Cosmology of F(T) gravity and k-essence

This a brief review on $F(T)$ gravity and its relation with k-essence. Modified teleparallel gravity theory with the torsion scalar has recently gained a lot of attention as a possible explanation of dark energy. We perform a thorough reconstruction analysis on the so-called $F(T)$ models, where $F(T)$ is some general function of the torsion term, and deduce the required conditions for the equivalence between of $F(T)$ models with pure kinetic k-essence models. We present a new class of models of $F(T)$-gravity and k-essence.


Introduction
Recent astrophysical data imply that the current expansion of the universe is accelerating [1]. There exist different candidates for this acceleration phase. The simplest one is the introduction of the Cosmological Constant Λ in the framework of General Relativity (ΛCDM model), namely an exotic form of energy (the dark energy) whose Equation of State (EoS) parameter w is equal to minus one and dynamically remains near this value, but in principle quintessence/phantom-fluid description is not excluded. Despite the fact that the ΛCDM is a good candidate to describe our universe, the finite but very small value of the Λ causes some well-known problems, like the difference between the order of Λ predicted by quantum field theory which is so called by fine-tuning, another problem is the time where such acceleration happen which is the coincidence problem. Further, the origin of dark energy is an unsolved question. Also, the existence of an early accelerated epoch, namely the inflation, introduces a new problem to the standard cosmology, and various proposals to construct acceptable inflationary model which exist like the scalar, spinor SU (2), (non-)abelian vector theory (SU (2)) U (1) and so on.
Another alternative approach to the dark energy puzzle is the modified gravity theories. A typical modified gravity is a generalization of Einstein's gravity, where some combination of curvature invariants is added into the classical Hilbert-Einstein action of General Relativity. This modification may lead to an accelerated era without invoking the dark energy. The simplest theory of modified gravity is the F (R) one, where the modification is given by a function of the Ricci scalar only. Another popular modification is given by the string-inspired Gauss-Bonnet modified theories, where a modification via the topological invariant four dimensional Gauss Bonnet G appears (see the recent reviews [2]- [13]). Also it can be represented by the f (R, T ) models where T is the trace of the energy-momentum tensor [14]- [16]. The field equations of these theories are much more complicated with respect to the case of General Relativity, since they are 4th order differential equations and it is so difficult to obtain the exact solutions.
Recently a new type of gravity model, the F (T )-gravity, has been proposed. Its field equations are 2nd order [17]- [18]. These models are based on the "teleparallel" equivalent of General Relativity (TEGR) [19]- [25], which, instead of using the curvature defined via the Levi-Civita connection, uses the Weitzenböck connection that has no curvature but only torsion (see Refs. [24]- [25] for applications to inflation). The fact that the field equations of F (T ) gravity are 2nd order makes these theories simpler than the ones where modification is via curvature invariants and it is of extreme interest a deeper investigation on this kind of models (see Refs. [26]- [40] for recent developments).
In this paper we give a brief review on F (T ) gravity and its relation with k-essence. We study some F (T ) -models and models of k-essence. In the following sections 2 and 3 we present some basic facts on F (T ) gravity. In the section 4, we study some models of F (T ) gravity for the FRW spacetime. Noether symmetry in F (T ) gravity was considered in the section 5. In Sec. 5, we consider the torsion-scalar model. We investigate k-essence and its models in Sec. 7 and in Sec. 8. Sec. 9 is devoted to the study of the relation between F (T ) gravity and k-essence and in Sec. 10 we present some generalizations of F (T ) gravity. In the last section we give conclusions and general remarks.

