Generalized second law of thermodynamic in modified teleparallel theory

This study is conducted to examine the validity of the generalized second law of thermodynamics (GSLT) in flat FRW for modified teleparallel gravity involving coupling between a scalar field with the torsion scalar T and the boundary term B=2∇μTμ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B=2\nabla _{\mu }T^{\mu }$$\end{document}. This theory is very useful, since it can reproduce other important well-known scalar field theories in suitable limits. The validity of the first and second law of thermodynamics at the apparent horizon is discussed for any coupling. As examples, we have also explored the validity of those thermodynamics laws in some new cosmological solutions under the theory. Additionally, we have also considered the logarithmic entropy corrected relation and discuss the GSLT at the apparent horizon.


I. Introduction
The rapid growth of observational measurements on expansion history reveals the expanding paradigm of the universe.This fact is based on accumulative observational evidences mainly from Type Ia supernova and other renowned sources [1].The expanding phase implicates the presence of a repulsive force which compensates the attractiveness property of gravity on cosmological scales.This phenomenon may be translated as the existence of exotic matter components and most acceptable understanding for such enigma is termed as dark energy (DE) having large negative pressure.Various DE models and modified theories of gravity have been proposed to incorporate the role of DE in cosmic expansion history (for review see [2]).
In contrast to Einstein's relativity and its proposed modifications where the source of gravity is determined by curvature scalar terms, another formulation is presented which comprises torsional formulation as gravity source [3].This theory is labeled as TEGR (teleparallel equivalent of general relativity) and it is determined by a Lagrangian density involving a zero curvature Weitzenböck connection instead of a zero torsion Levi-Civita connection with the vierbein as a fundamental tool.The Weitzenböck connection is an specific connection which characterizes a globally flat space-time endowed with a non-zero torsion tensor.Using that connection, one can construct an alternative and equivalent theory of GR.The later comes out since the scalar torsion only differs by a boundary term B = 2 e ∂ µ (eT µ ) with the scalar curvature by the relationship R = −T + B, making both variations of the Einstein-Hilbert and TEGR actions the same.Thus, these two theories have the same field equations.However, these two theories have different geometrical interpretations since in TEGR the torsion acts a force meanwhile in GR, the gravitational effects are understood due to the curved space-time.TEGR is then further extended to a generalized form by the inclusion of a f (T ) function in the Lagrangian density (as f (R) is the extension of GR) and it has been tested cosmologically by numerous researchers [4].It is important to mention that f (T ) and f (R) are no longer equivalent theories, and in order to consider the equivalent teleparallel theory of f (R), one needs to consider a more generalized function f (T, B), incorporating the boundary term in the action [5].In [6], the authors studied some cosmological features (reconstruction method, thermodynamics and stability) within f (T, B) gravity and in [7], some cosmological solutions were found using the Noether's symmetry approach.Additionally, it has been proved that when one consider Gauss-Bonnet higher order terms, an additional boundary term B G (related to the contorsion tensor) needs to be introduced to find the equivalent teleparallel modified Gauss-Bonnet theory f (R, G) [8].
Later, Harko et al. [9] proposed a comprehensive form of this theory by involving a non-minimal torsion matter interaction in the Lagrangian density.In a recent paper [10], Zubair and Waheed have investigated the validity of energy constraints for some specific f (T ) models and discussed the feasible bounds of involved arbitrary parameters.They also discussed the validity of generalized second law of thermodynamic in cosmological constant regime [11].
Another very studied approach in modified theories of gravity is to change the matter content of the universe by adding a scalar field in the matter sector.These models have been considered several times in cosmology, having different kind of scalar fields such as quintessence, quintom, k-essence, etc (See [12][13][14]).Moreover, we can also extend that idea by adding a coupling between the scalar field and the gravitational sector (See [15][16][17]) where cosmologically speaking, we can have new interesting results such as the possibility of crossing the phantom barrier.Motivated with these theories, another interesting modified theories of gravity have also been discussed in the literature [18] on cosmological landscape.Recently, Bahamonde and Wright [19] presented a new model of teleparallel gravity by introducing a scalar field non-minimally coupled to both the torsion T and the boundary term B = 2∇ µ T µ .It is shown that such theory can describe the non-minimal coupling to torsion and also non-minimal coupling to scalar curvature under certain limits.
