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(1)dShorizondt+dSinsidedt≥0.
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 ρde=3MplL-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-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, k-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,TG) [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=R2-4RμνRμν+RμνλσRμνλσ). 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/L2=-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(2H2+H˙) and G=24H2H2+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 S¨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(2)Stot˙=Sh˙+Sin˙≥0.Here Sh˙ gives entropy of horizon and entropy of matter inside horizon is represented by Sin˙. Now considering spatially flat FRW universe, the first Friedmann equation is(3)H2=κ23ρeff+Peff.Here ρeff and Peff 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(4)TdSin=PeffdV+dEin.Here Ein=ρeffV, V=4π/3Rh2, and T=1/2πRh which modified the above equation as follows:(5)Sin˙=8π2Rh3ρeff+PeffRh˙-HRh.For flat FRW universe, the Hubble horizon can be defined as(6)Rh=1H.By utilizing the above horizon, Peff=pde (for cold dark matter pm=0) and ρeff=ρd+ρm in (5), we can get(7)Sin˙=8π2Rh3ρeff+ρdωdeRh˙-1.From conservation equation, one can obtain(8)ρd˙=3H1+ωdρd⟹ωd=-1-ρd˙3Hρd.Substituting the value of ωde in (7), we get(9)Sin˙=8π2Rh3ρm-ρd˙3HRh˙-1.Moreover, Ricci-Gauss Bonnet dark energy can be defined as follows [67]:(10)ρd=36α2H2+H˙+23βHH2+H˙.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(11)Ωm=ρm3H2,Ωd=ρd3H2.According to first Friedman equation, we can obtain(12)Ωd+Ωm=1.Also, ρm can be evaluated by using conservation equation as follows:(13)ρm=ρm0a3,with ρm0=3H02Ωm0. By using this value of ρm, Ωm takes the following form:(14)Ωm=Ωm0H02a3H2.Using (12) and (14), we can find H as(15)H=H0Ωm0a31-Ωd.Differentiating H, we obtain(16)H˙=-H223-Ωd´1-Ωd,where prime denotes the differentiation with respect to x=lna. Also, differentiation of RA with respect to t leads to(17)Rh˙=-H˙H2=123-Ωd´1-Ωd.We get the following value of Ωde´ by differentiating (11):(18)Ωd´=ρd˙3H3+ρdH21+ρd3H21-Ωd-1.Now, Rh˙ takes the form(19)Rh˙=123-11-Ωdρd˙3H3+ρdH21+ρd3H21-Ωd-1.Also, Friedman first equation gives ρm=3H2-ρd and hence we can write(20)Sin˙=8π2Rh33H2-ρd-ρd˙3HRh˙-1.By inserting (6) in above equation, we have(21)Sin˙=8π2H33H2-ρd-ρd˙3HRh˙-1.By using value of Rh˙ from (19), we get(22)Sin˙=8π2H33H2-ρd-ρd˙3H12-121-Ωdρd˙3H3+ρdH2×1+ρd3H21-Ωd-1. 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(23)Sh=A4G.By using G=1,c=1, and A=4πRh2 being the area of horizon, we get(24)Sh=πRh2⟹S˙h=2πRhRh˙.By using the expressions of Rh and Rh˙, we have(25)S˙h=πH3-11-Ωdρd˙3H3+ρdH21+ρd3H21-Ωd-1.Equations (22) and (25) join to form(26)Stot˙=8π2H312-121-Ωde3-ρdeH21+ρde3H21-Ωde-1×6H2-ρde+πH3-11-Ωde1+ρde3H21-Ωde-1×3-ρdeH2, where Stot˙ represents the total entropy; that is, Stot˙=S˙in+S˙h.
Now, we assume the power law form of scale factor; that is, a=a0tn, where n and a0 appear as constant parameters. Under this assumption, the values of H and Rh turn out to be n/t, t/n respectively. In this way, Stot˙ reduces to(27)Stot˙=8π2tn33n2+U23n-112+U2n223n-1+πtn3+Un223n-1, where U=3(6α(2n2-n)+23βnn2-n). In order to analyze the clear picture of validity of GSLT for this entropy on the Hubble horizon, we plot Stot˙ 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 that Stot˙ remains positive with increasing value of t which confirms the validity of GSLT at apparent horizon with Bekenstein entropy.
