Thermodynamical study of FRW universe in Quasi-Topological Theory

By applying the unified first law of thermodynamics on the apparent horizon of FRW universe, we get the entropy relation for the apparent horizon in quasi-topological gravity theory. Throughout the paper, the results of considering the Hayward-Kodama and Cai-Kim temperatures are also addressed. Our study shows that whenever, there is no energy exchange between the various parts of cosmos, we can get an expression for the apparent horizon entropy in quasi-topological gravity, which is in agreement with other attempts followed different approaches. The effects of a mutual interaction between the various parts of cosmos on the apparent horizon entropy as well as the validity of second law of thermodynamics in quasi-topological gravity are also perused.

The thermodynamic consequences of such interactions in the Einstein framework are addressed in Ref. [27]. Recently, it is argued that since the origin of dark energy is unknown, some dark energy candidates may affect the Bekenstein entropy of apparent horizon in a flat FRW universe [28][29][30][31]. In addition, by following [32], one can see that all of the dark energy candidates and their interaction with other parts of cosmos may affect the apparent horizon of a flat FRW universe in the Einstein framework. There are additional terms to the Einstein tensor in modified theories of gravity. Since these additional terms can be interpreted as a geometrical fluid, a mutual interaction between the dark sides of cosmos may be an interaction between these geometrical and material terms [10]. Therefore, one may expect that such modifications to Einstein theory and their interaction with other parts of cosmos may also affect the horizon entropy. Indeed, there are some new attempts in which authors give a positive answer to this expectation and show that such geometrical fluids and their interaction with other parts of cosmos may also affect the horizon entropy of FRW universe [33][34][35][36].
String theory together with AdS/CFT correspondence conjecture give us the motivation to study spacetimes with dimensions more than 4 [37][38][39]. Moreover, brane scenarios also encourage us to study (4 + 1)-dimensional spacetimes [40,41]. The backbone of Einstein-Hilbert action is producing the second order field equations of motion. In looking for a suitable lagrangian for higher dimensions, which also keeps the field equations of motion for metric of second-order, one reaches to Lovelock lagrangian instead of the usual (n + 1) version of Einstein-Hilbert action [42]. It is shown that the corresponding gravitational field equations of static spherically symmetric spacetimes in the Einstein and Lovelock theories are parallel to the availability of the first law of thermodynamics on the horizons of considered metric [43]. For a FRW universe, where the field equations governs the universe expansion history, one can also reach at the corresponding Friedmann equations in the Einstein and Lovelock theories by applying the first law of thermodynamics on the apparent horizon of FRW universe [44]. Indeed, such generalization of Einstein-Hilbert action leads to additional terms besides the Einstein tensor in spacetimes with dimensions more than 4. Authors in [33] interprets such additional terms as the geometrical fluids and show that these terms and their interactions with other parts of cosmos affect the horizon entropy.
By considering cubic and higher curvature interactions, some authors try to find new lagrangian which keeps the field equation of motion for the metric of second-order [45][46][47]. Their theory is now called quasi-topological gravity which affects the gravitational field equations in spacetime with dimensions more than 4. Thermodynamics of static black holes are studied in this theory which leads to an expression for the horizon entropy [45][46][47].
Thermodynamics of static black holes are studied in this theory which leads to an expression for the horizon entropy [46][47][48][49][50]. Moreover, it is shown that if one assumes that the black hole entropy relation, derived in [46][47][48][49][50], is available for the apparent horizon of FRW universe, then by applying the unified first law of thermodynamics on the apparent horizon of FRW universe, one can get the corresponding Friedmann equations in quasi-topological gravity [51]. Thereinafter, authors point out the second law of thermodynamics and its generalized form [51]. In fact, this scheme proposes that, in quasi-topological gravity, one may generalize the black hole entropy to the cosmological setup by inserting the apparent horizon radii instead of the event horizon radii. Is it the only entropy relation for the apparent horizon? Does a mutual interaction between the geometrical fluid, arising from the terms besides the Einstein tensor in the field equations, and other parts of cosmos affect the horizon entropy?
