducible]. (42)

Utilization of (19) in the above equation leads to

[mathematical expression not reproducible]. (43)

Join (22) and (43) to obtain

[mathematical expression not reproducible]. (44)

In the presence of scale factor, the above expression turns out to be

[mathematical expression not reproducible]. (45)

By taking power Law entropy at apparent horizon, [[??].sub.tot] is plotted at apparent horizon as shown in Figure 7. With the same values for [alpha], [beta], and n, we have taken [mu] = 5 and [r.sub.c] = 2. Here the effectiveness of GSLT at apparent horizon is certified by positive moves of [[??].sub.tot] with increasing t. Differentiating the above equation, we get

[mathematical expression not reproducible]. (46)

Just like the 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.

3. Generalized Second Law of Thermodynamics at Event Horizon

In this section, we study GSL of thermodynamics at event horizon which is defined as [R.sub.h] = a(t) [[integral].sup.[infinity].sub.t] d[??]/a([??]). Its derivative with respect to time is given by [[??].sub.h] = H[R.sub.h] - l. The temperature we used in this section is T = bH/2[pi], where b is a constant. For the present case, rewriting (4) by using value of T and [[??].sub.h], we have the following equation for entropy inside horizon:

[mathematical expression not reproducible]. (47)

3.1. Bekenstein Entropy. Under this scenario, (24) can be written as

[[??].sub.h] = 2[pi][R.sub.h] (H[R.sub.h] - 1). (48)

The equation for [[??].sub.tot] can be obtained by using (43) and (48) as follows:

[mathematical expression not reproducible]. (49)

By putting values of scale factor and [R.sub.h] in the above equation, we have

[[??].sub.tot] = -8[[pi].sup.2]t/n[(n - 1).sup.2]b(3[n.sup.2] + U(2/3n - 1)) + 2[pi]t. (50)

Differentiating the above equation with respect to t, we get

[[??].sub.tot] = -8[[pi].sup.2]/n[(n - 1).sup.2]b(3[n.sup.2] + U(2/3n - 1)) + 2[pi]. (51)

Figure 9 contains the plot of [[??].sub.tot] by taking Bekenstein entropy at event horizon. Here we have taken [alpha] = 0.2, [beta] = 0.001, and n = 4. It is clear from figure that [[??].sub.tot] remains positive with increasing value of t. This confirms the validity of GSLT at event horizon with Bekenstein entropy. Figure 10 shows that [[??].sub.tot] < 0 for increasing values of n. Hence, at event horizon, the Bekenstein entropy fulfilled the condition of thermodynamic equilibrium.

3.2. Logarithmic Entropy. For this entropy at event horizon, (29) leads to

[mathematical expression not reproducible]. (52)

By using (47) and (52), the expression of [[??].sub.tot] can be written as

[mathematical expression not reproducible]. (53)

The following equation is obtained by using values of scale factor and [R.sub.h]

[[??].sub.tot] = -8[[pi].sup.2]t/n[(n - 1).sup.2]b(3[n.sup.2] + U(2/3n - 1)) + 2[pi]t/[(n - 1).sup.2] + 2[eta]/t + 2[xi][(n - 1).sup.2]/[pi][t.sup.3]. (54)

Differentiating the above equation with respect to t, we obtain

[[??].sub.tot] = -8[[pi].sup.2]/n[(n - 1).sup.2]b(3[n.sup.2] + U(2/3n - 1)) + 2[pi]/[(n - 1).sup.2] - 2[eta]/[t.sup.2] - 6[xi][(n - 1).sup.2]/[pi][t.sup.4]. (55)

Figure 11 presents the plot of [[??].sub.tot] by taking logarithmic entropy at event horizon. Here we have taken [eta] = 4 and [xi] = 6 along with the same values of [alpha], [beta], and n as in the abovementioned case. Clearly, [[??].sub.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 that [[??].sub.tot] < 0 for n = 1.5. Hence, for this case, thermodynamic equilibrium condition holds.

