Apparent horizon and gravitational thermodynamics of Universe in the Eddington-Born-Infeld theory

The thermodynamics of Universe in the Eddington-Born-Infeld (EBI) theory was restudied by utilizing the holographic-style gravitational equations that dominate the dynamics of the cosmical apparent horizon $\Upsilon_{A}$ and the evolution of Universe. We started in rewriting the EBI action of the Palatini approach into the Bigravity-type action with an extra metric $q_{\mu\nu}$. With the help of the holographic-style dynamical equations, we discussed the property of the cosmical apparent horizon $\Upsilon_{A}$ including timelike, spacelike and null characters, which depends on the value of the parameter of state $w_{m}$ in EBI Universe. The unified first law for the gravitational thermodynamics and the total energy differential for the open system enveloped by $\Upsilon_{A}$ in EBI Universe were obtained. Finally, applying the positive-heat-out sign convention, we derived the generalized second law of gravitational thermodynamics in EBI universe.


Introduction
Gravitational thermodynamics is quite an interesting question, which has attracted much attention. Recently, many studies have covered both the first and second laws of gravitational thermodynamics for the Friedmann-Robertson-Walker (FRW) Universe with a generic spatial curvature. The inspired work is the first law of thermodynamics for Universe by Cai and Kim [1], which is a part of the effort to seek the connections between thermodynamics and gravity [2] after discovering the black-hole thermodynamics [3,4]. In [1,5], Akbar and Cai reversed the formulation by rewriting the Friedmann equations into the heat balance equation and the unified first law of thermodynamics at the cosmical apparent horizon, for General Relativity (GR), Gauss-Bonnet, and Lovelock gravity. The results in [5] were soon generalized to other theories of gravity, such as the scalar-tensor gravity [6], ( ) gravity [7], braneworld scenarios [8][9][10], generic ( , , ∇ ∇ ) gravity [11], and Horava-Lifshitz gravity [12,13], to construct the effective total energy differentials by the corresponding modified Friedmann equations.
Inspired by the gravitational thermodynamics in these gravitational theories [5][6][7][8][9][10][11][12][13][14] and characteristics of the EBI action, we focused on generalizing the results to the EBI gravity. The Eddington-Born-Infeld action (EBI) was proposed in [15], which could mimic the presence of dark energy and dark matter in the expansion of Universe [15][16][17] so that EBI gravity can be regarded as a candidate for nonparticulate dark matter and dark energy, and it could also modify the Newton-Poisson equation that leads to flat rotation curves for galaxies. Generally, the EBI action is a Palatini-type action where the metric ] is not associated with the connection ] . However, by defining an extra metric ] to satisfy the condition in [15,18], the EBI action can be rewritten as the Bigravity-type action.
In this paper, we derived the holographic-style dynamical equations and discussed the properties of the cosmical apparent horizon Υ in EBI Universe, which rely on the contents inside the cosmical apparent horizon including the matter and the dark energy provided by the cosmological constant Λ and the spacetime self-coupling. Furthermore, we applied the Misner-Sharp energy, the Cai-Kim temperaturê , and the Hawking-Bekenstein entropy to obtain the unified first law for the gravitational thermodynamics and the total energy differential for the open system enveloped by Υ 2 Advances in High Energy Physics in EBI Universe. Finally, we derived the generalized second law of the nondecreasing entropy ( ) enclosed by Υ in EBI Universe.
This paper is organized as follows. In Section 2, we reviewed the cosmical apparent horizon and derived the holographic-style dynamical equations in the EBI theory. Then we discussed the properties of the cosmical apparent horizon. In Section 3, the unified first law of gravitational thermodynamics and the Clausius equation on Υ for an isochoric process in EBI Universe were discussed. And we derived the total energy differential enclosed by Υ in EBI Universe. In Section 4, the generalized second law of gravitational thermodynamics in the EBI Universe was derived. Conclusions and discussion are given in Section 5.