Plot of Stot˙ by taking Bekenstein entropy as entropy at apparent horizon, where time is measured in seconds.
To examine the thermodynamic equilibrium, we differentiate (27) to get Stot¨ given below(28)Stot¨=8π2n33n2+U23n-112+U2n223n-1+πn3+Un223n-1. We plot X=Stot¨ versus n in Figure 2 which shows that Stot¨<0 for the selected range of n. Hence, thermal equilibrium condition is satisfied for Bekenstein entropy at apparent horizon.
Plot of X=Stot¨ by taking Bekenstein entropy as 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:(29)Sh=A4G+ηlnA4G-ξ4GA+γ,where η, ξ, and γ are dimensionless constants. Differentiating with respect to t, we get(30)Sh˙=2πH+2ηH+2ξH3πRh˙,which takes the following form by inserting value of Rh˙ from (19):(31)Sh˙=πH+ηH+ξH3π3-11-Ωdρd˙3H3+ρdH21+ρd3H21-Ωd-1.
In the presence of logarithmic entropy, Stot˙ can be obtained by using (22) and (31):(32)Stot˙=8π2H33H2-ρd-ρd˙3H12-121-Ωd×ρd˙3H3+ρdH21+ρd3H21-Ωd-1+πH+ηH+ξH3π×3-11-Ωdρd˙3H3+ρdH21+ρd3H21-Ωd-1.
By substituting value of scale factor, the above equation reduces to(33)S˙tot=8π2tn33n2+U23n-112+U2n223n-1+πtn+ηnt+ξn3πt33+Un223n-1. Differentiating the above equation, we get(34)S¨tot=8π2n33n2+U23n-112+U2n223n-1+πn-ηnt2-3ξn3πt43+Un223n-1.Figure 3 presents the plot of S˙tot at apparent horizon by taking logarithmic entropy at apparent horizon. Here we have taken η=3.8 and ξ=3 along with the same values of α, β, and n as in the above-mentioned case. Here S˙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=Stot¨<0 with increasing value of t and n=1.5. Hence, for logarithmic entropy at apparent horizon, the condition of thermal equilibrium is satisfied.
Plot of Stot˙ by taking Logarithmic entropy as entropy at apparent horizon, where time is measured in second.
Plot of X=Stot¨ by taking Logarithmic entropy as entropy at apparent horizon, where time is measured in second.
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](35)Sh=1λln1+λA4G.It behaves as Bekenstein entropy for λ=0. Differentiating with respect to t, we obtain(36)Sh˙=2πHH2+λπRh˙.Using (19) in the above equation, we get(37)Sh˙=πHH2+λπ3-11-Ωdρd˙3H3+ρdeH21+ρd3H21-Ωd-1.Combine (22) and (37) to get(38)Stot˙=8π2H33H2-ρd-ρd˙3H12-121-Ωdρd˙3H3+ρdH2×1+ρd3H21-Ωd-1+πHH2+λπ3-11-Ωd×ρd˙3H3+ρdH21+ρd3H21-Ωd-1. For power law scale factor, we obtain(39)S˙tot=8π2tn33n2+U23n-112+U2n223n-1+nπtn2+πλt23+Un223n-1. The plot of S˙tot by taking Renyi entropy at apparent horizon is presented by Figure 5. Here α, β, and n have the same values like previous case and λ=1.5. In this case, Stot˙ behaves positively with the passage of time which verifies the validity of GSLT for the present case. Further, differentiating the above equation, we get(40)Stot¨=8π2n33n2+U23n-112+U2n223n-1+n3π-λπ2nt2n2+πλt223+Un223n-1.The plot of this expression is shown in Figure 6 which shows that S¨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.
Plot of Stot˙ by taking Renyi entropy as entropy at apparent horizon, where time is measured in seconds.
Plot of X=Stot¨ by taking Renyi entropy as entropy at apparent horizon, where time is measured in seconds.