General aspects of F (T ) gravity
The action of F (T ) -gravity reads [17,18,26] where Here T is the torsion scalar, e = det (e i µ ) = √ −g and L m is the matter Lagrangian. Here e i µ are the components of the vierbein vector field e A in the coordinate basis e A ≡ e µ A ∂ µ . Note that in the teleparallel gravity, the dynamical variable is the vierbein field e A (x µ ). To derive the equations of motion we consider the metric where g µν being the metric of space-time, η ab the Minkowski's metric, θ a the tetrads and e a µ and their inverses e µ a the tetrads basis. We note that the tetrad basis satisfy the relations The root of the metric determinant is given by The standard Weitzenbok's connection reads As a result, the covariant derivative, denoted by D µ , satisfies the equation Then the components of the torsion and the contorsion are given by Now we define another tensor from the components of torsion and the contorsion as Finally, we define the torsion scalar as usual Let us derive the equations of motion from the Euler-Lagrange equations. In order to use these equations we first write the quantities and where F T (T ) = dF (T )/dT and F T T (T ) = d 2 F (T )/dT 2 . Now we use the Euler-Lagrange equation Substituting the expressions (13) and (14) into the later equation we get the equations of motion of the F (T ) gravity (after mulltiplying by e −1 e a β /4) where is the gravitational energy momentum tensor. Of course, if we consider the TG case that is F (T ) = T then the gravitional equations reduce to which shows an equivalence between GR and TG since 3 The FRW space-time We will assume a flat homogeneous and isotropic FRW universe with the metric where t is cosmic time and a(t) is the scale factor. Then the modified Friedmann equations and the continuity equation read (see, e.g. [17], [18], [26]) This set can be rewritten as if we consider the following equivalent form of the action where f = F − T. Some properties of F (T ) -gravity were studied in [18]- [36]. The field equations (24)-(26) are equivalent toM wherê By using these equations we may construct high hierarchy of F (T ) gravity. For the case ρ m = p m = 0 such hierarchy is written asM where F 1 = F and (for n = 1, 2, 3) and so on.

Example 1: The M 13 -model
Let us consider the M 13 -model. Its Lagrangian is We consider the particular case where m = n = 1 and ν j = consts. Thus, By substituting these expressions into (2.9)-(2.10) we obtain where The effective EoS parameter is given by Let us set ν 1 = 1. Thus, and

Example 2: The M 21 -model
Our next example is the M 21 -model Now As a consequence, Eqs.(2.9)-(2.10) take the form One has The special case δ = 1/2 deserves a separate consideration. In this case the above equations take a simpler form For the effective energy density and pressure we get

Example 3: The M 22 -model
Now we consider the M 22 -model Thus so that Eqs.(2.9)-(2.10) take the form We have The EoS parameter reads

Example 4: The M 25 -model
In this subsection we will consider the M 25 -model where ξ = ln T . We take the case m = n = 1 and ν j = consts, namely Thus and In this case, Eqs.(2.9)-(2.10) lead to

Noether symmetry in F (T ) gravity
In this section we want to present a brief review on Noether symmetry in F (T ) gravity following to the paper [48]. Generally speaking, Noether symmetry is a power method to select models motivated at a fundamental level. It also allows to construct the exact solution of the model. We start from the point-like Lagrangian of F (T ) gravity: We now use the Euler-Lagrange equation: where q i are the generalized coordinates of the phase space and q i = a and T . Then we have Hence as f T T = 0 we obtain that is the Euler constraint of the dynamics. Next usingä/a = H 2 +Ḣ, we obtain i.e., the modified second Friedmann equation. Now let us consider the Hamiltonian corresponding to Lagrangian L [48]: so that Assuming that the total energy H = 0 (Hamiltonian constraint) and from Eq. (73), we get that is nothing but the first Friedmann equation. Now we want to present the Noether symmetry for F (T ) gravity in the FRW metric case. To do it, we introduce the generator of Noether symmetry as [48] where α = α(a, T ) and β = β(a, T ). As is well known, Noether symmetry exists if the equation has solution. Here L X L is the Lie derivative of the Lagrangian L with respect to the vector X.
The corresponding Noether charge reads as [48] From Eq. (79) and using the relationsα = (∂α/∂a)ȧ + (∂α/∂T )Ṫ ,β = (∂β/∂a)ȧ + (∂β/∂T )Ṫ , we come to the equation [48] 3αa Now we impose that the coefficients ofȧ 2 ,Ṫ 2 andȧṪ in Eq. (81) to be zero. Then we get [48] a ∂α As is known, the constraint (84) is sometimes called Noether condition. The corresponding Noether charge looks like [48] From Eq. (82) it follows that α = α(a). On the other hand, Eq. (84) gives us Hence we have which we recast as The last equations we split into two equations write as [48] nf − f T T = 0, These equations have the solutions [48] f (T ) = µT n , where µ and α 0 are real constants. So from Eq. (86), we get Finally we can conclude that the explicit non-zero solutions of f (T ), α and β exists that means Noether symmetry exists. Note that Noether symmetry allows us to construct the exact solution of a(t) for the given f (T ) model. For example, from Eq. (85) follows [48] a c1ȧ = c 2 , where Its solution reads as where c 3 = conts. This solution describes the accelerated expansion of the universe as where its prefactor (−1) 1+2n/3 · 3 2n c 2n/3 2 is not important. As is well-known, in order to get the expanding universe, the constraint n > 0 is required [48].