Dynamics of the Cosmical Apparent
Horizon in Eddington-Born-Infeld Gravity 2.1. Apparent Horizon. Physically, apparent horizons constitute the observable boundary which is the largest boundary of Universe in an instant. Mathematically, apparent horizons are many hypersurfaces where the outward expansion rate (ℓ) or the inward expansion rate ( ) is equal to zero. In general, the first kind of apparent horizons where (ℓ) = 0 and ( ) ̸ = 0 usually locates near the black holes, and the another kind of apparent horizon where ( ) = 0 and (ℓ) ̸ = 0 appears in the vicinity of the expanding boundary of Universe, called the cosmical apparent horizons. In this paper, we only discussed the cosmical apparent horizon via dynamic equations of Universe and thermodynamic methods.
In order to calculate the apparent horizon of the cosmology, we use the FRW metric to describe the spatially homogeneous and isotropic Universe [1,19] where ( ) is the scale factor of the evolution of Universe and the index denotes the normalized spatial curvature, with = {+1, 0, -1} corresponding to closed, flat, and open Universes, respectively. Using the spherical symmetry, the metric can be rewritten as where ℎ fl diag[−1, ( ) 2 /(1 − 2 )] represents the transverse 2-metric spanned by ( 0 = , 1 = ) and Υ fl ( ) stands for the astronomical circumference/areal radius. Based on the FRW metric, one can structure the following null tetrad adapted to the spherical symmetry and the null radial flow: corresponding to the metric signature (−, +, +, +). By calculating the Newman-Penrose spin coefficients fl − ] ∇ ] ℓ and fl ] ∇ ] , the outward expansion rate (ℓ) = −( + ) and the inward expansion rate ( ) = −( + ) are, respectively, given by where fl/ is the Hubble parameter of cosmic spatial expansion. The overdot denotes the derivative with respect to the comoving time .
For the expanding Universe ( > 0), the cosmical apparent horizon is given by derived from ( ) = 0 and (ℓ) > 0 corresponding to the unique marginally inner trapped horizon where Υ becomes a null vector with ] Υ ] Υ = 0 [20]. Then one derives the temporal derivative of (5) that is a kinematic equation of the cosmical apparent horizon.

The Holographic-Style Dynamical Equations in
Eddington-Born-Infeld Universe. The action of Eddington-Born-Infeld theory is given by [15][16][17][18] where is the Ricci scalar for the metric ] and represents the determinant of ] . ] is two-order Riemann curvature tensor dependent on the connection ] , provided by the Palatini approach. Λ is the cosmological constant and is an arbitrary constant. is the gravitational constant and L is the Lagrangian density of matter.

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Applying the Bigravity method [18] to replace the connection ] by the extra metric ] in the EBI theory, the action (7) can be rewritten into the Bigravity-type action where ] is the Ricci tensor for the extra metric ] and is the Ricci scalar for the extra metric ] . is a constant ( ≡ /ℓ 2 ) corresponding to ] and is the determinant of ] . Here, both ] and ] are innate metrics of spacetime and they are mutually independent. Hence, (1/ℓ 2 )√− ( ) can be regarded as the term from self-coupling of spacetime.
Varying the Bigravity-type action (8) with respect to the metric ] , we get the field equations [15] ] − where is the energymomentum tensor of the spacetime self-coupling.
The matter content of Universe is construed as the perfect fluid whose the energy-momentum tensor is where refers to the Equation-of-State (EoS) parameter of the perfect fluid. In order to study the cosmological property of the EBI Universe, we made ] be the FRW metric and assumed the extra metric ] [15,21] as where and are two undetermined positive functions independent with .
Depending on the field equation and the two metrics, we get the first Friedmann equatioṅ and the second Friedmann equatioṅ Eq. (14) can be rewritten into which is equivalent to With the help of (13) and (15), we obtaiṅ Based on (5) and (13), we get and substituting (17) into (6), we geṫ Eq. (16), (18), and (19) are the holographic-style dynamical equations of the cosmical apparent horizon [14], which means the evolution of Universe has a relation with the cosmical apparent horizon. If one takes = = 1 and Λ = 0, the holographic-style dynamical equations will return to the condition of the Einstein theory.
Furthermore, from (13) and (14), we can obtain the acceleration equation of the EBI Universë