Black hole thermodynamics suggests that there is a fundamental connection between gravitation and thermodynamics [20].Hawking radiation [21], the proportionality relation between the temperature and surface gravity, and also the connection between horizon entropy and the area of a black hole [22] support this idea.Jacobson [23] was the first to deduce the Einstein field equations from the Clausius relation T h d Ŝh = δ Q, together with the fact that the entropy is proportional to the horizon area.In the case of a general spherically symmetric spacetime, it was shown that the field equations can be constituted as the first law of thermodynamics (FLT) [24].The relation between the FRW equations and the FLT was shown in [25] for A /G.The investigation about the validity of thermodynamical laws in GR as well as modified theories has been carried out by numerous researchers in the literature [26].
In this work we are focused on the validity of thermodynamical laws in a modified teleparallel gravity involving a non-minimal coupling between both torsion scalar and the boundary term with a scalar field.The present paper is coordinated in this format.In Sec.II, we give a brief introduction of this theory and then we derive the respective field equation for flat FRW geometry with a perfect fluid as the matter contents.In Sec.III and IV, we formulate the first and second law of thermodynamics at the apparent horizon for any coupling.Additionally, as examples, we study the validity of the thermodynamics laws using new cosmological solutions based on power-law and exponential law cosmology (see appendix A).In Sec.V, we discuss the validity of GSLT for the entropy functional with quantum corrections.In each case, suitable limits of the parameters are chosen in order to visualize the validity of GSLT in quintessence, scalar field nonminimally coupled to torsion (known as teleparallel dark energy) and non-minimally coupled to the scalar curvature theories.Finally, in Sec.VI, a discussion of the work is presented.
In the following paper, the notation used is the same as in [19], where the tetrad and the inverse of the tetrad fields are denoted by a lower letter e a µ and a capital letter E µ a respectively with the (+, −, −, −) metric signature.

II. Teleparallel quintessence with a nonminimal coupling to a boundary term
In this study we consider the modified teleparallel model which involves scalar field non-minimally coupled to torsion T and a boundary term defined in terms of divergence of torsion vector B = 2 e ∂ µ (eT µ ), where e = det(e a µ ).The action of this theory is given by where L m determines the matter contents, κ 2 = 8πG, V (φ) is the energy potential and f (φ) and g(φ) are coupling functions.For simplicity, we will use the notation NMC-(B+T) to refer to this theory (f (φ) = g(φ) = 0).In Ref. [19], the authors consider a special case of this action where f (φ) = 1 + κ 2 ξφ 2 and g(φ) = κ 2 χφ 2 .Non-minimally coupled scalar field with the boundary term B (NMC-B) is recovered if ξ = 0. Using dynamical system techniques, the cosmology in NMC-B was studied in [19].If χ = 0, we get the same action as in [15], which is known as "teleparallel dark energy theory" (TDE).By choosing χ = −ξ we obtain scalar field models non-minimally coupled to the Ricci scalar that hereafter, for simplicity we will label as NMC-R [16].In addition, the so-called "Minimally coupled quintessence" theories arise when we take χ = ξ = 0 [17].
Variation of the action (1) with respect to the tetrad field yields the following field equations where ✷ = ∇ α ∇ α ; ∇ α is the covariant derivative linked with the Levi-Civita connection symbol and T ν a is the matter energy momentum tensor.
If we vary the action (1) with respect to the scalar field, yields the following modified Klein-Gordon equation Throughout this paper, prime denotes differentiation with respect to φ.We will assume the homogeneous and isotropic flat FRW metric in Euclidean coordinates defined as where a(t) represents the scale factor and the corresponding tetrad components are e i µ = (1, a(t), a(t), a(t)).It is important to mention that in despite that f (T ) gravity is not invariant under local Lorentz transformations, the later tetrad is a "good tetrad" to consider since it does not constraint its field equations [27].Hence, this tetrad can be used safely within this scalar field theory too.The energy-momentum tensor of matter is defined as where u µ is the four velocity of the fluid and ρ m and p m define the matter energy density and pressure respectively.Using the tetrad components for the flat FRW metric, the field equations ( 2) lead to Here, H = ȧ(t)/a(t) is the Hubble parameter and dots and primes denote differentiation with respect to the time coordinate and the argument of the function respectively.We can also rewrite these equations in a fluid representation, where κ 2 eff = κ 2 f (φ) , ρ eff = ρ m + ρ φ is the total energy density and p eff = p m + p φ is the total pressure.The energy density and the pressure of the scalar field ρ φ and p φ are respectively defined as follows In this theory the standard continuity equation reads Hereafter, we will assume a standard equation of state for the matter given by a barotropic equation p m = wρ m , with w being the state parameter.If we use the above equation, we can directly find that the energy density becomes where ρ 0 is an integration constant.It is proper to mention that for a flat FRW metric, the torsion scalar and the boundary term are and hence the Ricci scalar is recovered via Finally, the equation for the scalar field, the so-called Klein-Gordon equation takes the form Note that the Klein-Gordon equation can be also obtained directly from the field equations ( 6) and ( 7), so that it is not an extra equation.