The torsion-scalar model
In this section we would like to study the F (T ) gravity in the presence of matter whose lagrangian is where φ is a scalar field and V (φ) the potential depending on φ. The equations of motion assume the form where ǫ = 1 for the usual case and ǫ = −1 for the phantom case. From this system we get where dot denotes the derivative with respect to the time. If we compare these equations with (2.14)-(2.16) we have For simplicity we restrict ourself to the case F = αT + βT 0.5 . Thus, and Let us consider some examples.
6.1 Example 1: a = δ sinh m [µt] In our first example we consider the following form for the scale factor a = δ sinh m [µt].
As a consequence .
So we obtain and the potential takes the form ( tanh such that We finally get 6.3 Example 3: a = a 0 t n The next example is given by for which and The potential assumes the final form 7 The k-essence The action of k-essence reads [41]- [43] The corresponding (closed) set of equations for FRW metric (2.8) is where X = −(1/2)φ 2 . The equation for the scalar field φ is given by which corresponds to the equation (3.4). In the pure kinetic k-essence case we have K φ = 0 and from the last equation one has (see, e.g. [44])

Models of k-essence for FRW universe
In what follows we will present some new models of k-essence. All of they may give rise to cosmic acceleration.

Example 1: The M 12 -model
Let us consider the M 12 -model with the following Lagrangian where in general ν j = ν j (φ) = ν j (N ) and N = ln (aa −1 0 ). We study the case m = 0, n = 2, ν j = const. The M 12 -model becomes To find ν j and X we look for H in the form where µ j = consts [in general µ j = µ j (t)]. This solution corresponds to the scale factor a = a 0 e N .
Finally, we obtain the following parametric form of the M 12 -model (parametric pure kinetic k-essence) 8.2 Example 2: The M 1 -model Our next example is the M 1 -model, whose Lagrangian assumes the form where in general ν j = ν j (φ) = ν j (t). Let us explore this model for the case: m = 0, n = 2 and ν j = consts. In this case the M 1 -model takes the form To find ν j and X we look for the following form of H, where µ j = consts [in general µ j = µ j (t)]. As a consequence, we obtain the following explicit form of the k-essence Lagrangian We also have For X we get the following expression from which Finally, we reconstruct the M 23 -model [We recall that in general the M 23 -model is read as where ζ = ln X.]

Example 3: The M 24 -model
Here we present the M 24 -model where γ 3 = κ −1 α 6 m 2 λ 2 σ 6 , v = tanh[σt] and λ, σ, α, β, n, m are some constants. Solving the equation (3.3) we obtain from which we derive the scale factor as Note thatḢ The relation between F (T )-gravity and k-essence in the FRW universe In this section, we want to analyze the relation between modified teleparallel gravity and pure kinetic k-essence. In Appendix C, we will consider this relation in the context of general modified gravity theories.

Version-I
Let us consider the following transformation where T = −6H 2 . Thus Eqs.(2.12)-(2.14) take the form These are the equations of motion of pure kinetic k-essence. This result shows that the field equations of modified teleparallel gravity and pure kinetic k-essence are equivalent to each other. This equivalence permits to construct a new class of pure kinetic k-essence models starting from some models of modified teleparallel gravity. Let us see it for the following modified teleparallel gravity model: f (T ) = αT n [17]- [18]. In this case, we have Substituting these expressions into the equations (4.1)-(4.2) we get Let us consider some specific case. i) If the scale factor behaves as a = a 0 e g(t) so that H =ġ,Ḣ =g, K and X take the form If we now consider the simplest case g = t (it means,ġ = 1,g = 0), we get ii) A non-trivial model may be obtained from a = a 0 t m . In this case H = mt −1 ,Ḣ = −mt −2 , T = −6m 2 t 2 and K and X take the form Since t = (γ 5 X) 1 6m−4 we finally get the following pure kinetic k-essence model