3.3. Renyi Entropy. The following form is obtained from (34), by substituting value for [[??].sub.h]:

[[??].sub.h] = 2[pi]H/[H.sup.2] + [lambda][pi] (H[R.sub.h] - 1). (56)

Joining (47) and (56), we get

[mathematical expression not reproducible]. (57)

By using values of scale factor and [R.sub.h], the above equation reduces to

[[??].sub.tot] = -8[[pi].sup.2]t/n[(n - 1).sup.2]b (3[n.sup.2] + U(2/3n - 1)) + 2[pi]t/[(n - 1).sup.2] + [lambda][pi][t.sup.2]. (58)

Differentiating the above equation with respect to t, we get

[[??].sub.tot] = -8[[pi].sup.2]/n[(n - 1).sup.2]b (3[n.sup.2] + U(2/3n - 1)) + 2[pi][(n - 1).sup.2] - 2[[pi].sup.2][lambda]t[[pi].sup.2]/[([(n - 1).sup.2] + [lambda][pi][t.sup.2]).sup.2]. (59)

The plot of [[??].sub.tot] for Renyi entropy at event horizon is presented in Figure 13.

Here [alpha], [beta], and n have the same values like the previous case while [lambda] = 1.5. In this case, [[??].sub.tot] behaves positively with the passage of time which verifies the validity of GSLT. Figure 14 shows that the trajectories of [[??].sub.tot] remain negative for increasing of t with n = 1.5. This means that the present scenario obeys the condition for thermodynamic equilibrium.

3.4. Power Law Entropy. Under conditions of present section, (41) reduces to

[[??].sub.h] = [pi]/H (2 - [k.sub.[mu]](4 - [mu])[(4[pi]/[H.sup.2]).sup.1-[mu]/2])(H[R.sub.h] - 1). (60)

Joining (47) and (60), we get the following equation:

[mathematical expression not reproducible]. (61)

Inserting conditions for scale factor and [R.sub.h] in the above equation, we get

[[??].sub.tot] = -8[[pi].sup.2]t/n[(n - 1).sup.2]b (3[n.sup.2] + U(2/3n - 1)) + 2 - [mu](t/[r.sub.c](n - 1)).sup.2-[mu]]) [pi]t/(n - 1).sup.2]. (62)

The plot of this expression is displayed in Figure 15 with the same values for [alpha], [beta], and n while [mu] = 5 and [r.sub.c] = 2. Here the effectiveness of GSLT at event horizon is certified by positive moves of [[??].sub.tot] with increasing t. Differentiating with respect to t, we obtain

[[??].sub.tot] = -8[[pi].sup.2]/n[(n - 1).sup.2]b (3[n.sup.2] + U(2/3n - 1)) + 2[pi]/[(n - 1).sup.2] - [mu][pi](3 - [mu])[t.sup.2-[mu]]/[([r.sub.c](n - 1)).sup.2-[mu]][(n - 1).sup.2]. (63)

Figure 16 shows that the present scenario fulfils the thermodynamic equilibrium condition for power law entropy at event horizon.

4. Conclusion

The concept of thermodynamics in cosmological system originates through black hole physics. It was suggested [79] that the temperature of Hawking radiations emitting from black holes is proportional to their corresponding surface gravity on the event horizon. Jacobson [80] 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 = TdS, 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 TdS = dE + PdV (E, P, and V represent the internal energy, pressure, and volume of the spherical system) for any horizon [81]. 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 [82] or power law corrected [83-85] or logarithmic corrected [86] forms. Many people have explored the validity of GSLT of different systems including interaction of two fluid components like DE and dark matter [87-90], as well as interaction of three components of fluid [91-93] in the FRW universe by using simple horizon entropy of the universe. The thermodynamical analysis was widely performed in modified theories of gravity [94-97].

Motivated by the above-mentioned works, we have considered a newly proposed DE model named 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 was satisfied under certain conditions on constant parameters. The detailed results are as follows.

On Apparent Horizon. By utilizing usual entropy, GSLT on the apparent horizon was shown in Figure 1 which shows that [[??].sub.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 was displayed in Figure 3 which exhibits that GSLT remains valid for t [greater than or equal to] 1.5. However, Figure 4 shows that X = [[??].sub.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.

The plot of [[??].sub.tot] by taking Renyi entropy at apparent horizon was 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, [[??].sub.tot] is plotted at apparent horizon as shown in Figure 7. Here the effectiveness of GSLT at apparent horizon is certified by positive moves of [[??].sub.tot] with increasing t. Just like the 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 that [[??].sub.tot] < 0 for n = 1.5 which leads to the validity of thermal equilibrium condition.