III. Thermodynamics in Modified Teleparallel Theory
In this section we are interested to explore the general thermodynamic laws in the framework of the theory studied in [19], where the authors considered a quintessence theory non-minimally coupled between both a torsion scalar T and the boundary term B with the scalar field (NMC-B+T).The main aim of the next sections will be to formulate the first and second laws of thermodynamics in this theory.The complete and general thermodynamics law will be derived.After that, we will use the cosmological solutions found in the appendix A to study some interesting examples to visualize if them satisfy or not the thermodynamic laws.

A. First Law of Thermodynamics
This section is devoted to investigate the validity of the first law of thermodynamics in NMC-(B+T) at the apparent horizon for a flat FRW universe.The condition h αβ ∂ α R A ∂ β R A = 0 gives the radius R A of apparent horizon for flat FRW metric as The associated temperature is T h = κ sg /2π, where the surface gravity κ sg is given by [29] By using Eqs ( 9) and (III A) we easily get Now, multiplying both sides of this equation by a factor −2πR A T h = 1 − ṘA /(2HR A ), we can rewrite the above equation as follows where we have used that A = 4πR 2 A .From here, we can identify as the entropy as the quantity which is multiplied by T h , namely Now, we define energy of the universe within the apparent horizon.The Misner-Sharp energy is defined as E = R A /(2G ef f ), which can be written as In terms of volume V = 4πR 3 A /3, we obtain that the energy density is given by Taking differential of energy relation, we get Combining Eqs. ( 19) and ( 23), it results in By defining the work density, we get Here T (de)αβ h αβ is energy-density of the dark components.Using above definition of work density in Eq. ( 24), we obtain which can be re-written as where , which is the first law of thermodynamics in this teleparallel theory.The extra term dS p defined in Eq. ( 27) can be treated as an entropy production term in non-equilibrium thermodynamics.In gravitational theories such as Einstein, Gauss-Bonnet and Lovelock gravities [35], the usual first law of thermodynamics is satisfied by the respective field equations.In fact these theories do not involve any surplus term in universal form of first law of thermodynamics i.e., T dS = dE − W dV .Initially Akbar and Cai used this approach to discuss thermodynamic laws in f (R) gravity [32].It is shown that an additional entropy term produces as compared to other modified theories.Later Bamba et al. [33] developed the first law of thermodynamics in Palatini f (R), f (T ), f (R, φ, X) (where X = −1/2g µν ∇ µ φ∇ ν φ is the kinetic term of a scalar field φ) and f (R, φ, X, G) (where G = R 2 − 4R µν R µν + R µνρσ R µνρσ is the Gauss-Bonnet invariant) theories and formulated additional entropy production term.Similar approach is applied to discuss thermodynamic laws in f (R, T ), f (R, L m ) and f (R, T, R µν T µν ) theories, one can see that presence of non-equilibrium entropy production terms is necessary in such theories [36].Bamba et al. [33] have shown that one can manipulate the FRW equations in order to redefine the entropy relation, which results in equilibrium description of thermodynamics so that first law of thermodynamics takes the form T dS = dE − W dV .Moreover, in all these theories it has been that usual form form of first law of thermodynamics i.e., T dS ef f = dE − W dV can be obtained by defining the general entropy relation as sum of horizon entropy and entropy production term.
Here, we may define the effective entropy term being the sum of horizon entropy and entropy production term as S eff = S h + S p so that Eq.( 27) can be rewritten as where S eff is the effective entropy related to the contributions involves scalar field non-minimally coupled to torsion T and a boundary term at the apparent horizon of FRW spacetime.

