Dynamics in interacting scalar-torsion theory

In a spatially flat \ Friedmann--Lema\^{\i}tre--Robertson--Walker background space we consider a scalar-torsion gravitational model which has similar properties with the dilaton theory. This teleparallel model is invariant under a discrete transformation similar to the Gasperini-Veneziano duality transformation. Moreover, in the gravitational Action integral we introduce the Lagrangian function of a pressureless fluid source which is coupled to the teleparallel dilaton field. This specific gravitational theory with interaction in the dark sector of the universe is investigated by using methods of the dynamical system analysis. We calculate that the theory provides various areas of special interest for the evolution of the cosmological history. Inflationary scaling solutions and the de Sitter universe is recovered. Furthermore, we calculate that there exist an attractor which provides a stable solution where the two fluid components, the scalar field and the pressureless matter, contribute in the cosmological fluid. This solution is of special interest because it can describe the present epoch. Finally, the qualitative evolution of the cosmographic parameters is discussed.


INTRODUCTION
The detailed analysis of the recent cosmological data indicates that General Relativity may need to be modified in order to describe the observations, for a recent review we refer the reader to [1]. The late-time cosmic acceleration has been attributed to a fluid, the socalled dark energy, with has negative pressure and anti-gravity effects [2]. In order to explain the anti-gravitational effects, cosmologists have proposed the modification of the Einstein-Hiblert Action by using geometric invariants [3]. In this direction new geometrodynamical terms are introduced in the field equations which can drive the dynamics in order to explain, with a geometric approach, the observational phenomena [4,5].
Teleparallelism [6,7] includes a class of modify theories of gravity which have been widely studied the last years [8][9][10][11][12][13][14][15]. The fundamental invariant of teleparallelism is the torsion scalar of the antisymmetric connection which play the role important role, instead of the Levi-Civita connection in General Relativity. In previous studies it has been discussed that teleparallel gravity my violates the Lorentz symmetry [16], while Lorentz violation has not been observed yet, it is common in various subjects of quantum gravity [17]. However, recent studies has shown that Lorentz symmetry can be preserved in teleparallelism, see for instance [18,19]. Specifically, in [18] it was found that the introduction of a scalar field in the gravitational Action Integral of teleparallelism preserves the Lorentz symmetry. For a recent review on teleparallelism we refer the reader in [20]. In the literature, teleparallel cosmology has been widely studied. For instance, in f (T ) teleparallel theory the cosmological perturbations were investigated in [21,22], while in [22] it was found that f (T ) theory can mimicking dynamical dark energy models. The mechanism which describe the Higgs inflation era in scalar torsion theory was studied in [23]. An extension of the scalar torsion theories with the introduction of the boundary term was introduced in [24]. For other recent studies on teleparallelism we refer the reader in [25][26][27][28][29] and references therein.
In this study we focus on the scalar-torsion or teleparallel dark energy models [30][31][32] which can be seen as the analogue of the scalar-tensor theories in teleparallelism. In scalartorsion theory an scalar field is introduced in the gravitational Action Integral which is nonminimally coupled to the fundamental scalar of teleparallelism, the torsion scalar. There are various studies in the literature which indicate that the scalar-torsion theories can explain the recent cosmological observations [34,35]. In the following we consider the existence of a matter source with zero pressure in the gravitation Action which interacts with the scalar field [36][37][38][39]. In our analysis the interaction between the scalar field and the pressureless fluid are inspired by the interaction provided in the Weyl integrable theory [40,41]. The plan of the paper is as follows.
In Section 2 we introduce the model of our consideration which is the teleparallel dilaton model coupled to a pressureless fluid source. This model belongs to the family of scalartorsion theories, the field equations are derived. Section 3 includes the new material of this study. We performed a detailed study of the asymptotic dynamics for the gravitational field equations for the model of our consideration. We determine the stationary points and we investigate their stability as we discuss the physical properties of the exact solutions described by the stationary points. This analysis provide important information about the cosmological viability of the proposed theory. It is clear that our model can explain the mayor eras of the cosmological evolution and it can be used as a dark energy candidate.
Furthermore, in Section 4 we discuss the evolution of the cosmographic parameters as they are provided by our model. Finally, in Section 5 we summarize our results and we draw our conclusions.

