Thermodynamics of Gravitationally Induced Particle Creation Scenario in DGP Braneworld

Thermodynamics of Gravitationally Induced Particle Creation Scenario in DGP Braneworld

Abdul Jawad , Shamaila Rani and Salman Rafique
Department of Mathematics, COMSAT Institute of
Information Technology, Lahore-54000, Pakistan.
Abstract

In this paper, we discuss the thermodynamical analysis for gravitationally induced particle creation scenario in the framework of DGP braneworld model. For this purpose, we consider apparent horizon as the boundary of the universe. We take three types of entropy such as Bakenstein entropy, logarithmic corrected entropy and power law corrected entropy with ordinary creation rate . We analyze the first law and generalized second law of thermodynamics analytically for these entropies which hold under some constraints. The behavior of total entropy in each case is also discussed which implies the validity of generalized second law of thermodynamics. Also, we check the thermodynamical equilibrium condition for two phases of creation rate, that is constant and variable and found its validity in all cases of entropy.

Keywords: Gravitationally induced particle creation; Thermodynamics; Entropy corrections; DGP Brane-world.
PACS: 95.36.+d; 98.80.-k.

1 Introduction

It is believed that the universe undergoes an accelerated expansion due to mysterious form of force called dark energy (DE) which was firstly predicted by two independent teams of cosmologists [1, 2]. Both of them used distant type Ia supernova as standard candles to measure the expansion of the universe. This discovery was unexpected, because before this invention, cosmologists just think that the expansion of the universe would be decelerating because of the gravitational attraction of the matter in the universe. In the accelerated expansion of the universe, DE plays major role but its nature is still unknown. The simplest candidate of DE is the cosmological constant, but its composition and mechanism are unknown. More generally, the detail of its equation of state (EoS) and relationship with the standard model of particle physics continue to be investigated both through observationally and theoretically [3]. In order to explain this cosmic acceleration, various DE models and modified theories of gravity have been developed such as theory, theory, Brans-Dicke theory, dynamical Chern-Simons modified gravity, DGP Braneworld model, etc.

The concept of thermodynamics in cosmological system originates through black hole physics. It was suggested [4] that the temperature of Hawking radiations emitting from black holes is proportional to their corresponding surface gravity on the event horizon. Jacobson [5] 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) , where and 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 ( and represent the internal energy, pressure and volume of the spherical system) for any horizon [6].

The generalized second law of thermodynamics (GSLT) has been studied extensively in the scenario of expanding behavior of the universe. The GSLT states that the entropy of matter inside the horizon plus entropy of the horizon remains positive and increases with the passage of time [7]. In order to discuss GSLT, horizon entropy of the universe can be taken as one quarter of its horizon area [8] or power law corrected [9] or logarithmic corrected [10] forms. Many people have explored the validity of GSLT of different systems including interaction of two fluid components like DE and dark matter [11], as well as interaction of three components of fluid [12] in the FRW universe by using simple horizon entropy of the universe. The thermodynamical analysis widely performed in modified theories of gravity [13].

The gravitationally induced particle creation is another well-known mechanism which was firstly introduced by Schrodinger [14] on microscopic level. This mechanism was further extended by Parker et al. towards quantum field theory in curved spacetimes [15]-[19]. The macroscopic description of particle creation mechanism induced by gravitational field was presented by Prigogine et al. [20]. Later on, the covariant description of this mechanism was developed [21, 22] as well as the physical difference between particle creation and bulk viscosity was also given [23]. The particle creation process can be described with the inclusion of backreaction term in the Einstein field equations whose negative pressure may help in explaining the cosmic acceleration. In this way, various phenomenological models of particle creation have been presented [24]-[29]. It is also shown that phenomenological particle production [30]-[33] help in explaining the cosmic acceleration and paved the alternative way to the concordance CDM model.

The fact that we resides in a three-dimensional space embedded in an extra-dimensional world and five-dimensional models in which universe would be a hypersurface has attain a great attention. The four-dimensional Einsteins equations projected onto the brane have been explored by Shiromizu et al. [34]. The approaches which made on the basis of brane-world in the early-time cosmology favor a particular model of cosmic evolution featured by quadratic relations between the energy density and the Hubble parameter, dubbed quadratic cosmology [35].