B. Generalized Second Law of Thermodynamics
According to the generalised second law of thermodynamic (GSLT), the entropy of matter and energy sources inside the horizon plus the entropy associated with boundary of horizon must be non-decreasing.In previous section we have shown that usual first law of thermodynamics does not hold in this theory.Therefore to study GSLT, we would use the modified first law of thermodynamics.In fact the generalized entropy relation satisfies the condition where Ṡh represents the entropy associated with the horizon, Ṡp represents the entropy production term and Ṡin is the sum of all entropy components inside the horizon.Let us proceed with the modified first law of thermodynamics which can be represented as where R h represent the radius of the horizon, T in denote temperature for all the components inside the horizon and Q i is interaction term for ith component.Summing up the total entropy inside the horizon, we find Here, Now, let us study the validity of GSLT at the apparent horizon.In de Sitter space time R A = 1/H and the future event horizon becomes the same as the Hubble horizon.The time derivative of ( 20) results in Here, we set the thermal equilibrium with T in = T h , so that Eq. ( 32) implies After simplification, we get Hence, Eq.( 29) implies the relation of GSLT of the form In the following sections we will take a look if the GSLT is satisfied for some interesting cosmological solutions that can be constructed from our solutions found before (see Sec.A 1-A 3).
a Power-law potential with χ = 1   4   For this case, the energy potential reads, .
In Fig. 1, we present the evolution of energy density ρ and potential V (φ) for the power law case.It can be seen that V (φ) is decreasing function of time, where we set ξ ≤ 0.
Here, we discuss the specific cases NMC-B (ξ = 0) and TDE (χ = 0).The validity of GSLT for quartic potential is shown in Fig. 2. For NMC-B theory, the GSLT is valid in the range 0 < χ < 1  12 and in case of TDE we need 0 < ξ < 1  6 .These constraints are set in accordance with the condition of power law solutions n > 1.We also the evolution of GSLT in Fig. 3, where we set n = 2 and choose the particular values for χ and ξ.The black curve corresponds to NMC-B with χ = 1  18 and blue curve represents the plot of GSLT for TDE with ξ = 1 18 .
FIG. 3. Validity of GSLT for power-law potential with n = 2.
c Case m = 2/3 In this case one can recover the inverse potential i.e., V (φ) ∝ φ −1 .Here, we show the validity of GSLT in Figs. 4  and 5.In Fig. 4, left plot corresponds to NMC-B which shows the validity in the range − 1 8 < χ < 0 and right plot corresponds to TDE with − 2 3 < ξ < 0. It can be seen that GSLT is violated in case of TDE whereas it is valid in both NMC-B and NMC-R.For this case the potential takes the following form, For this model, we have fixed value of χ, so we can not discuss the TDE and NMC-B theories.In case of NMC-R (χ = −ξ), we find that this representation does not show realistic results as we need to fix n < 0 for inverse and quadratic potentials.
Moreover, in case of NMC-(B+R), we explore the validity for quartic potential, where we need to set ξ = −(3n+1) 12n .Finally, we have GSLT dependence on n, and validity is shown in Figure 6.
FIG. 6.Here, we show the validity for NMC-(B+R) theory with quartic potential.In this case we find that GSLT can be met for n > 0.5.In left plot we show the validity region, and its behavior for particular choices is shown on the right side.For black and blue curves correspond to n = 1, 20.
2. Specific model: Exponential solutions for f (φ) = 1 + κ 2 αe mφ and g(φ) = κ 2 βe kφ This section is devoted to analyse if the GSLT is satisfied for some exponential solutions found in Sec.A 3. Since mostly all the solutions are similar, we will only analyse the case where w = 2−φ0 3n − 1 (see Sec.A 3 b).In this case, the energy potential contains three exponentials that can represent different kind of potentials.For example, by taking , we find which is known as a double exponential potential.This kind of potential has been widely studied in the literature.From Fig. 7, it is observed that the scalar potential increases as the field increases.Thus the cosmic acceleration will be driven by the potential energy of the scalar field.
