Thermodynamics in modified Brans–Dicke gravity with entropy corrections

In this paper, we investigate the thermodynamics in the frame-work of recently proposed theory called modified Brans–Dicke gravity (Kofinas et al. in Class Quantum Gravity 33:15, 2016). For this purpose, we develop the generalized second law of thermodynamics by assuming usual entropy as well as its corrected forms (logarithmic and power law corrected) on the apparent and event horizons. In order to analyzed the clear view of thermodynamic law, the power law forms of scalar field and scale factor is being assumed. We evaluate the results graphically and found that generalized second law of thermodynamics holds in most of the cases.

The scalar tensor models of modified gravity have taken remarkable attention. The reason is that scalar fields for consistent condition appear in different branches of theoretical physics such as, the low energy limit of the string theory leads to a scalar degree of freedom. The Brans-Dicke (BD) theory [47,48] is a prototype of scalar tensor gravity which is based upon Dirac hypothesis. It relates scalar field (φ) with dynamical gravitational constant (G = G 0 φ ) and involves a tuneable constant coupling parameter ω. The scalar field is a fundamental feature of this gravity which is considered as a dark energy candidate. The parameter ω can adjust results according to the requirement and in the limit ω → ∞, it reduces BD theory into general relativity (GR).
The standard BD theory remains unable to probe cosmic evolution accurately. In this context, many researchers have generalized BD theory in different scenario like, selfinteracting potential model, model having time-dependent coupling parameter (ω) [49][50][51]. Brans-Dicke cosmological models with constant deceleration parameter in the form of particle creation [52]. The Friedman models (with zero curvature) under the effects of time dependent bulk viscosity [53]. Similarly, the modification of BD theory also involve interaction with dark matter, such as the concept of dissipative cold dark matter in BD gravity [54,55], model representing transfer of energy between BD gravity and dark matter [56,57]. Recently, Kofinas et al. [58] introduced the most generalized or corrected form of BD gravity by relaxing the standard conservation law of matter contribution (energy momentum of matter). He used a new dimensionless parameter ν in the theory. This new version of BD gravity has explored cosmic evolution in accords to observational data by involving matter-scalar field interaction.
In literature, the relation between gravitation and thermodynamics has been discussed extensively. Moreover, by inspiring the black hole theory, there is a deep connection between gravitation and thermodynamics. In GR, Hawking radiation [59] can be studied by using the proportionality relation between the surface gravity and temperature, and also developed the connection between the horizon entropy and area of thermodynamics. On the other hand, Jacobian in [60] was studied the Einstein field equations from the Clausis relation. By using this approach, the authors [61,62] have checked validity of generalized second law of thermodynamics (GSLT). In literature many researchers have been studied the validity of GSLT in GR as well as modified theory. However, it should be noted that the definition of entropy would rather be modified in order to include quantum effects motivated from the loop quantum gravity [63,64]. In this work we are focused on GSLT with modified BD theory involving dark matter and the dark energy with a scalar field.
The present paper is organized as follows. In Sect. 2 we give a brief description of the BD theory and then derive the late-time cosmic field equation with a perfect fluid. In Sect. 3, we formulate the Friedman equation of the late-time cosmological equation in the absence of ν = 0. In Sect. 4 we analyze the GSLT with entropy corrections. In this section we check validity of GSLT by using the logarithmic, power-law corrections at apparent as well as event horizon and obtain the graphical results. At the end, a discussion of this work is presented.