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](41)Sh=A4G1-kμA1-μ/2,with kμ=μ(4π)μ/2-1/(4-μ)rc(2-μ); rc is crossover length and μ appears as a constant.(42)Sh˙=πRh˙H2-kμ4-μ4πH21-μ/2.Utilization of (19) in the above equation leads to(43)Sh˙=π2H2-kμ4-μ4πH21-μ/23-11-Ωdρd˙3H3+ρdH2×1+ρd3H21-Ωd-1. Join (22) and (43) to obtain(44)Stot˙=8π2H33H2-ρd-ρd˙3H12-121-Ωdρd˙3H3+ρdH2×1+ρd3H21-Ωd-1+π2H2-kμ4-μ4πH21-μ/2×3-11-Ωdρd˙3H3+ρdH21+ρd3H21-Ωd-1. In the presence of scale factor, the above expression turns out to be(45)Stot˙=8π2tn33n2+U23n-112+U2n223n-1+πt2n2-μnrc/t2-μ3+Un223n-1. By taking power Law entropy at apparent horizon, Stot˙ is plotted at apparent horizon as shown in Figure 7. With the same values for α, β, and n, we have taken μ=5 and rc=2. Here the effectiveness of GSLT at apparent horizon is certified by positive moves of Stot˙ with increasing t. Differentiating the above equation, we get(46)Stot¨=8π2n33n2+U23n-112+U2n223n-1+πtn-πμ2nnrc2-μtμ-13+Un223n-1. 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.
Plot of Stot˙ by taking power law entropy as entropy at apparent horizon, where time is measured in seconds.
Plot of X=Stot¨ by taking power law entropy as entropy at apparent horizon, where time is measured in seconds.
3. Generalized Second Law of Thermodynamics at Event Horizon
In this section, we study GSL of thermodynamics at event horizon which is defined as Rh=a(t)∫t∞dt^/a(t^). Its derivative with respect to time is given by Rh˙=HRh-1. The temperature we used in this section is T=bH/2π, where b is a constant. For the present case, rewriting (4) by using value of T and Rh˙, we have the following equation for entropy inside horizon:(47)Sin˙=-8π2bHRh23H2-ρd-ρd˙3H.
3.1. Bekenstein Entropy
Under this scenario, (24) can be written as(48)Sh˙=2πRhHRh-1.The equation for Stot˙ can be obtained by using (43) and (48) as follows:(49)Stot˙=-8π2bHRh23H2-ρde-ρde˙3H+2πRhHRh-1.By putting values of scale factor and Rh in the above equation, we have(50)Stot˙=-8π2tnn-12b3n2+U23n-1+2πt.Differentiating the above equation with respect to t, we get(51)Stot˙=-8π2nn-12b3n2+U23n-1+2π.
Figure 9 contains the plot of S˙tot by taking Bekenstein entropy at event horizon. Here we have taken α=0.2, β=0.001, and n=4. It is clear from figure that S˙tot remains positive with increasing value of t. This confirms the validity of GSLT at event horizon with Bekenstein entropy. Figure 10 shows that S¨tot<0 for increasing values of n. Hence, at event horizon, the Bekenstein entropy fulfilled the condition of thermodynamic equilibrium.
Plot of Stot˙ by taking Bekenstein entropy as entropy at event horizon, where time is measured in seconds.
Plot of X=Stot¨ by taking Bekenstein entropy as entropy at event horizon.
3.2. Logarithmic Entropy
For this entropy at event horizon, (29) leads to(52)Sh˙=2πH+2ηH+2ξH3πRh˙.By using (47) and (52), the expression of Stot˙ can be written as(53)Stot˙=-8π2bHRh23H2-ρde-ρde˙3H+2πH+2ηH+2ξH3πRh˙.The following equation is obtained by using values of scale factor and Rh(54)Stot˙=-8π2tnn-12b3n2+U23n-1+2πtn-12+2ηt+2ξn-12πt3.Differentiating the above equation with respect to t, we obtain(55)Stot¨=-8π2nn-12b3n2+U23n-1+2πn-12-2ηt2-6ξn-12πt4.
Figure 11 presents the plot of Stot˙ by taking logarithmic entropy at event horizon. Here we have taken η=4 and ξ=6 along with the same values of α, β, and n as in the above-mentioned case. Clearly, Stot˙ 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 Stot¨<0 for n=1.5. Hence, for this case, thermodynamic equilibrium condition holds.
Plot of Stot˙ by taking logarithmic entropy as entropy at event horizon, where time is measured in seconds.
Plot of X=Stot¨ by taking logarithmic entropy as entropy at event horizon, where time is measured in seconds.