Another interesting example that can be constructed by taking φ 0 = 4 and α = 24κ 2 −m 2 48κ 4 , which gives us the following potential which is an interesting kind of potential studied among the literature (see [41,42]).The exponential function models generally lead to accelerating expansion behavior of the universe.Let us know study the GSLT for this kind of potential.Under this model, Eq. ( 36) becomes Ṡtot = πt where for simplicity we took κ = G = 1.Clearly, the above inequality will hold depending on the values of the parameters φ 0 , n and α.Since n > 1, if φ 0 ≥ −2 and α > 0, GSLT will be valid at any time.Additionally, if φ 0 ≤ −2 and α < 0, the GSLT will be always true.For all the other cases, the validity of GSLT will depend on time.Figs. 8  and 9 show the behaviour of the GSLT inequality for different values of the parameters.For φ 0 > 0 and α < 0, the inequality will be more constraint for bigger |φ 0 |.Additionally, it can be seen that GSLT will be always true at very late times for −2 < φ 0 < 0. .The entropy area relation involving quantum corrections leads to curvature corrections in the Einstein-Hilbert action [37].The logarithmic corrected entropy is defined through the relation [38,39] where α, β and γ are dimensionless constants, however the exact values of these constants are yet to be determined.These corrections arise in black hole entropy in loop quantum gravity due to thermal equilibrium fluctuations and quantum fluctuations [40].In [39] Sadjadi and Jamil investigated the validity of GSLT for FRW space-time with logarithmic correction.They found that in a (super) accelerated universe the GSL is valid whenever α(<) > 0 leading to a (negative) positive contribution from logarithmic correction to the entropy.In the following section, we will present the validity of the GSLT with modified entropy relations involving logarithmic corrections.First note that the time derivative of Eq. ( 41) yields In case of the apparent horizon, Eq. ( 41) implies Here, we used that the Hubble horizon is R A = 1/H.Now, by using Eq. ( 35) and (43), we get which can be represented in terms of effective components as follows The GSL with quantum corrections have natural implications in the studies of the very early universe since quantum corrections are directly linked with high energy and short distance scales.

V. Conclusions
The non-minimally coupling models in cosmology are frequently used to study guaranteed late time acceleration; phantom crossing and the existence of finite time future singularity.In this paper, the thermodynamic study is executed in modified teleparallel theory which involves scalar field non-minimally coupled to both the torsion and a boundary term [19].One interesting factor in this theory is that under suitable limit, one can recover very well-known theories of gravity as quintessence, teleparallel dark energy and non-minimally coupled scalar field with the Ricci scalar R.
In this paper, we investigated the GSLT for an expanding universe with apparent horizon.The laws of thermodynamics are universally valid and hence must apply in a modified form to the whole universe.The existence of a apparent horizon in a spacetime provides the opportunity to formulate the first law and the generalized second law of thermodynamics.According to the laws of thermodynamics, the first law must always be true for a physical system (provided it is non-dissipative), however the GSLT holds exactly using the apparent horizon.We obtained these laws in the non-minimal coupled teleparallel theory with a boundary term and scalar field.Our results generalize many previous GSLT studies while extending them to the logarithmic corrected entropy-area law.Moreover GSLT with quantum corrections have natural implications in the studies of the very early universe since quantum corrections are directly linked with high energy and short distance scales.
We explored the existence of power law solutions for two different choices of Lagrangian coefficients, which are functions of scalar field f (φ) and g(φ).In first case we set f (φ) = 1 + κ 2 ξφ 2 and g(φ) = κ 2 χφ 2 and choose the power law forms of scale factor and scalar filed to discuss possible forms of the power law potential V (φ) (see [28]).Here, One can recover the quadratic, quartic, inverse and Ratra-Peebles potentials.We also discuss the Brans-Dicke theory as a particular case in power law cosmology.Moreover, we also discuss the power law solutions for the choice of f (φ) = 1 + κ 2 αe mφ and g(φ) = κ 2 βe kφ with φ(t) = φ0 m log(t).Here, we explored different form of V (φ) depending on equation of state parameter.
We showed the behavior of matter density ρ and potential V (φ) for power law potential, it can be seen that both are decreasing functions of time as shown in Fig. 1.In our discussion on validity of GSLT, we considered the quartic and inverse in all the viable special cases like TDE, NMC-B, NMC-R and NMC-(B+R) and presented results in Figs.2-6.In Figs. 2 and 3, we find that GSLT is true for NMC-B (with 0 < ξ < 1  6 ) and TDE (with 0 < χ < 1  12 ).For inverse potential, we find that GSLT is valid for NMC-B (with −1 8 < χ < 0) and TDE (with − 2 3 < ξ < 0), whereas it is violated for TDE as shown in Figs. 4 and 5.For the choice of χ = 1  4 , we found the validity region only in case of NMC-(B+R) with n > 1 2 as shown in Fig. 6.For exponential forms of Lagrangian coefficients, we found that GSLT is always true for (φ 0 ≤ −2 & α < 0) and (φ 0 ≥ −2 & α > 0) whereas for other choice of parameters validity of GSLT is time dependent.Finally, we also formulated the GSLT for logarithmic entropy corrections.