Energy conditions in $F(T,\Theta)$ gravity and compatibility with a stable de Sitter solution

We study a new type of the modified teleparallel gravity of the form $F(T,\,\Theta)$ in which $T$, the torsion scalar, is coupled with $\Theta$, the trace of the stress-energy tensor. In a perturbational approach, we study the stability of the solutions and as a special case we find a condition for stability of the de Sitter phase. Then we adopt a suitable form for $F(T,\Theta)$ that realizes a stable de Sitter solution so that the stability condition creates a specific constraint on the parametric space of the model. Finally, the energy conditions in the framework of $F(T,\Theta)$ gravity is investigated.


Introduction
Einstein's general relativity is a completely geometrical theory so that gravitation is described not as a force, but as a geometric deformation of flat Minkowski space-time. In this point of view, the gravitational field creates a curvature in space time that its action on the particles is determined by allowing them to follow the geodesics of the space time. In this approach, trajectories is described by the geodesic equation not the force equation [1]. On the other hand, in 1928, Einstein in an attempt to build a unified gauge theory of gravitation and electrodynamics presented the other theory of gravity, the so-called teleparallel gravity [2]. In this theory torsion, the antisymmetric part of connection, is non-zero and torsion instead of curvature describes the gravitational interaction. In teleparallel gravity, tetrad (or vierbein) fields form the (pseudo) orthogonal bases for the tangent space at each point of flat space time. Similar to the metric tensor in general relativity, here tetrad play the role of the dynamical variables. Teleparallel gravity also uses the curvature-free Weitzenböck connection instead of Levi-Civita connection of general relativity to define covariant derivatives [3]. In spite of the such fundamental conceptual differences between teleparallel theory and general relativity, it has been shown that teleparallel Lagrangian density only differs with Ricci scalar by a total divergence [4,5].
In general relativity, the dark energy puzzle can be addressed by introducing additional geometrical degree of freedom into the theory, that is called F (R) modified gravity. In F (R) gravity the late time acceleration of the universe is catched by dark geometry instead of dark energy [6]. The modification of gravity in teleparallel gravity is accomplished by supplementing an additional torsion term into Einstein-Hilbert Lagrangian [7]. The F (T ) gravity has interesting properties that the field equations are of second order, unlike F (R) gravity which is of fourth order in the metric approach. In this context, F (T ) models have been extensively used in cosmology to explain the late time cosmic speed-up expansion without the need of dark energy [7,8].
In this paper we construct a generalization of F (T ) modified gravity by considering coupling between torsion scalar T and trace of the stress-energy tensor Θ via a general function as F (T, Θ). Then we investigate stability of the de Sitter solution (when subjected to homogeneous perturbations) in this framework. In this sense, we obtain a stability condition for the de Sitter phase in the general F (T, Θ) theories. Then we propose a specific F (T, Θ) model and show that the stability condition can be expressed as a constraint equation between the parameters of the model. We also consider the constraints imposed by the energy conditions and investigate whether the parameters ranges of the proposed model are consistent with the stability conditions. We note that since homogeneous and isotropic perturbations can be considered as the route to determine the stability of different modified gravity theories (see for instance [9]), the full anisotropic analysis of the cosmological perturbations is not considered here.
The paper is organized as follows: in Section II the general features of the F (T, Θ) theories is explored by writing the corresponding modified Einstein equations. In Section III the evolution equations of corresponding perturbations in FRW background is introduced. In section IV we devote to the study of stability around the de Sitter solution. In section V we present the energy conditions in F (T, Θ) gravity and compare the results with the obtained constraints from the stability conditions. We close the paper by giving our conclusions in Section VI.