Our aim in this paper is to study the thermodynamics of apparent horizon of FRW universe in the quasi-topological gravity theory to provide proper answers for the above mentioned questions. In this investigation, we point to the results of considering the Hayward-Kodama and Cai-Kim temperatures for the apparent horizon. In order to achieve this goal, by looking at the higher order curvature terms, arising in quasi-topological gravity, as a geometrical fluid and applying the unified first law of thermodynamics on the apparent horizon, we get an expression for the apparent horizon entropy in this theory, while the geometrical fluid does not interact with other parts of cosmos. Moreover, we focus on a FRW universe in which the geometrical fluid interacts with other parts of cosmos. This study signals us to the effects of such mutual interaction on the apparent horizon entropy of a FRW universe with arbitrary curvature for quasi-topological gravity theory. As the non-interacting case, the second law of thermodynamics is also studied in an interacting universe.
The paper is organized as follows. In the next section, we give an introductory note about the quasi-topological gravity, the corresponding Friedmann equations and some properties of FRW universe, including its apparent horizon radii, surface gravity and thus the corresponding Hayward-Kodama temperature. Section(III) devotes to a breif discussion about the Unified First Low of thermodynamics (UFL). In section (IV), bearing the Hayward-Kodama temperature together with the unified first law of thermodynamics in mind, we study the effect of a geometrical(curvature) fluid on the apparent horizon entropy in a FRW universe with arbitrary curvature parameter for Quasi-topological theory. The results of considering the Cai-Kim temperature are also addressed. Thereinafter, we generalize our study to a universe in which the geometrical fluid interacts with other parts of cosmos. The results of attributing the Cai-Kim temperature to the apparent horizon in an interacting universe are also pointed out in section (IV). Section (V) includes some notes about the availability of second law of thermodynamics in an interacting cosmos described by quasi-topological gravity theory. The final section is devoted to summary and concluding remarks.

II. A BREIF OVERVIEW OF QUASI-TOPOLOGICAL GRAVITY
A natural generalization of the Einstein-Hilbert action to higher dimensional spacetime, and higher order gravity with second order equation of motion, is the Lovelock action The action of 4-th order quasi-topological gravity in (n + 1) dimensions can be written as follows which not only works in five dimensions but also yield second-order equations of motion for spherically symmetric spacetimes.
In action (2), L 1 = R is just the Einstein-Hilbert Lagrangian, L 2 = R abcd R abcd −4R ab R ab + R 2 is the second order Lovelock (Gauss-Bonnet) Lagrangian and L matter is the Lagrangian of the matter field. X 3 is the curvature-cubed Lagrangian given by [46] and X 4 is the fourth order term of quasi-topological gravity [47] the coefficients c i in the above term are given by   In the context of universal thermodynamics, our universe should be a non-stationary gravitational system while from the cosmological point of view, it should be homogeneous and isotropic. Therefore, the natural choice is the FRW Universe, a dynamical spherically symmetric spacetime, having only inner trapping horizon (the apparent horizon), which is described by the line element ) and dΩ 2 is the metric of (n − 1)-dimensional unit sphere.
Varying the action (2) with respect to metric leads to [51] This is the Friedmann equation of arbitrary order quasi-topological cosmology where H is the Hubble parameter. From now on, we sets G n+1 = 1 for simplicity. Moreover,μ i 's are dimensionless parameters as followŝ The dynamical apparent horizon, is determined by the relation h ab ∂ ar ∂ br = 0, wherer = a(t)r . It is a matter of calculation to show that the radius of the apparent horizon for the FRW universe is [52] In cosmological context, various definitions of temperature are used to get the corresponding  [53][54][55][56][57][58]. Therefore, the surface gravity at the apparent horizon of the FRW universe has the following which leads to for the Hayward-Kodama temperature of apparent horizon. Then, using the Cai-Kim temperature and the Clausius relation we get the entropy of apparent horizon. In this approach, the horizon temperature is [44,61] and for an infinitesimal time interval, the horizon radius will have a small change and we can use the dr A = 0 approximation [44,61].