The plot of [[??].sub.tot] for Renyi entropy at event horizon is presented in Figure 13. It is observed that [[??].sub.tot] behaves positively with the passage of time which verifies the validity of GSLT. Figure 14 shows that the trajectories of [[??].sub.tot] remain negative for increasing of t with n = 1.5. This means that the present scenario obeys the condition for thermodynamic equilibrium. The plot of [[??].sub.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.

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

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

References

[1] P. C. Davies, "Cosmological horizons and entropy," Classical and Quantum Gravity, vol. 5, no. 10, pp. 1349-1355, 1988.

[2] Q. Huang and M. Li, "The holographic dark energy in a non-flat universe," JCAP, no. 0408, p. 013, 2004.

[3] G. Izquierdo and D. Pavon, "Dark energy and the generalized second law," Physics Letters B, vol. 633, no. 4-5, pp. 420-426, 2006.

[4] J. D. Bekenstein, "Black holes and entropy," Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 7, pp. 2333-2346, 1973.

[5] S. W. Hawking, "Particle creation by black holes," Communications in Mathematical Physics, vol. 43, no. 3, pp. 199-220, 1975, Erratum in: Communications in Mathematical Physics, vol. 46, p. 206, 1976.

[6] T. Padmanabhan, "Gravity and the thermodynamics of horizons," Physics Reports, vol. 406, no. 2, pp. 49-125, 2005.

[7] T. Padamanabhan, Phys. Rep, vol. 7, Article ID 046901, 2010.

[8] G. W. Gibbons and S. W. Hawking, "Cosmological event horizons, thermodynamics, and particle creation," Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 15, no. 10, pp. 2738-2751, 1977.

[9] M. D. Pollock and T. P. Singh, "On the thermodynamics of de SITter spacetime and quasi-de SITter spacetime," Classical and Quantum Gravity, vol. 6, no. 6, pp. 901-909, 1989.

[10] R. Brustein, "Generalized second law in cosmology from causal boundary entropy," Physical Review Letters, vol. 84, no. 10, pp. 2072-2075, 2000.

[11] D. Pavon, "The generalised second law and extended thermodynamics," Classical and Quantum Gravity, vol. 7, no. 3, pp. 487-491, 1990.

[12] T. Hooft, "Dimensional reduction in quantum gravity," [gr-qc/9310026].

[13] L. Susskind, "The world as a hologram," Journal of Mathematical Physics, vol. 36, no. 11, pp. 6377-6396, 1995.

[14] E. Witten, "Anti de Sitter space and holography," Advances in Theoretical and Mathematical Physics, vol. 2, no. 2, pp. 253-291, 1998.

[15] R. Bousso, "The holographic principle," Reviews of Modern Physics, vol. 74, no. 3, pp. 825-874, 2002.

[16] R. Easther and D. Lowe, "Holography, cosmology, and the second law of thermodynamics," Physical Review Letters, vol. 82, no. 25, pp. 4967-4970, 1999.

[17] N. Kaloper and A. Linde, "Cosmology versus holography," Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 60, no. 10,103509, 7 pages, 1999.

[18] S. P. de Alwis, "Brane worlds in 5D and warped compactifications in IIB," Physics Letters. B. Particle Physics, Nuclear Physics and Cosmology, vol. 603, no. 3-4, pp. 230-238, 2004.

[19] C. Gao, F. Wu, X. Chen, and Y.-G. Shen, "Holographic dark energy model from Ricci scalar curvature," Physical Review D: Particles, Fields, Gravitation and Cosmology, no. 79, Article ID 043511, p. 043, 2009.

[20] L. N. Granda, A. Oilveros, and B. Phys. Lett, "Infrared cut-off proposal for the Holographic density," Phys. Lett. B669, no. 671, pp. 275-277, 2008.

[21] S. Chattopadhyay, "Interacting Ricci dark energy and its statefinder description," The European Physical Journal Plus, no. 126, p. 130, 2011.

[22] T. K. Mathew, J. Suresh, and D. Divakaran, "Modified holographic ricci dark energy model and state finder diagnosis in flat universe," Int. J. Mod. D, no. 22, Article ID 1350056, 2013.

[23] M. S. Berger and H. Shojaei, "Interacting dark energy model for the expansion history of the Universe," Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 74, no. 4, 2006.

[24] B. Wang, Y. Gong, and E. Abdalla, "Transition of the dark energy equation of state in an interacting holographic dark energy model," Physics Letters B, vol. 624, no. 3-4, pp. 141-146, 2005.