F (T, Θ) Gravity
In this section, firstly a general F (T, Θ) function is considered for Lagrangian density of the action as follows where κ 2 = 8πG. e = √ −g is determinant of the vierbein e i µ and T is the torsion scalar. g is the determinant of the metric tensor and the metric of the space-time g µν is related to the vierbein by g µν = η ij e i µ e j ν . Here we use the Greek alphabet (µ, ν, ρ, ... = 0, 1, 2, 3) to denote indices related to spacetime, and the Latin alphabet (i, j, k, ... = 0, 1, 2, 3) to denote indices related to the tangent space. The ordinary matter part of the action is shown by L m and the corresponding stress-energy tensor is The connection that is used in general relativity, is the Levi-Civita connection This connection leads us to nonzero spacetime curvature but zero torsion [10]. In teleparallel gravity, tetrad fields give rise to a connection namely the Weitzenböck connection, instead of the Levi-Civita connection, which is defined bỹ One of the consequences of this definition is that the covariant derivative, D µ , of the tetrad fields vanishes identically: This equation leads us to a zero curvature but nonzero torsion [10]. We define the torsion and contortion by respectively, that the contortion is expressed as the interrelation between Weitzenböck and Levi-Civita connections [4]. Now, one can define super-potential as follows to obtain the torsion scalar which is used as the Lagrangian density in formulation of the teleparallel theories. The generalized field equations are extracted by varying of the expression (1) with respect to the vierbein field e i ν as follows where Ξ µρ i = ∂Θ/∂ e i µ,ρ and Υ µ i = 1 4 ∂Θ/∂e i µ . Note that F T and F Θ (F T T and F ΘΘ ) are the first (second) derivatives of the F (T, Θ) with respect to T and Θ, respectively. Here, it is assumed that the Lagrangian of matter is only in terms of e i µ , and so Ξ µρ i is zero. Since Θ = e j α Θ α j , one can say Υ µ i is written as Υ . So the field equations (10) can be rewritten as follows On the other hand, Ω µ i by definition (2) takes the form In this paper, we consider a perfect fluid form for the stress-energy tensor of the matter as Θ i µ = (ρ + p)u i u µ − p e i µ , where ρ is the energy density, p is the pressure and u µ describes the four-velocity. We also assume that the matter Lagrangian takes the form L m = −ρ [11] (see [12] for the case of L m = −p). Thus, with these assumptions Ω µ i is rewritten as Now in a flat FRW background, ds 2 = dt 2 − a 2 (t) dX 2 with scale factor a(t), the field equations for F (T, Θ) gravity are given by and where a dot denotes derivative with respect to time. Torsion scalar as a function of the Hubble parameter H =˙a a is expressed by Using Eqs. (14) and (15), one can obtain the modified Friedmann equations as follows and where ρ T and p T are energy density and pressure contribution of torsion scalar, respectively. ρ (T,Θ) and p (T,Θ) are the energy density and pressure contribution of the coupling between torsion and stress-energy tensor, respectively. These quantities are defined as follows and From Eqs. (19) − (22), we can define gravitationally effective form of dark energy density ρ DE = ρ T + ρ (T,Θ) and pressure p DE = p T + p (T,Θ) , so that equation of state parameter is defined as In which follows, we want to rewrite the field equations (11) to a suitable form for our purpose in section 5. To this end, we firstly multiply g µσ e i ν in both sides of (11), so that the coefficient of the term F T takes the following form where we have used the following relation On the other hand, by Eq. (7), the Riemann tensor for the Levi-Civita connection is written in the following form then the corresponding Ricci tensor is written as By using K ρ µν given in Eq. (8) and the relations (25), and also by considering S µ ρµ = K µ ρµ = T µ µρ one obtains [10,13,14] therefore, one reaches to where G µν = R µν − (1/2)g µν R is the Einstein tensor. Finally, combining Eqs. (24) and (29), one can rewrite the field equations for F (T, Θ) gravity as follows In upcoming sections, we use the trace of Eq. (30) as an independent relation to simplify the field equation. Since A µ µ = − 1 2 (R + 2T ), the mentioned trace can be expressed as where B = B µ µ , Ω = Ω µ µ and Θ = Θ µ µ [10].