III. UNIFIED FIRST LAW OF THERMODYNAMICS AND HORIZON EN-TROPY
The unified first law of thermodynamics can be expressed as [52,59,60] dE = AΨ + W dV (11) where E is the total baryonic energy content of the universe inside an n-sphere of volume V , while A is the area of the horizon. The energy flux Ψ is termed as the energy supply vector and W is the work function which are respectively defined as The term W dV in the first law comes from the fact that we have a volume change for the total system enveloped by the apparent horizon. For a pure de Sitter space, ρ = p, and the work term reduces to the standard pdV , thus we obtain exactly the standard first law of It is a matter of calculation to show that Eq. (12) can be written as and thus The Friedmann equation of FRW Universe in quasi-topological gravity represents by (6). In modified gravity theories, the modified Friedmann equations can be written as In the above equations ρ t is the total energy density which includes of non-interacting two fluid systems, one is the usual fluid of energy density ρ and thermodynamic pressure p, while the second one is termed as effective energy density ρ e due to curvature contributions and its corresponding pressure p e . So, we have where and where we have defined ε =˙r A 2Hr A , in whichṙ A = dr A dt , and in terms of the horizon radiusr A , we have [63] Finally, we should note that by combining Eqs. (19) and (20) with each other, we geṫ ρ e +nH(ρ e +p e ) = 0. It means that Eqs. (19) and (20) are only valid if theρ e +nH(ρ e +p e ) = 0 andρ + nH(ρ + p) = 0 conditions be simultaneously available. These results demonstrate that there is no energy exchange between the geometrical and material fluids.

A. Non-interacting case
The terms due to the curvature, formally play the role of a further source term in the field equations which effect is the same as that of an effective fluid of purely geometrical origin.
Indeed, since such terms modify the Einstein theory, they lead to change the Bekenstein relation for the entropy of black hole horizon [46][47][48][49], which is in agreement with the result obtained by applying the first law of thermodynamics on the black hole horizon [50]. In order to study the thermodynamics of apparent horizon of FRW universe in quasi-topological gravity, authors in [51] assumed that the black hole entropy expression, derived in [46][47][48][49][50], is also valid for the apparent horizon of FRW universe. In addition, by applying the UFL of thermodynamics on the apparent horizon they could get the Friedmann equation in the quasi-topological gravity [51]. In fact, their recipe is a way of proposing an expression for the apparent horizon entropy instead of a way for deriving the corresponding entropy. Therefore, the inverse of their recipe may indeed be considered theoretically as a more acceptable way of getting the apparent horizon entropy. Finally, based on approach [51], it seems that their proposal for the apparent horizon entropy is independent of any interaction between the cosmos sectors. It is also worthy to note that the mutual interaction between the geometrical (T e µν ) and non-geometrical (T m µν ) sectors of cosmos in various modified theories of gravity, affect the apparent horizon entropy [33][34][35][36]. In fact, depending on the scales, it is such a curvature fluid which may play the role of dark energy. In Einstein general relativity framework, a dark energy candidate and its interaction with other parts of cosmos affects the Bekenstein entropy [28][29][30][31][32]. Therefore, the results obtained in [33][34][35][36] are in line with those of [28][29][30][31][32]. In order to get the effects of mutual interaction between the geometrical and non-geometrical parts of cosmos, we need to establish a recipe to find out the horizon entropy by starting from the Friedmann equations and applying the UFL of thermodynamics on the apparent horizon of FRW universe. The energy conservation law leads to the continuity equation in the forṁ For a non interacting system it breaks down tȯ ρ + nH(ρ + p) = 0 (23) ρ e + nH(ρ e + p e ) = 0, which implies that Eqs. (19) and (20) are available. Differentiating Eq. (16), we reach at Bearing Eq. (23) in mind, we have Multiplying both sides of the above equation in (−T ), one obtains Assuming the total energy content of the universe inside an n-sphere of radiusr A is E = ρV , where V = Ω nr n A is the volume enveloped by an n-dimensional sphere. Taking differential form of the total energy, after using the continuity equation (23), we obtain which leads to Substituting Eq. (28) into (26), simple calculations leads to and consequently, where we have used dV = nΩ nr n−1 A dr A . In this equation the work density W = ρ−p 2 is regarded as the work done when the apparent horizon radius changes fromr A tor A + dr A .