[25] S. Umezu, T. Hatta, and H. Ohmori, "Fundamental characteristics of bioprint on calcium alginate gel," Japanese Journal of Applied Physics, vol. 52, no. 5, Article ID 05DB20, 2013.

[26] J. Zhou, B. Wang, Y. Gong, and E. Abdalla, "The generalized second law of thermodynamics in the accelerating universe," Physics Letters B, vol. 652, no. 2-3, pp. 86-91, 2007.

[27] M. R. Setare, "Interacting holographic dark energy model and generalized second law of thermodynamics in non-flat universe," JCAP, no. 023, p. 0701, 2007.

[28] A. Sheykhi and B. Wang, "Generalized second law of thermodynamics in Gauss-Bonnet braneworld," Physics Letters B, vol. 678, no. 5, pp. 434-437, 2009.

[29] A. Sheykhi and B. Wang, "Generalized second law of thermodynamics in warped DGP braneworld," Modern Physics Letters A, vol. 25, no. 14, pp. 1199-1210, 2010.

[30] K. M. Wong, "Study of the electronic structure of individual free-standing Germanium nanodots using spectroscopic scanning capacitance microscopy," Japanese Journal of Applied Physics, vol. 48, no. 8, Article ID 085002, 2009.

[31] I. Zlatev, L. Wang, and P. J. Steinhardt, "Quintessence, cosmic coincidence, and the cosmological constant," Physical Review Letters, vol. 82, no. 5, pp. 896-899, 1999.

[32] B. Z. Kopeliovich and A. V. Tarasov, "Gluon shadowing in heavy flavor production off nuclei," Nuclear Physics A, vol. 710, no. 1-2, pp. 180-217, 2002.

[33] V. Sahni, "The cosmological constant problem and quintessence," Classical and Quantum Gravity, vol. 19, no. 13, pp. 3435-3448, 2002.

[34] T. Chiba, T. Okabe, and M. Yamaguchi, "Kinetically driven quintessence," Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 62, Article ID 023511, 2000.

[35] M. Setare, "Interacting generalized Chaplygin gas model in non-flat universe," The European Physical Journal C, vol. 52, no. 3, pp. 689-692, 2007.

[36] A. E. Bernardini and O. Bertolami, " Lorentz violating extension of the standard model and the ," Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 77, no. 8, 2008.

[37] S. D. H. Hsu, "Entropy bounds and dark energy," Physics Letters B, vol. 594, no. 1-2, pp. 13-16, 2004.

[38] M. Li, "A model of holographic dark energy," Physics Letters B, vol. 603, no. 1-2, pp. 1-5, 2004.

[39] K. Bamba, S. Capozziello, S. Nojiri, and S. D. Odintsov, "Dark energy cosmology: the equivalent description via different theoretical models and cosmography tests," Astrophys. Space Sci, no. 342, pp. 155-228, 2012.

[40] J. Amoros, J. De Haro, and S. D. Odintsov, "Bouncing loop quantum cosmology from F(T) gravity," Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 87, no. 10, Article ID 104037, 2013.

[41] E. V. Linder, "Erratum: Einstein's other gravity and the acceleration of the universe [Phys. Rev. D 81,127301 (2010)]," Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 82, no. 10, Article ID 109902, 2010.

[42] M. Jamil, D. Momeni, and R. Myrzakulov, "Stability of a nonminimally conformally coupled scalar field in F(T) cosmology," The European Physical Journal C, vol. 72, article 2075, 2012.

[43] E. H. Baffou, A. V. Kpadonou, M. E. Rodrigues, M. J. S. Houndjo, and J. Tossa, "Cosmological viable f(R, T) dark energy model: dynamics and stability," Astrophysics and Space Science, vol. 356, no. 1, pp. 173-180, 2015.

[44] M. J. S. Houndjo and O. F. Piattella, "Reconstructing f(R, T) gravity from holographic dark energy," International Journal of Modern Physics D, vol. 21, no. 3, Article ID 1250024, 2012.

[45] S. Nojiri and S. D. Odintsov, "Modified Gauss-Bonnet theory as gravitational alternative for dark energy," Physics Letters B, vol. 631, no. 1-2, pp. 1-6, 2005.