Perturbations of the flat FRW solutions
Now we study the homogenous and isotropic perturbations around a specific cosmological solution for the model described by the action (1). First, we obtain the perturbed equations for the most general case. Then as a specific case, the de Sitter solution will be studied in which follows. Let us assume a general solution in the FRW cosmological background, which is described by a Hubble parameter H =H(t) . This solution for a particular F (T, Θ) model satisfies equation (14). The matter fluid is assumed to be in the form of a perfect fluid with a constant equation of state p = ω ρ, in which the matter energy density ρ satisfies the standard continuity equationρ Then the evolution of the matter energy density is obtained by solving the continuity equation Where is expressed in term of particular solutionH(t). In order to investigate the perturbations around the solutions H =H(t) and energy density (34), small deviations from the Hubble parameter and the energy density evolution is considered as [15] where δ(t) and δ m (t) correspond to the isotropic deviation of the background Hubble parameter and the matter energy density, respectively. In which follows, to study the behavior of the perturbations in linear regime, the F (T, Θ) function is expanded in the power ofT andΘ evaluated at the solution H =H(t) as: where a bar indicates the value of F (T, Θ) function and its derivatives evaluated at T =T and Θ =Θ . By substituting Eqs. (35) and (36) into the Friedmann equation (14), one can obtain an expression for the perturbations δ(t) in linear approximation as follows It seems that Eq. (37) is an algebraic equation for δ(t), but since Θ as trace of the stress-energy tensor itself is expressed in terms of H andḢ, one should find a differential equation for δ(t) .
For this purpose, we firstly contract the field equations (11) by e i µ to find It is easy to show that Also by using expression (13) and Ω µ µ = e i µ Ω µ i , one obtains So, the expression for Θ can be deduced as follows Now one can substitute Eqs. (35) and (36) into the expression for Θ and then Θ in Eq. (37) in order to get the corresponding differential equation for δ(t) as follows where C 0 ,1 ,2 and C m depend on the F (T, Θ) and its derivatives which are explicitly written in the following and Also, there is another perturbed equation which is obtained from the matter continuity equation (33) and perturbed expressions (35), as followṡ In general relativity, the stability equation (42) is reduced to an algebraic relation between geometrical and the matter perturbations. For higher order theories of gravity, the evolution of the perturbations is in general determined by a system composed of ordinary differential equations (42) and (47). Equation (42) is a non-homogeneous and linear first order differential equation. To solve this differential equation, one firstly rewrites it in the standard form and then finds an integrating factor. Multiplying the standard equation by integrating factor, δ(t) can be obtained as Hence, for a FRW cosmological solution, the stability of the solution can be investigated in the framework of F (T, Θ) gravity by solving the equations (42) and (47). In the next section, we will illustrate the previous discussions by considering theories which include the de Sitter solution as the simplest cosmological solution.

The stability of the de Sitter solution
The de Sitter solution is one of the simplest cosmological solutions which can realize the latetime accelerated phase of the universe expansion as well as the inflationary epoch. On the other hand, the existence of a stable de Sitter solution helps the theory to be cosmologically viable. Therefore, we study the stability of the de Sitter solution, where H (0) is a constant. Since the de Sitter solution is a vacuum solution, the perturbations is depend on only the underlying gravitational theory. According to the differential equation for the perturbations, Eq. (42), now we have 51) where the notation (0) indicates the value of each function evaluated in the de Sitter phase T = T (0) and Θ = Θ (0) . The general solution of equation (50) demonstrates the dynamical behavior of the gravitational perturbations as follows where α is an arbitrary integration constant. As we see, the growth of the gravitational perturbations tends to the constant value − with the stability condition: As we see the stability of the de Sitter solution depends explicitly on the values of the function F (T, Θ) and its derivatives at T (0) and Θ (0) . In order to display the previous calculations, we consider the F (T, Θ) function as follows (see for instance [15]) where k 1 and k 2 are positive or negative coupling constants. Now one can easily solve Eq. (14) to find Θ n (0) = Then by using Eqs. (16) and (41) (in the linear regime), we obtain Now, the combination of the last two equations gives the following de Sitter solution As a specific example, we consider the case with m = 2 3 and n = 1 . Thus the de Sitter solution takes the following form As we see, in this model there is a determinate district for k 2 and it is that for a universe with a well-defined de Sitter expansion, k 2 must be negative. Now, the stability condition (53) by using (56) gives Substituting (16) and (58) into (59), the stability condition is rewritten as −576 It is clear that this inequality is incorrect. In other words, in the model with m = 3 2 and n = 1 the perturbation grows exponentially and the de Sitter solution becomes unstable. So, it is not cosmologically a viable model. The other example is a model with m = 6 7 and n = 1 (we note that these choices are restricted from the energy conditions viewpoint as we see later). In this case, the de Sitter solution takes the form H (0) = (−32 k 7 2 ) − 1 12 . Again for a universe with a well-defined de Sitter expansion, k 2 must be negative. Here the stability condition (53) by using (56) and (58)  Thus the stability condition reduces to Note that for the models with n = 1, the stability condition has no dependence to the parameter k 1 . Generally, to recover the teleparallel equivalent of the General Relativity, k 1 should be positive.
In the upcoming section, we discuss the energy conditions in the general F (T, Θ) theories, then we specify a kind of the F (T, Θ) function in the spirit of (54) and finally we investigate whether the energy conditions can be satisfied in the context of the constraint (61) or not.