3.3. Renyi Entropy
The following form is obtained from (34), by substituting value for Rh˙:(56)Sh˙=2πHH2+λπHRh-1.Joining (47) and (56), we get(57)Stot˙=-8π2bHRh23H2-ρde-ρde˙3H+2πHH2+λπHRh-1.By using values of scale factor and Rh, the above equation reduces to(58)Stot˙=-8π2tnn-12b3n2+U23n-1+2πtn-12+λπt2.Differentiating the above equation with respect to t, we get(59)Stot¨=-8π2nn-12b3n2+U23n-1+2πn-12-2π2λt2n-12+λπt22.The plot of Stot˙ for Renyi entropy at event horizon is presented in Figure 13.
Plot of Stot˙ by taking Renyi entropy as entropy at event horizon.
Here α, β, and n have the same values like the previous case while λ=1.5. In this case, Stot˙ behaves positively with the passage of time which verifies the validity of GSLT. Figure 14 shows that the trajectories of Stot¨ remain negative for increasing of t with n=1.5. This means that the present scenario obeys the condition for thermodynamic equilibrium.
Plot of X=Stot¨ by taking Renyi entropy as entropy at event horizon.
3.4. Power Law Entropy
Under conditions of present section, (41) reduces to(60)Sh˙=πH2-kμ4-μ4πH21-μ/2HRh-1.Joining (47) and (60), we get the following equation:(61)Stot˙=-8π2bHRh23H2-ρde-ρde˙3H+πH2-kμ4-μ4πH21-μ/2HRh-1.Inserting conditions for scale factor and Rh in the above equation, we get(62)Stot˙=-8π2tnn-12b3n2+U23n-1+2-μtrcn-12-μπtn-12.
The plot of this expression is displayed in Figure 15 with the same values for α, β, and n while μ=5 and rc=2. Here the effectiveness of GSLT at event horizon is certified by positive moves of Stot˙ with increasing t. Differentiating with respect to t, we obtain(63)Stot¨=-8π2nn-12b3n2+U23n-1+2πn-12-μπ3-μt2-μrcn-12-μn-12.Figure 16 shows that the present scenario fulfils the thermodynamic equilibrium condition for power law entropy at event horizon.
Plot of Stot˙ by taking power law entropy as entropy at event horizon.
Plot of X=Stot¨ by taking power law entropy as 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 Stot˙ 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≥1.5. However, Figure 4 shows that X=Stot¨<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 S˙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, Stot˙ is plotted at apparent horizon as shown in Figure 7. Here the effectiveness of GSLT at apparent horizon is certified by positive moves of Stot˙ 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 Stot¨<0 for n=1.5 which leads to the validity of thermal equilibrium condition.
The plot of Stot˙ for Renyi entropy at event horizon is presented in Figure 13. It is observed that Stot˙ behaves positively with the passage of time which verifies the validity of GSLT. Figure 14 shows that the trajectories of Stot¨ 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 Stot˙ 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.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
DaviesP. C.Cosmological horizons and entropy198851013491355MR96498010.1088/0264-9381/5/10/013Zbl0646.53076HuangQ.LiM.The holographic dark energy in a non-flat universe20040408013IzquierdoG.PavonD.Dark energy and the generalized second law20066334-542042610.1016/j.physletb.2005.12.040MR2195773Zbl1247.83275BekensteinJ. D.Black holes and entropy1973723332346MR036634310.1103/PhysRevD.7.2333Zbl1369.830372-s2.0-33750485765HawkingS. W.Particle creation by black holes1975433199220Erratum in: Communications in Mathematical Physics, vol. 46, p. 206, 197610.1007/bf02345020Zbl06700760PadmanabhanT.Gravity and the thermodynamics of horizons200540624912510.1016/j.physrep.2004.10.003MR2115111PadamanabhanT.20107046901GibbonsG. W.HawkingS. W.Cosmological event horizons, thermodynamics, and particle creation1977151027382751MR045947910.1103/PhysRevD.15.27382-s2.0-33646044994PollockM. D.SinghT. P.On