TELEPARALLEL DILATON MODEL
The gravitational model of our consideration is an extension of the teleparallel dilaton theory known as scalar-torsion theory where the gravitational Action Integral is defined in which T is the torsion scalar of the antisymmetric curvatureless connection, φ x k is a scalar field with potential function V (φ) , ω is a nonzero constant, L m is the Lagrangian function for the additional matter source and e is the determinant of the vierbein fields.
Action Integral (1) belongs to the family of gravitational models known as teleparallel dark energy models, or scalar torsion models [30,31,42,43]. Scalar-torsion models can be seen as the analogue of scalar-tensor models in teleparallelism, in which instead of the Ricci scalar the torsion scalar T is used for the definition of the Action Integral. The gravitational field equations are of second-order, however, under conformal transformations the scalartorsion theories are now equivalent with the quintessence model [42]. Under a conformal transformation the scalar-torsion Action Integral is equivalent with that of a modified higherorder theory known as f (T, B) where B is the boundary term differs the torsion T and the Ricci scalar [42]. That makes the scalar-torsion and scalar-tensor theories to be totally different which means that there is not any conformal transformation which may connect the solutions of the two theories. In this study, we assume that the matter source and the scalar field are interacting, that is, from (1) the mixed term e − φ 2 L m exists.
In the case of a spatially flat Friedmann-Lemaître-Robertson-Walker metric (FLRW) background space the gravitational field equations for the (1) in the case of vacuum are invariant under a discrete transformation with origin the O (d, d) symmetry [44]. The resulting discrete transformation is a generalization of the Gasperini-Veneziano scale-factor duality transformation for the dilaton field in scalar tensor theory [45]. The existence of this discrete transformation it is important for the study of the pre-big bang period of the universe as it is described by string cosmology. However, when parameter ω is small, the Gasperini-Veneziano transformation is recovered. However, in general the pre-big bang period for the teleparallel dilaton theory differs from that of sting cosmology by a term provided by the nonzero constant ω.
In this study we assume the scalar field φ interact with the matter source. For the latter, we assume that of a pressureless fluid source, dust fluid, with energy density ρ m , known as dark matter. Models with interaction in the dark sector of the universe has drawn the attention of cosmologists the last decade [46][47][48][49][50]. Indeed, there are various theoretical studies which shows that such models are viable, while from the analysis of the cosmological data it seems that the interacting models are supported by the observations [51][52][53][54]. The Action Integral (1) can be seen as the teleparallel extension of some scalar tensor models coupled to dark matter [40,41].
For the background space (2) it follows T = 6H 2 , H =˙a a , hence, from the Action integral (1) and for a dust fluid for the matter source we derive the modified gravitational field equations while for the matter source and the scalar field it followṡ In the case where ρ m = 0 and V (φ) = V 0 , the discrete transformation which keeps invariant the field equations is However, in the presence of the matter source the field equations do not remain invariant under the action of the discrete transformation. Moreover, when ω is near to zero the discrete transformation (7) becomes the Gasperini-Veneziano duality transformation [45]. The discrete transformation symmetry is determined which helps us to solve the Wheeler-DeWitt equation [44].
Additionally, from equation (3) we define the energy density for the scalar field , thus in order ρ φ ≥ 0 and do not have ghost terms we assume ω > 0. The pressure component p φ of the scalar field from equation (4) it is defined as In the following Section we study the general evolution for the cosmological history as it is provided by the dynamical system (3)-(6).

ASYMPTOTIC DYNAMICS
We define the new dimensionless variables where the field equations are written as the following algebraic-differential system in which the new independent variable is dt = Hdτ .
In addition, in the new variables the equation of state parameter for the effective fluid As far as the scalar field potential is concerned, we consider the exponential potential For the exponential potential we infer that λ = λ 0 is a constant and Γ (λ) = 1 such that the rhs of equation (12) is always zero. Thus, for the exponential potential the dimension of the dynamical system is two. Moreover, in order the matter source to be physically accepted, it follows that Ω m is constraint as, 0 ≤ Ω m ≤ 1, from (9) it follows that the variables (x, y) are constrained similarly as x 2 + y 2 ≤ 1. Furthermore, we observe that the equations are invariant under the change of variables y → −y, which means that without loss of generality we can work on the branch y ≥ 0.

Stationary points for the exponential potential
We proceed our analysis by considering the exponential scalar field potential. For arbitrary value of the free parameter λ the dynamical system (10) Yes under conditions

Attractor
The stationary points A ± 2 = (±1, 0) describe universes where only the scalar field contributes in the cosmological solution, that is, Ω m A ± 2 = 0. Because y A ± 2 = 0, it means that the scalar field potential does not contribute in the total cosmological evolution, while only the kinematic part contributes, that is, V (φ) <<φ 2 . The effective equation of state Furthermore, the asymptotic solution at A + 2 describes an accelerated universe when ω > 4 3 . On the other hand, the asymptotic solution at the stationary point A − 2 gives w ef f A − 2 > 1. As far as the stability properties of the points are concerned, we derive the eigenvalues Thus, point A + 2 is a saddle point when λ < 1 and ω < 1 3 (1 − λ) 2 , otherwise A + 2 is a source. Point A − 2 a saddle point when λ > 1 and ω < 1 3 (1 − λ 2 ) . The stationary point is physically accepted when ω > The eigenvalues of the linearized system are derived . We infer that point A 4 is always an attractor.
The results for the stationary points and their physical properties are summarized in Table I, while in Table II we summarize the stability conditions for the stationary points.

COSMOGRAPHIC PARAMETERS
The cosmographic approach is a model independent construction way of the cosmological physical variables [56]. Specifically, the scale factor it is written in the expansion form where H 0 is the value of the Hubble function of today, q 0 is the deceleration parameter of today, j 0 and s 0 are the present value for the jerk and snap parameters. The H, q, j, s are kinematical quantities which are extracted directly from the spacetime. The kinematic quantities are defined as [57,58]  , or in terms of the Hubble function they can be written in the equivalent form Thus, from the evolution of the cosmographic q, j, s, one can understand the expansion of the universe, the rate of acceleration and its derivative. In Figs. 3 and 4 we present the qualitative evolution of the cosmographic parameters for the model of our consideration for initial conditions near to the matter dominated era and for the values of the free parameters (λ, ω) same with that of the numerical solutions of Fig. 2. From the evolution of the cosmographic parameters presented in Figs. 3 and 4 we observe that while the q 0 value for the ΛCDM is recovered that it is not true for the jerk and snap parameters which in general have different evolution.
From the qualitative evolution we observe that our model can predict values for the cosmographic parameters as they given by the cosmological constraints [59].

CONCLUSIONS
In this study we considered a scalar-torsion model known as teleparallel dilaton theory coupled to a pressureless fluid source, which we assume that it describes the dark matter. [61]. Consequently, the present model of study has interesting properties which can explain the cosmic history and deserves further study. In addition, in the absence of the dust fluid we recall that the theory admits a discrete symmetry which can be used to study the cosmic evolution in the pre-big bang era as in string cosmology. However, in contrary to string cosmology and the dilaton field for this teleparallel model the pre-big bang era is not recovered as reflection of the present epoch.
In a future study, we plan to use the cosmological observations in order to constraint this specific theory as a dark energy candidate.