Energy conditions
The Raychaudhuri equation is origin of the strong and null energy conditions together with the requirement that gravity is attractive for the space time manifold that is endowed by a metric g µν . For a congruence of timelike geodesics with tangent vector field u µ , Raychaudhuri equation as the temporal variation of expansion θ [16] is defined as follows where θ is expansion parameter, σ µν and ω µν are, respectively, shear and rotation associated with the congruence defined by the vector field u µ . In the case of null vector field n µ the temporal version of the expansion is given by Note that the Raychaudhuri equation is a purely geometric equation and hence, it is not restricted to a specific theory of gravitation. Since the shear tensor is purely spatial σ µν σ µν > 0, thus, for any hypersurface of orthogonal congruence (ω µν = 0), the conditions for gravity to be attractive, become SEC : By using the field equations of any gravitational theory, one can relate the Ricci tensor to the energy-momentum tensor Θ µν . Thus, the combination of the field equations and Raychaudhuri equation sets a series of physical conditions for the energy-momentum tensor. By employing (64) and (65) in the general relativity framework, one can restrict the energy-momentum tensor as follows and R µν n µ n ν = Θ µν n µ n ν ≥ 0 where for a perfect fluid with density ρ and pressure p , this expression reduces to the well-known forms of the SEC and NEC in general relativity: Note that in the inequalities (66) and (67) we have set κ 2 = 1. In which follows we continue to use this convention.

The energy condition in F (T, Θ) gravity
The Raychaudhuri equation together with attractor character of the gravitational interaction have led us to Eqs. (66) and (67). These relations hold for any theory of gravity. In which follows, we apply this approach to drive the strong energy condition (SEC) and null energy condition (NEC) in F (T, Θ) gravity. First we rewrite the field equations (30) as follows From this equation and the trace of the field equations, Eq. (32), we have where Now in a FRW background, from Eqs. (6) and (9) along with Eq. (31), we obtain For a perfect fluid of density ρ and pressure p, Θ µν = e α i g αν Θ i µ , and by taking u µ = (1, 0, 0, 0) and n µ = (1, a, 0, 0), we obtain the T µν and its trace T as follows and Here we can use equations (64) and (65) together with Eqs. (75) and (76), for a general F (T, Θ) gravity, to achieve the strong and null energy conditions respectively as As one may expect, the energy conditions in the general relativity framework i.e. Eq. (68), can be recovered as a particular case of SEC and NEC in the context of F (T, Θ) gravity if we set F (T, Θ) = T . By defining an effective energy-momentum tensor in the context of F (T, Θ) gravity, SEC and NEC can also be recasted in the form of that of GR ( ρ ef f + 3p ef f ≥ 0 and ρ ef f + p ef f ≥ 0 , respectively). In this respect we can drive weak energy condition (WEC) and dominant energy condition (DEC). The effective energy-momentum tensor in the framework of F (T, Θ) gravity is defined as follows (similar to F (T ) gravity in Ref. [14]) ρ ef f and p ef f can be derived via the effective energy-momentum tensor by the following definitions Thus, using the effective energy-momentum tensor approach, the weak energy condition (WEC) in F (T, Θ) gravity (ρ ef f ≥ 0) is written as Finally one can write the dominant energy condition (ρ ef f > |p ef f | ≥ 0 ) as follows In the next section, we test one of the F (T, Θ) models in the context of the energy conditions we derived. In this way we obtain a constraint on the parametric space of the model.