Recently, by assuming the gravitationally induced particle scenario with constant specific entropy and arbitrary particle creation rate (), thermodynamics on the apparent horizon for FRW universe has been discussed [36]. They have investigated the first law, GSLT and thermodynamical equilibrium by assuming the EoS for perfect fluid and put forward various constraints on for which thermodynamical laws hold. Our aim is to discuss the thermodynamical analysis on the apparent horizon for gravitationally induced particle creation scenario with ordinary creation rate by assuming entropies (Bakenstein entropy or usual entropy, logarithmic corrected entropy and power law corrected entropy) in DGP braneworld model. The scheme of the paper is as follows: In the next section, we will present the basic equations of DGP brane-world, particle creation rate and cosmological parameters. Section 3, 4, 5 contain the discussion of thermodynamic quantities as well as its laws corresponding to usual, logarithmic and power law corrected entropies, respectively. The last section comprises of concluding remarks on our results.

2 Basic Equations

A most particular version was proposed by Dvali et al. [37], in which the four dimensional FRW universe is enclosed in a five dimensional Minkoski bulk with infinite size. The gravitational laws were obtained by adding an Einstein-Hilbert term to the action of brane computed with the brane’s intrinsic curvature. The presence of such a term in the action is generically induced by quantum corrections coming from the bulk gravity and its coupling with matter living on the brane and must be included for a large class of theories for self-consistency [38, 39]. Here, we consider -brane embedded in a space-time with an intrinsic curvature term included in the brane whose action can be written as

 S\textmd(5)=−12κ2∫d5X√−~g~R+∫d5X√−~gLm, (1)

where is the brane curvature term, given by

 Lm=−12μ2∫d4x√−gR, (2)

and . The Eq.(1) represents the Einstein-Hilbert action in five dimensions for a five-dimensional metric (bulk metric) of Ricci scalar . Similarly, Eq.(2) indicates the Einstein-Hilbert action for the induced metric on the brane with appeared as its scalar curvature. From Eq.(1), we can get modified Friedmann equation as [40]

 H2+ka2=(√ρ3M2p+14r2c+ϵ2rc)2, (3)

where is Hubble parameter with is the scale factor. Also, , the subscripts and represent the energy densities corresponding to dark matter and DE respectively, is the crossover length which represents the scale that has length away from which gravity starts opening into the bulk [40]. Moreover it is the distance scale follow the comparison among and effects of gravity and can be written as [40]

 r\textmdc=M2p2M35, (4)

where stands for the Planck mass and is the Planck mass.

For the spatially flat DGP braneworld , Eq.(3) reduces to

 H2−ϵr\textmdcH=ρ3M2p. (5)

There exist two different branches for the DGP model depending on the sign of . These are as follows:

• For , there is a de Sitter solution for Eq.(5) with constant Hubble parameter, i.e., in the absence of any kind of energy or matter field on the brane (i.e., ). However, this branch faces some problems like ghost instabilities [41].

• For , the accelerated expansion of the universe can only be explained through the inclusion of DE component in the DGP scenario.

We consider the latter case in the present work. The equation of continuity for this model will become

 ˙ρ+Θ(ρ+P+Π)=0, (6)

where is a particle creation pressure which represents the gravitationally induced process of particle creation and is the fluid expansion. Differentiating Eq.(5) and replacing the value of using Eq.(6), we obtain

 ˙H=−H(ρ+p+Π)M2p(2H−ϵrc). (7)

The respective EoS for this model is giving by with . The non-conservation of the total rate of change of number of particles, with comoving volume and is the number density of particle production in an open thermodynamical system yields

 ˙n+Θn=nΓ, (8)

where is a particle creation rate has negative and positive phases. Negative represents the particle destruction and positive describes the elimination of particles. Furthermore, a non-zero produces effective bulk viscous pressure [42]-[48].

Now using the Eqs.(6), (8) and Gibbs relation, we get

 Tds=d(ρn)+pd(1n). (9)

An equation related to the creation pressure and the creation rate has the form

 Π=−ΓΘ(ρ+p). (10)

Under traditional assumption that the specific entropy of each particle is constant, i.e., the process is adiabatic or isentropic. This implies a dissipative fluid is similar to a perfect fluid with a non-conserved particle number. To discuss cosmological parameters, we insert Eqs.(10) and in Eq.(7), to obtain

 ˙HH2=−3γ(H−ϵr\textmdc)(1−Γ3H)(2H−ϵrc). (11)

The deceleration parameter can be written as

 q=−˙HH2−1=3γ(H−ϵrc)(1−Γ3H)(2H−ϵrc)−1. (12)

The effective EoS parameter for this model turns out to be

 ω\textmdeff=p+Πρ=γ(1−Γ3H)−1. (13)

This parameter has ability to explain the different phases of the universe on the basis of , i.e., if , then we have quintessence era of the universe (), if then effective EoS parameter represents the phantom era of the universe () while for , effective EoS parameter exhbits the cosmological constant ().