Modified Brans-Dicke field equations
The action for standard BD theory in terms of Jorden frame is given by [47,48] where L m represents the matter Lagrangian depending on scalar field ψ, √ −gd 4 x denotes four dimensional volume, ω is the coupling constant which depends on dimensionless parameter λ( = 0) as ω = 2−3λ 2λ , φ shows the scalar field and R is the Ricci scalar. The corresponding field equations are Here τ μ ν is the energy-momentum tensor for matter and τ μ μ is its trace. The BD theory has proposed in a generalized simple form by relaxing conservation equations (5) [58]. In this modified form, the energy momentum tensor of scalar field is constructed through terms that contain φ itself or two derivatives of one or two scalar fields. Thus, the generalized field equations contain modified form of Eqs. (3) and (5) given as while the remaining equations remain same. Here, ν is an arbitrary function of integration which numerical values as well as sign can be determined experimentally. For ν = 0 the above system of equations reduces to the standard BD model ((2)-(5)). The FRW metric is where a is the scale factor and k = −1, 0, 1 represent spatial curvatures. According to the symmetry of scale factor, the scalar field behaves as a function of time, i.e., φ = φ(t).
For matter distribution, we take energy momentum tensor of perfect fluid as τ μ ν = diag(−ρ, p, p, p) with ρ(t) be its energy density and p(t) its pressure. Equations (2), (4), (6) and (7) along with above metric lead to For ν = 0, the above define equations reduce to standard BD model. Integrating Eq. (11), using p = ωρ, we get where ρ * is an integration constant. This obtain relation of energy density represents a direct coupling of scalar field and matter density. In effective scenario, the field Eqs. (9) and (10) can be rewritten as 2Ḣ Here the effective energy density and pressure are defined by

Generalized second law of thermodynamics
In this section, we will study the validity of GSLT in the framework of modified BD theory on the apparent and event horizons. According to GSLT, the combination of entropy of horizon and entropy of all matter sources inside horizon does not decrease with time. Mathematically, it can be expressed aṡ whereṠ h represents the entropy related to the horizon anḋ S in is the sum of all entropies inside the horizon. Let us start from Gibb's equations which evolutes the entropy of matter and energy source to the pressure inside the horizon, given as which leads to Thus the total entropy inside the horizon can be written as Also, the Hawking entropy relation is defined by where A = 4π R 2 h is the area of horizon.

Apparent horizon
Let us assume FRW universe which comprises of apparent horizon and is defined as a null space with vanishing expansion. For spatially flat FRW metric, the radius and temperature are defined as Substituting the value of area at apparent horizon in Eq. (23) and taking the derivative, we geṫ which leads to By re-arranging above equation, we geṫ The radius at apparent horizon in the form of ρ eff can be defined as Using the derivative of Eq. (28) into Eq. (27), the entropy of fluid turns out to bė After simplification, we geṫ Now we would like to check the validity of the GSLT of the system which is enclosed by apparent horizon. Hence, the total entropy turns out to bė By invoking the values of ρ eff andρ eff in above equation, we geṫ The derivatives of Eqs. (16)-(18) becomė Using Eqs. (30)-(32), GSLT takes the forṁ

Event horizon
In this subsection, we study GSLT at event horizon which can be written aṡ The radius of event horizon is defined as In this case, we utilized the following temperature Using relations (34) into Eq. (36), we geṫ Similarly, the entropy at inside the horizon in case of event iṡ ThusṠ tot can be obtained (for event horizon) from Eqs. (38), (37) and (33) as followṡ The plot ofṠ tot versus t at event horizon is shown in Fig. 2. The values of constant parameters are same as utilized in figure. It can be seen from Fig. 2 that all trajectories ofṠ tot are increasing function of time and remain positive. Hence, GSLT satisfied for the present system enclosed by event horizon.

Entropy corrections with event horizon
In GR, entropy area relation with motivation of quantum loop gravity leads to the curvature corrections in the Hilbert-Einstein action [65,66]. In this section, we will discuss two types of corrections.

Logarithmic correction
Recently, logarithmic correction has studied for black hole in quantum gravity due to the fluctuation of thermal equilibrium [67][68][69]. Then, Sadjadi and Jamil [70] explored the GSLT for flat FRW metric with logarithmic correction. The logarithmic corrected entropy is expressed by the given relation where α, β and γ are dimensionless constants. Taking the derivative of above equation, we geṫ By following the same procedure, inserting Eqs. (40), (38) into (33), we geṫ Its illustrated form leads tȯ In order to observe the validity of GSLT for logarithmic correction at event horizon, we plot it in Fig. 3. We plotṠ tot versus t by taking different values of α, β as α = 3.8, β = 3 and using values of other parameters same. This figure shows that the trajectories ofṠ tot remains positive and exhibits the increasing behavior at the present as well as initial epoch while turn negative after some interval of time. Hence, GSLT holds at initial epoch for all cases of m while does not satisfy at later epoch.