[46] S. Nojiri, S. D. Odintsov, A. Toporensky, and P. Tretyakov, "Reconstruction and deceleration-acceleration transitions in modified gravity," General Relativity and Gravitation, vol. 42, no. 8, pp. 1997-2008, 2010.

[47] K. Bamba, S. D. Odintsov, L. Sebastiani, and S. Zerbini, "Finite-time future singularities in modified Gauss-Bonnet and f(R, G) gravity and singularity avoidance," The European Physical Journal C, vol. 67, no. 1, pp. 295-310, 2010.

[48] G. Kofinas, G. Leon, and E. N. Saridakis, "Dynamical behavior in F(T, Tg) cosmology," Classical and Quantum Gravity, vol. 31, no. 17, Article ID 175011, 2014.

[49] G. Kofinas and E. N. Saridakis, "Cosmological applications of F(T,Tg) gravity," Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 90, no. 8, Article ID 084045, 2014.

[50] S. Chattopadhyay, A. Jawad, D. Momeni, and R. Myrzakulov, "Pilgrim dark energy in F(T,TG) cosmology," Astrophysics and Space Science, vol. 353, no. 1, pp. 279-292, 2014.

[51] A. Jawad, S. Rani, and S. Chattopadhyay, "Modified QCD ghost f(T, Tg) gravity," Astrophysics and Space Science, vol. 360, p. 37, 2015.

[52] A. Jawad and U. Debnath, "New agegraphic pilgrim dark energy in f(T, Tg) gravity," Communications in Theoretical Physics, vol. 64, no. 2, pp. 145-150, 2015.

[53] A. Jawad and A. Majeed, "Correspondence of pilgrim dark energy with scalar field models," Astrophysics and Space Science, vol. 356, pp. 375-381, 2015.

[54] T. Harko, F. S. N. Lobo, G. Otalora, and E. N. Saridakis, "F(T, T) gravity and cosmology," Journal of Cosmology and Astroparticle Physics, vol. 2014, no. 12, article no. 021, 2014.

[55] I. G. Salako, A. Jawad, and S. Chattopadhyay, "Holographic dark energy reconstruction in (Formula presented.)gravity," Astrophysics and Space Science, vol. 358, no. 1, article no. 13, 2015.

[56] G. Gupta, E. N. Saridakis, and A. A. Sen, "Nonminimal quintessence and phantom with nearly flat potentials," Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 79, no. 12, 2009.

[57] Y-F. Cai et al., Phys. Reports, no. 493, p. 1, 2010.

[58] E. N. Saridakis, P. F. Gonzalez-Diaz, and C. L. Siguenza, "Unified dark energy thermodynamics: varying w and the -1-crossing," Class. Quant. Grav, vol. 26, Article ID 165003, pp. 1-5, 2009.

[59] A. Jawad and A. Majeed, "Correspondence of pilgrim dark energy with scalar field models," Astrophysics and Space Science, vol. 356, no. 2, pp. 375-381, 2015.

[60] A. Jawad, "Cosmological analysis of pilgrim dark energy in loop quantum cosmology," The European Physical Journal C, vol. 75, p. 206, 2015.

[61] A. Jawad, S. Rani, and M. Saleem, "Cosmological study of reconstructed f(T) models," Astrophysics and Space Science, vol. 362, no. 4, article no. 63, 2017.

[62] A. Jawad, S. Rani, I. G. Salako, and F. Gulshan, "Pilgrim dark energy models in fractal universe," International Journal of Modern Physics D, vol. 26, Article ID 1750049, 2017.

[63] A. Jawad, S. Rani, and N. Azhar, "Entropy corrected holographic dark energy models in modified gravity," International Journal of Modern Physics D, vol. 26, Article ID 1750040, 2017.

[64] A. Jawad, S. Rani, I. G. Salako, and F. Gulshan, "Cosmological study in loop quantum cosmology through dark energy model," International Journal of Modern Physics D: Gravitation, Astrophysics, Cosmology, vol. 26, no. 2, 1750007, 16 pages, 2017.

[65] S. Rani, A. Jawad, I. G. Salako, and N. Azhar, "Non-flat pilgrim dark energy FRW models in modified gravity," Astrophysics and Space Science, vol. 361, p. 286, 2016.

[66] A. Jawad, S. Chattopadhyay, and S. Rani, "Viscous pilgrim f(T) gravity models," Astrophysics and Space Science, vol. 361, no. 7, article no. 231, 2016.