Constraining F (T, Θ) models from energy conditions
We firstly list the energy conditions in terms of the phenomenological parameter of deceleration q = −ä a H −2 = −(1 +Ḣ H 2 ). The positivity of the Newtonian gravitational constant requires also the constraint F T > 0. With these notifications, the energy conditions are rewritten as follows NEC : We note that all the above conditions depend on the present value of pressure p 0 , so for simplicity we assume p 0 = 0 . Then we should adopt a specific function for F (T, Θ) to obtain the constraints on the parametric space of the considered model from the point of view of the energy conditions. On the other hand, we know that in order for a theoretical model to be cosmologically viable, it should satisfy at least the weak energy condition. This leads us to the mentioned constraints on parametric space of the model. Here we again consider F (T, Θ) = k 1 T + k 2 T m Θ n as our background gravitational model. The weak energy condition together with Eqs. (55) and (56) is satisfied by By restricting the parameter m values, one can constrain the parameter k 1 . Also to recover the teleparallel equivalent of the General Relativity, k 1 should be positive. Now one can obtain three ranges for m as 1 2 < m < 1 , m > 1 and m < 1 2 in which the constraint (87) is rewritten as follows 1. The case with m < 1 2 and 1 < m : Here by considering the condition for recovery of General Relativity, that is, k 1 > 0, we are led to the other constraint on the parameter n and m as 5 ≤ 5n 2(1 − m) < 6 .
2. The case with 1 2 < m < 1 : Now the mentioned condition, k 1 > 0, leads us to the following constraint: So, in the model with m = 2 3 and n = 1 which is considered in section 4, the weak energy condition can be realized with condition k 1 ≤ 2Ω m 0 , but the de Sitter solution is unstable in this case. Nevertheless, we could find the suitable values for m and n for which the weak energy condition can be realized as well as the stable de Sitter solution. This can be done if we set m = 6 7 and n = 1 for instance. In this case the weak energy condition holds if k 1 ≤ 150 23 Ω m 0 and the de Sitter phase is stable. Also one can investigate consistency of the null energy condition in the de Sitter phase. Note that one should set the value of q 0 = −1 for the de Sitter phase, so that the coefficient of the term F T 0 T 0 in Eq. (85) vanishes. The NEC in the de Sitter phase imposes a constraint on the parameters m and n as follows n 12(m − 1) + 5n ≤ 1 3 On the other hand, the constraints of WEC on the m and n (along with the positivity of k 1 ) which have already been mentioned, can be used to obtain a more restricted ranges of the parameters m and n . For example, in the second case in which 1 2 < m < 1, the NEC in the de Sitter solution imposes the following constraints So, in the model with m = 2 3 and n = 1, in spite of the realization of the WEC with condition k 1 ≤ 2Ω m 0 , the null energy condition can not be satisfied. While in the model with m = 6 7 and n = 1 in the de Sitter phase, both of the WEC (with k 1 ≤ 150 23 Ω m 0 ) and NEC are realized as well as the stable de Sitter solution. Thus, the later model is cosmologically viable.

Conclusion
In this work we discussed the cosmological viability of an alternative gravitational theory, namely, the modified F (T, Θ) gravity, where T is the Torsion scalar and Θ is the trace of the energy-momentum tensor. The viability of the model is based on the existence of a stable de Sitter solution and the realization of all the energy conditions or at least some of them. In a perturbational approach, we have obtained a differential equation for δ(t) . As a special case, we analyzed the differential equation for the de Sitter solution and we obtained a condition for the stability of this solution. Then we focused on the case where the algebraic function F (T, Θ) is cast into F (T, Θ) = k 1 T + k 2 T m Θ n , where k 1 , k 2 , m and n are input parameters. We firstly adopted the case with m = 2 3 and n = 1 and we have shown that the perturbations in the model grow with time exponentially. Then we considered the other case with m = 6 7 and n = 1. This model realizes a stable de Sitter phase with the condition k 2 < −0.76. Note that for simplicity we have adopted the value n = 1, because by this choice one gets rid of the dependence of the stability condition to the parameter k 1 . Finally we investigated the energy conditions in the F (T, Θ) models. We focused on the fact that WEC is the main condition for the cosmological viability of the theory to obtain a constraint on the parameters m , n and k 1 . Then by assuming that the parameter k 1 should be positive to recover the teleparallel equivalent of the General Relativity, we achieved the more restricted parametric space for m and n . In the next step, the adopted values for m and n (in the stability discussion) are applied. We have shown that the case with m = 2 3 , n = 1 and the other case with m = 6 7 and n = 1 can realize the WEC along with k 1 < 2Ω m 0 and k 1 ≤ 150 23 Ω m 0 , respectively. In the last step we considered the cosmological viability of the model from the point of view of the NEC . Since the purpose of our study was the comparison of the energy conditions with the stability of the de Sitter phase, we considered NEC at q = −1 (in the de Sitter solution). Here we obtained the more complete constraint on the m and n , so that it entails both WEC and NEC . As we saw, the case with m = 6 7 and n = 1 realizes NEC too and is cosmologically a viable gravitational theory.