In the following, we will discuss first and second thermodynamical laws in the presence of particle creation rate on the apparent horizon.

3 Thermodynamical Analysis with Usual Entropy

For flat FRW universe, Hubble parameter coincides with the apparent horizon as . Differentiating the apparent horizon with respect to time, we get

 ˙RA=−˙HH2=3γ(H−ϵrc)(1−Γ3H)(2H−ϵrc). (14)

The Bekenstein entropy and Hawking temperature of the apparent horizon are given by

 S\textmdA=A4=R2A8andT\textmdA=12πRA=4RA, (15)

where . The first law of thermodynamics at the horizon can be obtained through the Clausius relation as

For the sake of convenance, we consider . The differential is the amount of energy crossing the apparent horizon can be evaluated as [49]

 −dE\textmdA=12R3\textmdA(ρ+p)Hdt=3γ(H−ϵr\textmdc)2Hdt. (17)

From Eq.(15), the differential of surface entropy at apparent horizon yields

 dSA=3γ4H(H−ϵrc)(1−Γ3H)(2H−ϵrc), (18)

Thus, turns out to be

 Ξ=3γ(H−ϵr)(1−Γ3H2H−ϵrc−12H). (20)

From this relation, it can be seen that first law of thermodynamics holds (i.e., ) at the apparent horizon for .

Next, we will discuss the GSLT and thermodynamical equilibrium of a system containing perfect fluid distribution bounded by apparent horizon in DGG brane-world scenario. For GSLT, the total entropy of the system can not be decrease, i.e., . In this relation, and appear as the entropy at apparent horizon and the entropy of cosmic fluid enclosed within the horizon, respectively. The Gibbs equation is given by

 T\textmdfdS\textmdf=dE\textmdf+pdV, (21)

where is the temperature of the cosmic fluid and is the energy of the fluid (. The evolution equation for fluid temperature having constant entropy can be described as [50]

 ˙T\textmdfT\textmdf=(Γ−Θ)∂p∂ρ. (22)

Eq.(11) leads to which inserting in Eq.(22) gives the following equation

 ln(T\textmdfT\textmd0)=2(γ−1)γ∫dHH  ⇒  T\textmdf=T\textmd0(H2−ϵr\textmdcH)(γ−1)γ, (23)

where is the constant of integration. The differential of the fluid entropy can be obtained by using the Eq.(21) as follows

 dS\textmdf = (24) × (1−12Hr\textmdc−3γ2H(H−ϵr\textmdc))dt.

Using Eqs.(18) and (24), we get the rate of change of total entropy as

 ˙ST = (25) × (12Hϵr\textmdc−3γ2H(H−ϵr\textmdc)−1)],

where . We discuss the validity of GSLT on the basis of such that

• : The GSLT holds if the following constraint

satisfies. This shows that the GSLT satisfies in the qunitessence era of the evolving universe.

• : For this case, we have the constraint

which implies the GSLT holds in phantom era of the universe.

• : This case implies in the cosmological constant era.

Replacing to and integrating Eq.(25), we get

 S\textmdT = S\textmdA+S\textmdf (26) = −18H2ϵ2rcT0(1−γ)(1−ϵH2rc)−1γ(1−H2rcϵ)−1γ[6γ2ϵ2 2F1(−1+γγ , −1γ,2−1γ,ϵH2rc)(H2−ϵrc)1γ(1−H2rcϵ)1γ+2(H2−ϵrc)1γr2c{−4Hϵ × (−1+γ) 2F1(−12,−1γ,12,H2rcϵ,)(1−ϵH2rc)1γ+γ(6H3(−1+γ) × 2F1(12,−1+γγ32,H2rcϵ)(1−ϵH2rc)1γ+ϵ2 2F1(−1+γγ,−1γ,2−1γ , ϵH2rc)(1−H2rcϵ)1γ)}+2H2r3c(H2−ϵrc)1γ(−1+γ)(4H 2F1(12 , −1+γγ,32,H2rcϵ)(1−ϵH2rc)1γ+γϵ(− 2F1(−1γ,−1γ,−1+γγ,ϵH2rc) × (1−H2rcϵ)1γ+(1−ϵH2rc)1γ(−1+(1−H2rcϵ)1γ)))+(−1+γ)ϵ × rc{6γH(H2−ϵrc)1γ(−2(1−ϵH2rc)1γ 2F1(−12,−1γ,12,H2rcϵ)−Hγ × (1−H2rcϵ)1γ 2F1(−1γ,−1γ,−1+γγ,ϵH2rc)+(1−ϵH2rc)1γ × (−1+(1−H2rcϵ)1γ)−ϵ(1−ϵH2rc)1γ(1−H2rcϵ)1γ}].