[67] E. N. Saridakis, "Ricci-Gauss-Bonnet holographic dark energy," https://arxiv.org/abs/170709331.

[68] K. A. Meissner, "Black-hole entropy in loop quantum gravity," Classical and Quantum Gravity, vol. 21, no. 22, pp. 5245-5251, 2004.

[69] A. Gosh and P. Mitra, "Log correction to the black hole area law 2005," Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 71, Article ID 027502, 2004.

[70] A. Chatterjee and P. Majumdar, "Universal canonical black hole entropy," Physical Review Letters, vol. 92, no. 14, Article ID 141301, 2004.

[71] R. Bonerjee and S. K. Modak, "Exact differential and corrected area law for stationary black holes in tunneling method," JHEP, no. 0905, p. 36, 2009.

[72] S. K. Modak, "Corrected entropy of BTZ blackhole in tunneling approach," Physics Letters B, vol. 671, no. 1, pp. 167-173, 2009.

[73] M. Jamil and M. U. Farooq, "Interacting holographic dark energy with logarithmic correction," Journal of Cosmology and Astroparticle Physics, vol. 2010, no. 3, p. 001, 2010.

[74] H. M. SadjadiandM. Jamil, Europhys. Lett, vol. 9, p. 69001, 2010.

[75] T. S. Biro and V. G. Czinner, "A q-parameter bound for particle spectra based on black hole thermodynamics with Renyi entropy," Physics Letters. B. Particle Physics, Nuclear Physics and Cosmology, vol. 726, no. 4-5, pp. 861-865, 2013.

[76] N. Komatsu and J. C, Eur. Phys. J. C, p. 77, 2017.

[77] S. Das, S. Shankaranarayanan, and S. Sur, "Power-law corrections to entanglement entropy of horizons," Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 77, no. 6, 2008.

[78] N. Radicella, D. Pavon, and B. Phys. Lett, Phys. Lett. B, p. 121, 2010.

[79] S. W Hawking, "Particle creation by black holes," Communications in Mathematical Physics, vol. 43, no. 3, pp. 199-220, 1975.

[80] T. Jacobson, "Thermodynamics of spacetime: the Einstein equation of state," Physical Review Letters, vol. 75, p. 1260, 1995.

[81] T. Padmanabhan, "Classical and quantum thermodynamics of horizons in spherically symmetric spacetimes," Classical and Quantum Gravity, vol. 19, no. 21, pp. 5387-5408, 2002.

[82] R. G. CaiandS. P. Kim, "First law of thermodynamics and Friedmann equations of Friedmann-Robertson-Walker universe," Journal of High Energy Physics, vol. 2005, article 050, 2005.

[83] R. Banerjee and S. K. Modak, "Quantum tunneling, blackbody spectrum and non-logarithmic entropy correction for Lovelock black holes," Journal of High Energy Physics, vol. 2009, no. 11, pp. 073-073, 2009.

[84] H.-X. He, "Gauge symmetry and transverse symmetry transformations in gauge theories," Communications in Theoretical Physics, vol. 52, no. 2, pp. 292-294, 2009.

[85] S. Banerjee, R. K. Gupta, and A. Sen, "Logarithmic corrections to extremal black hole entropy from quantum entropy function," JHEP, no. 147, p. 1103, 2011.

[86] A. Sheykhi and M. Jamil, "Power-law entropy corrected holographic dark energy model," General Relativity and Gravitation, vol. 43, no. 10, pp. 2661-2672, 2011.

[87] K. Karami, S. Ghaffari, and M. M. Soltanzadeh, "The generalized second law of gravitational thermodynamics on the apparent and event horizons in FRW cosmology," Classical and Quantum Gravity, vol. 27, no. 20, Article ID 205021, 2010.

[88] K. Karami, "Comment on "Interacting holographic dark energy model and generalized second law of thermodynamics in a nonflat universe", by M.R. Setare (JCAP 01 (2007) 023)," Journal of Cosmology and Astroparticle Physics, vol. 2010, no. 1, article no. 015, 2010.

[89] A. Sheykhi, "Thermodynamics of interacting holographic dark energy with the apparent horizon as an IR cutoff," Classical and Quantum Gravity, vol. 27, no. 2, Article ID 025007, 2010.