the thermodynamics of de SITter spacetime and quasi-de SITter spacetime198966901909MR99904310.1088/0264-9381/6/6/014Zbl1248.83041BrusteinR.Generalized second law in cosmology from causal boundary entropy2000841020722075MR174503910.1103/PhysRevLett.84.2072Zbl0949.83084PavonD.The generalised second law and extended thermodynamics19907348749110.1088/0264-9381/7/3/022MR1038627HooftT.Dimensional reduction in quantum gravity[gr-qc/9310026]SusskindL.The world as a hologram1995361163776396MR135591310.1063/1.531249Zbl0850.000132-s2.0-33645954409WittenE.Anti de Sitter space and holography199822253291MR1633012Zbl0914.5304810.4310/ATMP.1998.v2.n2.a2BoussoR.The holographic principle200274382587410.1103/RevModPhys.74.825MR1925130Zbl1205.830252-s2.0-0036659444EastherR.LoweD.Holography, cosmology, and the second law of thermodynamics1999822549674970MR169741110.1103/PhysRevLett.82.4967Zbl0951.830612-s2.0-0001326090KaloperN.LindeA.Cosmology versus holography19996010103509, 7MR1757604de AlwisS. P.Brane worlds in 5D and warped compactifications in IIB20046033-4230238MR2105004GaoC.WuF.ChenX.ShenY.-G.Holographic dark energy model from Ricci scalar curvature20097904304351110.1103/PhysRevD.79.043511GrandaL. N.OilverosA.Phys. LettB.Infrared cut-off proposal for the Holographic density2008671275277ChattopadhyayS.Interacting Ricci dark energy and its statefinder description2011126130MathewT. K.SureshJ.DivakaranD.Modified holographic ricci dark energy model and state finder diagnosis in flat universe2013221350056BergerM. S.ShojaeiH.Interacting dark energy model for the expansion history of the Universe200674410.1103/PhysRevD.74.043530WangB.GongY.AbdallaE.Transition of the dark energy equation of state in an interacting holographic dark energy model20056243-4141146UmezuS.HattaT.OhmoriH.Fundamental characteristics of bioprint on calcium alginate gel201352510.7567/JJAP.52.05DB2005DB202-s2.0-84880906191ZhouJ.WangB.GongY.AbdallaE.The generalized second law of thermodynamics in the accelerating universe20076522-3869110.1016/j.physletb.2007.06.067MR2335905Zbl1248.83190SetareM. R.Interacting holographic dark energy model and generalized second law of thermodynamics in non-flat universe20070230701SheykhiA.WangB.Generalized second law of thermodynamics in Gauss-Bonnet braneworld200967854344372-s2.0-6774913554610.1016/j.physletb.2009.06.075SheykhiA.WangB.Generalized second law of thermodynamics in warped {DGP} braneworld2010251411991210MR2652999Zbl1193.83049WongK. M.Study of the electronic structure of individual free-standing Germanium nanodots using spectroscopic scanning capacitance microscopy200948810.1143/JJAP.48.085002085002ZlatevI.WangL.SteinhardtP. J.Quintessence, cosmic coincidence, and the cosmological constant199982589689910.1103/PhysRevLett.82.8962-s2.0-0001544612KopeliovichB. Z.TarasovA. V.Gluon shadowing in heavy flavor production off nuclei20027101-218021710.1016/s0375-9474(02)01124-7SahniV.The cosmological constant problem and quintessence200219133435344810.1088/0264-9381/19/13/3042-s2.0-0012856318Zbl1003.83049ChibaT.OkabeT.YamaguchiM.Kinetically driven quintessence20006210.1103/PhysRevD.62.023511023511SetareM.Interacting generalized Chaplygin gas model in non-flat universe200752368969210.1140/epjc/s10052-007-0405-5BernardiniA. E.BertolamiO.Lorentz violating extension of the standard model and the200877810.1103/PhysRevD.77.085032HsuS. D. H.Entropy bounds and dark energy20045941-2131610.1016/j.physletb.2004.05.020LiM.A model of holographic dark energy20046031-21510.1016/j.physletb.2004.10.014BambaK.CapozzielloS.NojiriS.OdintsovS. D.Dark energy cosmology: the equivalent description via different theoretical models and