The plot between total entropy and parameter is shown in Figure 1 for three values of by setting constant values as , , . We observe that for all the values of which leads to the validity of GSLT.

3.1 Thermal Equilibrium Scenario

Further, we will discuss the thermal equilibrium scenario in the present case. For thermodynamical equilibrium, the entropy function attains a maximum value and satisfies the condition . For this purpose, we consider two cases of particle creation rate ().

Case 1: Γ = constant

Firstly, we consider particle creation rate as a constant. Under this scenario, differentiating Eq.(25) w.r.t time, it results the following second order differential equation

 ¨ST = −3γ(1−Γ3H)(−ϵrc+H)λ˙H2H(−ϵrc+2H)2+3γ(1−Γ3H)λ˙H4H(−ϵrc+2H)+γΓ(1−Γ3H) (27) × (−ϵrc+H)λ˙H4H3(−ϵrc+2H)−3γ(1−Γ3H)(−ϵrc+H)λ˙H4H2(−ϵrc+2H)+3γ4H(−ϵrc+2H) × (1−Γ3H)(−ϵrc+H)(4(−ϵrc+H2)1−γγT0(−1+ϵ2rcH−3γ2H × (−ϵrc+H))˙H+4(1−γ)HγT0(−ϵHrc+H2)−1+1−γγ(−1+ϵ2Hrc − 3γ(−ϵrc+H)2H)(−ϵrc+2H˙H)+4H(−ϵHrc+H2)1−γγ ×

where . The plot between versus for three values of with constant values of , . , as shown in Figure 2. One can observe that the thermodynamical equilibrium condition holds for all values of with specific ranges of . For example, for , thermodynamical equilibrium holds for the range and does not obey for . For , thermal equilibrium holds for the range and does not showing the validity for . However, for , thermodynamic equilibrium condition holds for the range and disobey for the range .

Case 2: Γ=Γ(t)––––––––––

Here we take as variable parameter, i.e., , for which Eq.(25) becomes

 ¨ST = −3γλ(−ϵrc+H)(1−Γ3H)˙H2H(−ϵrc+2H)2+3γλ(1−Γ3H)˙H4H(−ϵrc+2H)−3γλ(1−Γ3H)˙H (28) × (−ϵrc+H)4H2(−ϵrc+2H)+3γ(−ϵrc+H)(1−Γ3H)4H(−ϵrc+2H)(4(−ϵHrc+H2)1−γγ × (−1+ϵ2rcH−3γ(−ϵrc+H)2H)˙HT0+4(1−γ)HγT0(−ϵHrc+H2)−1+1−γγ × 4H(1−γ)(−1+ϵ2rcH−3γ(−ϵrc+H)2H)(−ϵ˙Hrc+2H˙H) + 4H(−ϵHrc+H2)1−γγ(−ϵ˙H2rcH2−3γ˙H2H+3γ(−ϵrc+H)˙H2H2T0) + 3γλ(−ϵrc+H)(Γ˙H3H2−˙Γ3H)4H(−ϵrc+2H),

where

 ˙Γ=−9γ(Γ3H)(H−ϵrc)(1−Γ3H)2H−ϵrc+3HΓ(1−Γ3H).

The plot of versus for three values of as shown in Figure 3 by keeping the same constant values as in previous case. We observe that the thermodynamic equilibrium holds for all values of with different ranges of . For example, for , thermodynamic equilibrium holds for the range and does not satisfying . For , it leads to the validity of thermodynamic equilibrium for the range and does not valid for . However, for , thermodynamic equilibrium holds for the range and does not satisfying within .

4 Logarithmic Corrected Entropy

Quantum gravity allows the logarithmic corrections in the presence of thermal equilibrium fluctuations and quantum fluctuations [51]-[57]. The logarithmic entropy corrections can be defined as

 S\textmdA=A4L2\textmdp+αlnA4L2\textmdp+β4L2\textmdpA, (29)

where and are constants whose values are still under consideration. The differential form of above equation leads to

 dS\textmdA=−3γ(H−ϵr\textmdc)(1−Γ3H)(2H−ϵr\textmdc)(14HL2\textmdp+2αH−16βH3L2\textmdp)dt, (30)

which gives

In view of this entropy, the quantity takes the form

 Ξ=3γ(H−ϵrc)(−12H+(1−Γ3H)(2H−ϵrc)(1L2\textmdp+8H2α−64βH4L2\textmdp)). (32)

It can be observed from Eq.(32) that the first law of thermodynamics holds when . To discuss the GSLT for logarithmic corrected entropy of horizon, we obtain the total entropy by using Eqs.(24) and (30) as follows

 ˙ST = (33) × (H2−ϵr\textmdcH)1−γγ(1−12Hϵr\textmdc−3γ2H(H−ϵr\textmdc))).