Version-II
Let us rewrite Eqs.(2.12)-(2.14) as where We introduce the following two functions K and X, These functions belongs to the system of the equations (4.3)-(4.6).
9.2 Specific case case: φ = φ 0 + ln a ± √ 12 One specific interesting case is given by It deserves separate investigation. In fact for this caseφ = ± √ 12H so that X = −0.5φ 2 = −6H 2 = T . The corresponding continuity equation is or, in terms of T , where ρ ′ = 2T f T − f, p ′ = f andρ ′ + 3H(ρ ′ + p ′ ) = 0. Let us split the equation (2.13) into two separate equations, Eq.(4.27) is automatically satisfied since it is just an another form for the continuity equation (4.26). So we finally obtain the equation system for F (T ) -gravity, which takes the form After the identification T = X = −6H 2 and f = 2κ 2 K, we recover equations (4.3)-(4.6) . So we can conclude that for the special case (4.24) both F (T ) -gravity and pure kinetic k-essence are equivalent to each other at least at the level of the dynamical equations. Some remarks can be observed from the continuity equation (4.25) [=(4.26)=(4.27)]. Two integrals of motion (I jT = 0) appear: Their general solution is given by Finally we would like to present an exact solution for both F (T )-gravity and pure kinetic k-essence. Let us consider the ΛCDM model for which a −3 = − 1 2ρ0 (T + 2Λ) = − 1 2ρ0 (X + 2Λ) so that which is the M 32 -model. This is the exact solution of the equations of motion of pure kinetic k-essence and F (T ) -gravity simultaneously.

F (R, T ) gravity
We have just considered one generalization of F (T ) in the presence of scalar field. In this section we would like to present another possible generalization of F (T ) gravity, namely the so-called F (R, T ) gravity.

The M 37 -model
The action of M 37 -gravity is given by [36] where L m is the matter Lagrangian, ǫ i = ±1 (signature) and Here u = u(x i ; g ij ,ġ ij ,g ij , ...; f j ) and v = v(x i ; g ij ,ġ ij ,g ij , ...; g j ) are some functions to be defined. Now we work in the FRW universe with the metric (2.9). In this case the curvature and torsion scalars can be written as where, u = u(t, a,ȧ,ä, ... a , ...; f i ) and v = v(t, a,ȧ,ä, ... a , ...; g i ) are some real functions, H = (ln a) t , while f i and g i are some unknown functions related with the geometry of the spacetime. By introducing the Lagrangian multipliers we can now rewrite the action (224) as where λ and γ are Lagrange multipliers. If we take the variations with respect to T and R of this action we get Therefore, the action (185) can be rewritten as Then the corresponding point-like Lagrangian reads as We finally obtain the following equations of the M 37 -model [36]: ρ + 3H(ρ + p) = 0. Here and The M 37 -model (224) admits some interesting particular and physically important cases. Let us see some example. i) F (R) -gravity. If the model is independent of the torsion, namely F = F (R, T ) = F (R), and we assume that u = 0, the action (224) takes the form where is the curvature scalar. We work with the FRW metric (225). In this case R assumes the form We rewrite the action as where the Lagrangian is given by The corresponding field equations of F (R) gravity read ρ + 3H(ρ + p) = 0.
ii) F (T ) -gravity. Now we assume that the function F = F (R, T ) is independent of the curvature scalar R and v = 0. In this case we get the modified teleparallel gravity -F (T ) gravity. Its gravitational action is where e = det (e i µ ) = √ −g. The torsion scalar T is defined as where For a spatially flat FRW metric (225), we have that the torsion scalar assumes the form The action (213) can be written as where the point-like Lagrangian reads The equations of F(T) gravity look like ρ + 3H(ρ + p) = 0.

The M 43 -model
In this subsection we consider the M 43 -model which is one of the representatives of F (R, T ) gravity. The action of M 43 -model reads as where L m is the matter Lagrangian, ǫ i = ±1 (signature), R is the curvature scalar, T is the torsion scalar. Let us consider the spacetime where the curvature and torsion are written by using the connection G λ µν as a sum of the curvature and torsion, namely Here Γ j iµ is the Levi-Civita connection and K j iµ is the contorsion. The quantities Γ j iµ and K j iµ are defined as Γ l jk = and respectively. Here the components of the torsion tensor are given by The curvature R ρ σµν is defined as where the Riemann curvature is defined in the standard waȳ Now we introduce the curvature and torsion scalars, where Now the M 43 -model is written in the form of (224). Now we want to present the M 43 -model for the spatially flat FRW spacetime. In this case the metric assumes the form ds 2 = −dt 2 + a 2 (t)(dx 2 + dy 2 + dz 2 ), where a(t) is the scale factor. In this case, the non-vanishing components of the Levi-Civita connection are where h and f are some real functions. Note that the indices of the torsion tensor are raised and lowered with the metric, namely T ijk = g kl T ij l .
In this way, we have derived the M 43 -model as one of geometrical realizations of F (R, T ) gravity by starting from the pure geometrical point of view.