The Characters of the Cosmical Apparent Horizon.
In general, the cosmical apparent horizon is not null surface, which is different from the event and particle horizon. The equation of the cosmical apparent horizon in comoving coordinates is [22] Its normal has components 4

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The norm squared of the normal vector is where = Υ / . Substituting (20) and (18), we get where we consider that is only the quadratic function H( ) representing the inner product of the normal vector of the cosmical apparent horizon. The quadratic function H( ) has two zero points, = 0 that shows the normal vector is a null vector and the apparent horizon Υ is a null surface. It coincides with the cosmological event horizon Υ = ∫ ∞ −1̂, which is a future-pointed null causal boundary [19,22]. And it shares the signature of isolated black-hole horizons [23]. ], > 0 that shows is a spacelike vector and Υ is the timelike surface. Υ has the signature (−, +, +) that shares the signature of a quasi-local timelike membrane in black-hole physics [20,24].

Thermodynamics of the Holographic-Style Dynamical Equations in the Eddington-Born-Infeld Universe
Based on the holographic-style dynamical equations (18), (19), and (16) in Section 2, we continue to investigate the thermodynamics about the cosmical apparent horizon. Firstly, we define the effective energy within a sphere of radius Υ (surface area = 4 Υ 2 and volumê= (4/3) Υ 3 ): = , which is the entire energy enveloped by the cosmical apparent horizon Υ (takêto represent the volume in order to distinguish the function ).

Unified First Law of Thermodynamics. Applying the Misner-Sharp mass/energy
fl (Υ/2 )(1 − ℎ Υ Υ) [26,27] to be the effective energy and substituting ℎ = diag[−1, 2 /(1 − 2 )], one obtains and For the EBI Universe, substituting (16) and (18) into (25), the total energy differential in the ( , Υ) coordinates is obtained where = 4 Υ 2 . Similarly, we can obtain which is the total energy differential in the ( , ) coordinates. From the above two equations, we can know that the EoS parameter from spacetime self-coupling is negative, because of the energy density term "(1/8 ℓ 2 )√ / " and the intensity of pressure "−(1/8 ℓ 2 )√ / ". The unified first law of (equilibrium) thermodynamics is given by proposed by Hayward [28]. is the work density, given by Here, and Ψ are invariant. Moreover, the definitions of and Ψ are valid for all spherically symmetric spacetimes and FRW spacetime.
In the EBI theory, the field equations can be rewritten into where we define an effective energy-momentum tensor From the above equation, we can consider that the effective energy-momentum tensor includes the part of the dark energy corresponding to the terms with Λ and ( , ). Then we can generalize Hayward's unified first law of (equilibrium) thermodynamics into the EBI Universe by taking ( ) ] to replace ( ) ] . Imitating the definitions of and Ψ [28], we define wherẽfl We consider that the FRW metric ] is physically subsistent, which is used to raise or descend the index here, and another metric ] is an extra metric provided by the primordial mechanism of Universe. Based on this, we get and (40) SubstitutingΨ ,Ψ , and̃into (34), we get which is the expression of in ( , ) coordinates. Naturally, because of the invariance of̃andΨ, we can rewrite these in the ( , Υ) coordinates, given by wherẽ It illustrates our hypothesis of replacing ( ) ] by ( ) ] is reasonable that (43) and (44) are, respectively, identical to (28) and (27). The unified first law for gravitational thermodynamics of Universe is totally different from the first law in the black-hole thermodynamics [14]. Eq. (43) and (44) are both called "the unified first law" for the EBI Universe's gravitational thermodynamics.