cosmography tests2012342155228AmorósJ.De HaroJ.OdintsovS. D.Bouncing loop quantum cosmology from F(T) gravity2013871010403710.1103/PhysRevD.87.1040372-s2.0-84878531636LinderE. V.Erratum: Einstein’s other gravity and the acceleration of the universe [Phys. Rev. D 81, 127301 (2010)]2010821010990210.1103/physrevd.82.109902JamilM.MomeniD.MyrzakulovR.Stability of a non-minimally conformally coupled scalar field in F(T) cosmology201272, article 207510.1140/epjc/s10052-012-2075-1BaffouE. H.KpadonouA. V.RodriguesM. E.HoundjoM. J. S.TossaJ.Cosmological viable f(R,T) dark energy model: dynamics and stability2015356117318010.1007/s10509-014-2197-z2-s2.0-84957539870HoundjoM. J. S.PiattellaO. F.Reconstructing f(R, T) gravity from holographic dark energy201221310.1142/S02182718125002411250024Zbl1250.83048NojiriS.OdintsovS. D.Modified Gauss–Bonnet theory as gravitational alternative for dark energy20056311-21610.1016/j.physletb.2005.10.010Zbl1247.83292NojiriS.OdintsovS. D.ToporenskyA.TretyakovP.Reconstruction and deceleration-acceleration transitions in modified gravity201042819972008MR267045610.1007/s10714-010-0977-5Zbl1197.830972-s2.0-77954762339BambaK.OdintsovS. D.SebastianiL.ZerbiniS.Finite-time future singularities in modified Gauss-Bonnet and f(R,G) gravity and singularity avoidance201067129531010.1140/epjc/s10052-010-1292-82-s2.0-77951879898KofinasG.LeonG.SaridakisE. N.Dynamical behavior in F(T,TG) cosmology2014311717501110.1088/0264-9381/31/17/1750112-s2.0-84906751876KofinasG.SaridakisE. N.Cosmological applications of F(T,TG) gravity201490808404510.1103/physrevd.90.084045ChattopadhyayS.JawadA.MomeniD.MyrzakulovR.Pilgrim dark energy in F(T,TG) cosmology2014353127929210.1007/s10509-014-2029-1JawadA.RaniS.ChattopadhyayS.Modified QCD ghost f(T,TG) gravity201536037JawadA.DebnathU.New agegraphic pilgrim dark energy in f(T,TG) gravity201564214515010.1088/0253-6102/64/2/145MR3445726JawadA.MajeedA.Correspondence of pilgrim dark energy with scalar field models2015356375381HarkoT.LoboF. S. N.OtaloraG.SaridakisE. N.F(T, T) gravity and cosmology2014201412, article no. 0212-s2.0-8491851977910.1088/1475-7516/2014/12/021SalakoI. G.JawadA.ChattopadhyayS.Holographic dark energy reconstruction in (Formula presented.)gravity20153581, article no. 132-s2.0-8493093910210.1007/s10509-015-2406-4GuptaG.SaridakisE. N.SenA. A.Nonminimal quintessence and phantom with nearly flat potentials2009791210.1103/PhysRevD.79.123013CaiY-F.20104931SaridakisE. N.Gonzalez-DiazP. F.SiguenzaC. L.Unified dark energy thermodynamics: varying w and the -1-crossing20092615165003JawadA.MajeedA.Correspondence of pilgrim dark energy with scalar field models2015356237538110.1007/s10509-014-2206-2JawadA.Cosmological analysis of pilgrim dark energy in loop quantum cosmology201575206JawadA.RaniS.SaleemM.Cosmological study of reconstructed f(T) models20173624, article no. 632-s2.0-8501461995610.1007/s10509-017-3040-0JawadA.RaniS.SalakoI. G.GulshanF.Pilgrim dark energy models in fractal universe2017261750049MR3651194JawadA.RaniS.AzharN.Entropy corrected holographic dark energy models in modified gravity2017261750040JawadA.RaniS.SalakoI. G.GulshanF.Cosmological study in loop quantum cosmology through dark energy model20172621750007, 16MR3605873Zbl1359.83040RaniS.JawadA.SalakoI. G.AzharN.Non-flat pilgrim dark energy FRW models in modified gravity201636128610.1007/s10509-016-2868-z2-s2.0-84981263394JawadA.ChattopadhyayS.RaniS.Viscous pilgrim f(T) gravity models20163617, article no. 2312-s2.0-8497584261210.1007/s10509-016-2814-0SaridakisE. N.Ricci-Gauss-Bonnet holographic dark energyhttps://arxiv.org/abs/1707.09331MeissnerK. A.Black-hole entropy in loop quantum