The Clausius relation is [32] T dS A = δQ m = AΨ, where δQ m is the energy flux crossing horizon during the universe expansion. Eqs. (11), (30) and (31), we have Using Eq. (23) leads to Now, by either inserting Eq. (19) into Eq. (32) or Eq. (20) into Eq. (33), and integrating the result, we get for the horizon entropy. This result is in full agreement with previous proposal in which authors assumed that the event horizon entropy is extendable to the apparent horizon of FRW universe [51]. S 0 is an integration constant and we can consider it zero without lose of generality. It is also useful to mention here that we considered the Clausius relation in the form T dS A = δQ m to obtain this result, while authors in [51] took into account the T dS A = −δQ m form of the Clausius relation. This discrepancy is due to our different way of defining the horizon temperature. We have used the Hayward-Kodama definition of temperature (9), while authors in [51] used its absolute value to avoid attributing negative temperatures on the apparent horizon and thus the Hawking radiation. where δQ m is the energy flux crossing the horizon during the infinitesimal time interval dt, we get Substituting dρ from Eq. (24) into the above equation, we are led to where we have used V = Ω nr n A and the Cai-Kim temperature (T = 1 2πr A ) [44,61]. Finally, we obtain Consequently the modified entropy on the event horizon has the explicit form Which is compatible with the results obtained by considering the Hayward-Kodama temperature (34). As before, since entropy is not an absolute quantity, without lose of generality we can set S 0 to zero. Setting n = 3 and i = 1 in the above result, leads to the Bekenstein-Hawking entropy formula.

B. Interacting universe
In order to take the interaction into account, for the quasi-topological gravity, consider the energy-momentum conservation law in the formṡ andρ e + nH(ρ e + p e ) = −Q, where Q is the mutual interaction between different parts of the cosmos. The latter means that Eqs. (19) and (20) are not simultaneously valid for an interacting universe. Using Eqs. (24) and (39), we get Multiplying both sides of the above equation by (−T ), one obtains where we have usedṙ A dt = adr A + r A Hdt. Multiplying the result by the factor n(n−1)Ωnr n+1 A 4 , we arrive at Simple calculations leads to The right hand side of Eq. (44) is nothing but the energy flux (δQ m ) crossing the apparent horizon (14). Bearing the Clausius relation (31) in mind, we get Now, using Eq. (40) to obtain which leads to which is in agreement with the results obtained by authors [33] for the Lovelock theory but in a different way. In fact, since in the interacting universe ρ e and ρ e +p e cannot simultaneously meet Eqs. (19) and (20), respectively, we can not use them to integrate from the RHS of Eqs. (45) and (46) In the above equation, first two terms of the right hand side is exactly the same as Eq. (37).
In other words, It is crystal clear that the second term of RHS of this equation counts the effects of mutual interaction between the geometrical and non-geometrical fluids on the horizon entropy.
Loosely speaking, the above expression for horizon entropy is no longer the usual Bekenstein entropy formula, rather there are two correction terms which are due to higher order curvature terms, arising from the quasi-topological nature of gravity theory, along with the mutual interaction between the geometrical and non-geometrical parts. Note that in this situation, due to equation Eq. (40), the ρ e + p e term does not meet Eq. (20). As another example, consider a situation in which ρ e + p e satisfies Eq. (20), which also means that, due to equation Eq. (40), ρ e does not obey Eq. (19). For this case, by combining Eqs. (40) and (45), and integrating the result, we obtain which means that the mutual interaction Q does not affected the horizon entropy in this special case.

The Cai-Kim approach
The rest of this section devotes to achieve Eq. (46) by using the Cai-Kim approach [44,61]. For an interacting universe the UFL is in the form [31] dS Using Eqs. (24) and (40), we have Substituting it into Eq. (51), leads to which is in full agreement with Eq. (46).