The GSLT will hold under these constraints.

• For the case , the GSLT satisfy in the quintessence region of the universe if the following constraint holds

 4L2\textmdp+2α > 16βH2L2\textmdp+(HT0)−1(1−12Hϵr\textmdc−3γ2H(H−ϵr\textmdc)) × (H2−ϵr\textmdcH)1−γγ.
• For this case, we obtain the following constraint

 4L2\textmdp+2α < 16βH2L2\textmdp+(HT0)−1(1−12Hϵr\textmdc−3γ2H(H−ϵr\textmdc)) × (H2−ϵr\textmdcH)1−γγ.

which implies the GSLT holds in phantom era of the universe.

• The case means in the cosmological constant era.

The expression of total entropy in the form of Hubble parameter is given by

 S\textmdT = S\textmdA+S\textmdf (34) = −8H2βL2\textmdp+2ln(H)(α+2L2\textmdp)−γ(H(H−ϵrc))1γ2H3T0.

The plot of total entropy versus with respect to three values of is shown in Figure 4 with constant values as , , . It is observed that the total entropy is positive, i.e, which leads to the validity of GSLT for all values of .

4.1 Thermal Equilibrium Scenario

Now we will discuss the thermodynamic equilibrium by assuming two cases for particle creation rate as follows:

Case 1: Γ = constant

In this way, the second order differential equation can be obtained from Eq.(33) for as a constant

 ¨ST = −6γ(1−Γ3H)(H−ϵrc)(2H−ϵrc)(4HL2\textmdp−λ2+2αH−16L2\textmdpβH3)˙H+3γ (35) × (1−Γ3H)(4HL2\textmdp−λ2+2αH−16L2\textmdpβH3)˙H(2H−ϵrc)+γΓ(H−ϵrc) × × (4L2\textmdp˙H−(1−γ)(1−3γ(H−ϵrc)2H−ϵ2Hrc)(H2−ϵHrc)−1+1−γγγT0 × +

where

 λ2=(1−3γ(H−ϵrc)2H−ϵ2Hrc)(H2−ϵHrc)1−γγT0.

The plot between and for three values of by fixing the constant values , and as shown in Figure 5. It can be seen that thermodynamical equilibrium is obeying the condition for all values of which leads to the thermal equilibrium.

Case 2: Γ=Γ(t)––––––––––

Taking as a function of , Eq.(33) yields

 ¨ST = −6γ(1−Γ3H)(H−ϵrc)(2H−ϵrc)(4HL2\textmdp−λ2+2αH−16L2\textmdpβH3)˙H+3γ (36) × (1−Γ3H)(4HL2\textmdp−λ2+2αH−16L2\textmdpβH3)˙H(2H−ϵrc)+3γ(1−Γ3H)(2H−ϵrc) × (H−ϵrc)(4L2\textmdp˙H−(1−3γ(H−ϵrc)2H−ϵ2Hrc)(H2−ϵHrc)−1+1−γγγT0 × (1−γ)(2H˙H−ϵ˙Hrc)−(−3γ˙H2H+3γ(H−ϵrc)2H2+ϵ˙H2H2rc)T0 × (H2−ϵrc)1−γγ+2α˙H−48L2\textmdpβH2β˙H)+3γ(H−ϵrc) ×

Figure 6 reperesents the plot between and for three values of for variable for same constant values. Figure 6 indicates that the trajectories of corresponding to all the values of ensure the validity of thermodynamical equilibrium.

5 Power Law Corrected Entropy

The power-law correction to the entropy-area law comes from association of the wave-function of the scalar field between the ground state and the exited state [58]-[60]. The correction term is also more significant for higher excitations. It is important to note that the correction term decreases faster with A and hence in the semi-classical limit (large area) the entropy-area law is recovered. The power entropy can b given as

 (37)

where is dimensionless constant and is the crossover scale. From Eq.(37) the differential of surface entropy at horizon can be expressed as

 (38)

which gives