[90] N. Mazumder and S. Chakraborty, "Validity of the generalized second law of thermodynamics of the universe bounded by the event horizon in holographic dark energy model," General Relativity and Gravitation, vol. 42, no. 4, pp. 813-820, 2010.

[91] M. Jamil, E. N. Saridakis, and M. R. Setare, "Thermodynamics of dark energy interacting with dark matter and radiation," Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 81, no. 2, Article ID 023007, 6 pages, 2010.

[92] K. Karami, A. Abdolmaleki, N. Sahraei, and S. Ghaffari, "Thermodynamics of apparent horizon in modified FRW universe with power-law corrected entropy," JHEP, no. 150, p. 1108, 2011.

[93] K. Karami, Eur. Phys. Lett, vol. 93, 2011.

[94] M. Jamil, E. N. Saridakis, and M. R. Setare, "The generalized second law of thermodynamics in Horava-Lifshitz cosmology," Journal of Cosmology and Astroparticle Physics, no. 11, article 32, 20 pages, 2010.

[95] K. Bamba, C.-Q. Geng, and S. Tsujikawa, "Thermodynamics in modified gravity theories," International Journal of Modern Physics D, vol. 20, no. 8, p. 1363, 2011.

[96] K. Bamba, C. Geng, and S. Tsujikawa, "Physics Letters B," International Journal of Modern Physics D, no. 688, p. 101, 2010.

[97] S. Rani, T. Nawaz, and A. Jawad, "Thermodynamics in dynamical Chern-Simons modified gravity with canonical scalar field," An International Journal of Astronomy, Astrophysics and Space Science, vol. 361, no. 9, p. 285, 2016.

Ayesha Iqbal (1) and Abdul Jawad (iD) (2)

(1) Department of Mathematics, Government College University, Faisalabad, Pakistan

(2) Department of Mathematics, COMSATS Institute of Information Technology, Lahore 54000, Pakistan

Correspondence should be addressed to Abdul Jawad; jawadab181@yahoo.com

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

Academic Editor: Chao-Qiang Geng

Caption: Figure 1: Plot of [[??].sub.tot] by taking Bekenstein entropy as entropy at apparent horizon, where time is measured in seconds.

Caption: Figure 2: Plot of X = [[??].sub.tot] by taking Bekenstein entropy as entropy at apparent horizon.

Caption: Figure 3: Plot of [[??].sub.tot] by taking Logarithmic entropy as entropy at apparent horizon, where time is measured in second.

Caption: Figure 4: Plot of X = [[??].sub.tot] by taking Logarithmic entropy as entropy at apparent horizon, where time is measured in second.

Caption: Figure 5: Plot of [[??].sub.tot] by taking Renyi entropy as entropy at apparent horizon, where time is measured in seconds.

Caption: Figure 6: Plot of X = [[??].sub.tot] by taking Renyi entropy as entropy at apparent horizon, where time is measured in seconds.

Caption: Figure 7: Plot of [[??].sub.tot] by taking power law entropy as entropy at apparent horizon, where time is measured in seconds.

Caption: Figure 8: Plot of X = [[??].sub.tot] by taking power law entropy as entropy at apparent horizon, where time is measured in seconds.

Caption: Figure 9: Plot of [[??].sub.tot] by taking Bekenstein entropy as entropy at event horizon, where time is measured in seconds.

Caption: Figure 10: Plot of X = [[??].sub.tot] by taking Bekenstein entropy as entropy at event horizon.

Caption: Figure 11: Plot of [[??].sub.tot] by taking logarithmic entropy as entropy at event horizon, where time is measured in seconds.

Caption: Figure 12: Plot of X = [[??].sub.tot] by taking logarithmic entropy as entropy at event horizon, where time is measured in seconds.

Caption: Figure 13: Plot of [[??].sub.tot] by taking Renyi entropy as entropy at event horizon.

Caption: Figure 14: Plot of X = [[??].sub.tot] by taking Renyi entropy as entropy at event horizon.

Caption: Figure 15: Plot of [[??].sub.tot] by taking power law entropy as entropy at event horizon.

Caption: Figure 16: Plot of X = [[??].sub.tot] by taking power law entropy as entropy at event horizon.

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Title Annotation: | Research Article |
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Author: | Iqbal, Ayesha; Jawad, Abdul |

Publication: | Advances in High Energy Physics |

Article Type: | Report |

Geographic Code: | 1USA |

Date: | Jan 1, 2018 |

Words: | 7073 |

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