gravity200421225245525110.1088/0264-9381/21/22/015MR2103252Zbl1062.830562-s2.0-10044267716GoshA.MitraP.Log correction to the black hole area law 2005200471027502ChatterjeeA.MajumdarP.Universal canonical black hole entropy2004921410.1103/physrevlett.92.141301141301MR2069487Zbl1267.83053BonerjeeR.ModakS. K.Exact differential and corrected area law for stationary black holes in tunneling method2009090536ModakS. K.Corrected entropy of BTZ black hole in tunneling approach2009671116717310.1016/j.physletb.2008.11.043MR2487977JamilM.FarooqM. U.Interacting holographic dark energy with logarithmic correction20102010300110.1088/1475-7516/2010/03/001SadjadiH. M.JamilM.2010969001BiroT. S.CzinnerV. G.A q-parameter bound for particle spectra based on black hole thermodynamics with Renyi entropy20137264-5861865MR3118019KomatsuN.CJ.201777DasS.ShankaranarayananS.SurS.Power-law corrections to entanglement entropy of horizons200877610.1103/PhysRevD.77.064013RadicellaN.PavonD.Phys. LettB.2010121HawkingS. W.Particle creation by black holes1975433199220MR038162510.1007/BF02345020Zbl06700760JacobsonT.Thermodynamics of spacetime: the Einstein equation of state199575126010.1103/PhysRevLett.75.1260Zbl1020.83609PadmanabhanT.Classical and quantum thermodynamics of horizons in spherically symmetric spacetimes2002192153875408MR193992310.1088/0264-9381/19/21/306Zbl1011.830182-s2.0-0036417749CaiR. G.KimS. P.First law of thermodynamics and Friedmann equations of Friedmann-Robertson-Walker universe20052005, article 05010.1088/1126-6708/2005/02/050BanerjeeR.ModakS. K.Quantum tunneling, blackbody spectrum and non-logarithmic entropy correction for Lovelock black holes200920091107307310.1088/1126-6708/2009/11/073HeH.-X.Gauge symmetry and transverse symmetry transformations in gauge theories2009522292294MR258294010.1088/0253-6102/52/2/21Zbl1183.81125BanerjeeS.GuptaR. K.SenA.Logarithmic corrections to extremal black hole entropy from quantum entropy function20111471103SheykhiA.JamilM.Power-law entropy corrected holographic dark energy model2011431026612672MR284336310.1007/s10714-011-1190-xZbl1228.831482-s2.0-80053101689KaramiK.GhaffariS.SoltanzadehM. M.The generalized second law of gravitational thermodynamics on the apparent and event horizons in FRW cosmology201027202-s2.0-7864957948510.1088/0264-9381/27/20/205021205021Zbl1202.83134KaramiK.Comment on ''Interacting holographic dark energy model and generalized second law of thermodynamics in a non-flat universe", by M.R. Setare (JCAP 01 (2007) 023)201020101, article no. 0152-s2.0-7614913542510.1088/1475-7516/2010/01/015SheykhiA.Thermodynamics of interacting holographic dark energy with the apparent horizon as an IR cutoff201027210.1088/0264-9381/27/2/025007025007MR2578710Zbl1184.83074MazumderN.ChakrabortyS.Validity of the generalized second law of thermodynamics of the universe bounded by the event horizon in holographic dark energy model2010424813820MR260278710.1007/s10714-009-0881-zZbl1187.830812-s2.0-77950461401JamilM.SaridakisE. N.SetareM. R.Thermodynamics of dark energy interacting with dark matter and radiation2010812610.1103/PhysRevD.81.023007023007KaramiK.AbdolmalekiA.SahraeiN.GhaffariS.Thermodynamics of apparent horizon in modified FRW universe with power-law corrected entropy20111501108KaramiK.201193JamilM.SaridakisE. N.SetareM. R.The generalized second law of thermodynamics in Hořava-Lifshitz cosmology201011, article 322010.1088/1475-7516/2010/11/032MR2741452BambaK.GengC.-Q.TsujikawaS.Thermodynamics in modified gravity theories2011208136310.1142/S02182718110195422-s2.0-79961155971BambaK.GengC.TsujikawaS.Physics Letters B2010688101Zbl1263.83123RaniS.NawazT.JawadA.Thermodynamics in dynamical Chern-Simons modified gravity with canonical scalar field20163619285MR3533543