Clausius Equation on the Cosmical Apparent Horizon for an Isochoric Process.
Having obtained the unified first law =Ψ +̃̂in EBI Universe, we are interested in the region enclosed by the cosmical apparent horizon Υ .
where = 4 Υ 2 . The left-hand side of (47) can be manipulated into [14] One applies the geometrically defined Hawking-Bekenstein entropy [4,29] (in the units ℏ = and the Cai-Kim temperature [1, 30] where is the specific condition of (44) when Υ = Υ . Eq. (51) is actually the Clausius equation for equilibrium and reversible thermodynamic processes as same as the situation in GR [14]. After considering the EBI theory, the Clausius equation is generalized to include dark energy from the cosmological constant Λ and the effect the self-coupling of spacetime ( , ), which may explain the problems about the cosmic expansion.
Finally, for the open system enveloped by Υ , we substitute the unified first law (44) and the Clausius equation (51) into the total energy differential where the "-" sign shows the positive-heat-out sign convention that means heat emitted by the open system takes positive values ( = ( ) > 0 ) rather than the traditional positive-heat-in thermodynamic sign convention [14].
Comparing (52) with the total energy differential of the Einstein gravity, there are two extra terms (Λ/8 )̂and (1/8 ℓ 2 )√ /̂A, which originate from dark energy of the cosmological constant and the spacetime self-coupling, respectively. It means that there are the matter's transition and the dark energy's fluxion on both sides of the cosmical apparent horizon during the expansion of Universe.

Generalized Second Laws of Thermodynamics in Eddington-Born-Infeld Universe
With the help of the first holographic-style dynamical equation (18), the effective energy ( = Υ 3 /2 Υ 2 ) can be rewritten into where = + Λ/8 + (1/8 ℓ 2 )√ / . Then we can rewrite the Friedmann equations intȯ where we define that = −Λ/8 −(1/8 ℓ 2 )√ / . Based on two above equations, one can obtain the continuity equation in the EBI theorẏ In many papers [31][32][33], the entropy of the cosmic energy-matter content with temperaturêis always determined by the traditional Gibbs equation =̂− . In order to generalize to the EBI theory, we keep the same form and redefine it into the positive-heat-out sign convention for consistency with the horizon entropy [14], given by From =̂+̂, we obtain Based on the continuity equation (56), we obtain When Υ = Υ , one can get ] .
Physically, temperature is positive (̂> 0) and the present Universe is expanding ( > 0). If Λ satisfies the precondition (Λ > −(1/ℓ 2 )√ / − 8 ) shown in Section 2, we can exactly obtain that , the present Universe is an accelerated expanding Universe. Hence, if < 3 − 8 ℓ 2 √ / , has the possibility to be a positive number to produce an accelerated expanding Universe, which is different from the result of Einstein Universe [14].
In a word, the physical effective entropy (  ]. Noteworthily, ( ) is composed of the matter's entropy and the dark energy's entropy.
Secondly, based on the holographic-style dynamical equations, we obtained two forms of the total energy differential in the ( , Υ) coordinates and ( , ) coordinates. And we proved that these two forms of the total energy differential can be derived from the unified first laws of the gravitational dynamics =Ψ+̃̂by redefining the effective energymomentum tensor ( ) ] in Hayward's approach [28]. Thirdly, we derived the total energy differential for the open system enveloped by Υ , = −̂+ [ + Λ/8 + (1/8 ℓ 2 )√ / ]̂, where the two extra terms (Λ/8 )̂and (1/8 ℓ 2 )√ /̂are, respectively, corresponding to the dark energy and the spacetime selfcoupling in EBI Universe. It illustrates that not only the matter's transition but also the dark energy's fluxion arises on the cosmical apparent horizon Υ with the expansion of Universe.
In addition to all the above we would like to point out that the method of the theories we mentioned in our paper satisfying the equilibrium thermodynamics does not mean this method fits all possible theories. Actually, nonequilibrium thermodynamics might be also a good way in finding new gravitational theories [34][35][36], which would be a further task for us to study.

Data Availability
No data were used to support this study.

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