Power law correction
Power law correction for event horizon can be defined as [71,72] where and r c is the crossover scale and α is dimensionless constant. The second term in (42), as a power-law correction to the entropy, has been appeared from the scalar field of the wavefunctions between two excitation states. The higher the excitation state is the more significant than the correction term.
The time rate of change of entropy for power-law correction is given aṡ The total rate of change of entropy for power law correction at event horizon verses t is shown in Fig. 4. It can be observed thatṠ tot shows increasing behavior for all values of m at early, present as well as later epoch. However, GSLT holds for all cases for t > 3.2.

Entropy corrections with apparent horizon
In this section, we investigate the validity of GSLT of the system by assuming entropy corrections on the apparent horizon.

Logarithmic correction
In case of apparent horizon, Eq. (39) implieṡ By replacing the values of R A andṘ A in Eq. (45), we obtaiṅ which leads tȯ The display of total rate of change of entropy for logarithmic correction on apparent horizon versus cosmic time is shown in Fig. 5. All constant parameters are same as utilized in previous plots. It can be observed thatṠ tot shows increasing behavior and remains positive for all cases of m. This exhibits its validity for the present scenario.

Power law correction
The rate of change of power law correction at apparent horizon is given bẏ The plot of GSLT for the power law correction on the apparent horizon is shown in Fig. 6. It is observed that the trajectories ofṠ tot is increasing for all values of m and remain positive. Thus GSLT is satisfied for the present scenario on the apparent horizon.

Conclusion
We have investigated the thermodynamics in the frame-work of recently proposed theory called modified Brans-Dicke gravity [58]. For this purpose, we have developed the GSLT by assuming usual entropy as well as its corrected forms (logarithmic and power law corrected) on the apparent and event horizons. In order to analyzed the clear view of thermodynamic law, we have assumed the power law forms of scalar field and scale factor is being assumed. We have evaluated the results graphically and summarized them in the following. • In phase one, we have analyzed GSLT on the apparent as well as event horizon by assuming usual entropy. In this case, we have observed that GSLT holds for m = 1 becauseṠ Atot ≥ 0, while does not remain valid for m = 2 and m = 3 becauseṠ Atot > 0 (Fig. 1). The plot oḟ S tot versus t at event horizon has shown in Fig. 2. It has been seen from Fig. 2 that all trajectories ofṠ tot are increasing function of time and remain positive. Hence, GSLT satisfied for the present system enclosed by event horizon. • In second phase, the validity of GSLT on the apparent as well as event horizon has been analyzed by assuming logarithmic entropy correction. It has been observed from Fig. 3 that the trajectories ofṠ tot (on the event horizon) remains positive and exhibits the increasing behavior at the present as well as initial epoch while turn negative after some interval of time. Hence, GSLT holds at initial epoch for all cases of m while does not satisfy at later epoch. The display of total rate of change of entropy for logarithmic correction on the apparent horizon versus cosmic time is shown in Fig. 5. It can be observed thaṫ S tot shows increasing behavior and remains positive for all cases of m. This exhibits its validity for the present scenario. • In third phase, the validity of GSLT on the event as well as apparent horizon has been analyzed by assuming power law entropy correction. The total rate of change of entropy for power law correction at event horizon verses t is shown in Fig. 4. It has observed thatṠ tot shows increasing behavior for all values of m at early, present as well as later epoch. However, GSLT holds for all cases for t > 3.2. The plot of GSLT for the power law correction on the apparent horizon is shown in Fig. 6. It has observed that the trajectories ofṠ tot is increasing for all values of m and remain positive. Thus GSLT is satisfied for this scenario on the apparent horizon.