V. SECOND LAW OF THERMODYNAMICS
The time evolution of the entropy in a non-interacting universe governed by quasitopological gravity, is already considered by authors in [51]. Now, we are interested in examining the validity of the second law of thermodynamics in an interacting universe.
This low states that the horizon entropy should meet the dS A dt ≥ 0 condition [62]. For this propose, from Eq. (46) we have However, using Eqs. (40) and (41), one getṡ which leads to the Raychaudhuri equatioṅ Substituting it into Eq. (54), we arrive at The above result indicate that the second law of thermodynamics ( dS A dt ≥ 0) is valid for the apparent horizon under the condition (ρ+p) > 0 and if we define the state parameter ω = p ρ , this law is valid whenever the state parameter obeys ω ≥ −1.
After giving a brief review of the quasi-topological gravity theory, by considering a FRW universe, we pointed out to the corresponding Friedmann equation in this modified gravity theory. In section (II), we mentioned the apparent horizon of FRW universe as the proper causal boundary, its surface gravity and the Hayward-Kodama temperature of this hypersurface. Motivated by recent works on the thermodynamics of horizon of FRW universe in some modified gravity theories [33][34][35][36], we considered the terms other than Einstein tensor as a fluid with energy density ρ e and pressure p e . In fact, since these terms may play the role of dark energy and because a dark energy candidate may also affect the horizon entropy [28][29][30][31][32], such justification at least is not forbidden. Then, we showed that if we either use the Hayward-Kodama or the Cai-Kim temperature, the same result for the horizon entropy is obtainable (Eqs. (34) and (38)) which is in agreement with previous work [50], in which authors by using different definition for the horizon temperature and generalizing the black hole entropy to the cosmological horizon setup, could propose the same expression as that of us for the apparent horizon entropy in quasi-topological gravity. Moreover, it is useful to note that Eq. (34) is valid only if there is no energy-momentum exchange between the geometrical (T e µν ) and non-geometrical (T m µν ) fluids. Finally, we generalized our investigation to an interacting FRW universe with arbitrary curvature parameter (k) in quasi-topological gravity. We got a relation for the horizon entropy (Eq. (47)) which showed the effects of an energy-momentum exchange between the geometrical and non-geometrical fluids on the apparent horizon entropy. Thereinafter, we studied the special case in which ρ e meets Eq. (19) to perceive clearly the effects of such mutual interaction on the horizon entropy. We got Eq. (49) for the horizon entropy, which showed two terms besides the Bekenstein entropy due to higher order curvature terms, arising from the quasi-topological nature of model, and mutual interaction between the geometrical and non-geometrical fluids. Our investigation also showed that whenever ρ e + p e satisfies Eq. (20), the mutual interaction Q does not affect the horizon entropy and therefore, the result of non-interacting case is obtainable. We have also pointed out the validity of second law of thermodynamics in an interacting case and found out that, independent of curvature parameter (k), whenever ρ + p ≥ 0, the second law is obtainable.
Our study shows that, in quasi-topological gravity theory, a geometrical fluid with energy density ρ e and pressure p e , independent of its nature, affects the apparent horizon entropy as S A = A 4 − 2πnΩ n Hr n+1 A (ρ e + p e )dt, (58) which is in full agreement with other studies [28][29][30][31][32][33][34][35][36]. Since such correction in quasitopological gravity has geometrical nature, it is indeed inevitable. Moreover, the effects of mutual interaction between the dark energy candidate and other parts of cosmos is stored in the second term of RHS of this equation. Indeed, based on our approach, one may find the apparent horizon entropy of FRW universe with arbitrary curvature parameter (k) in modified theories of gravity by interpreting the terms other than Einstein tensor in the corresponding Friedmann equation as a geometrical fluid and applying the UFL of thermodynamics on the apparent horizon. At the end, we should stress that in the absence of a mutual interaction, the results of interacting universes in quasi-topological theory converge to those of the non-interacting case.