Constraining cosmological dynamics of scalar-tensor models of dark energy in teleparallel gravity

We consider a scalar-tensor theory in teleparallel gravity where a general function of the scalar field, f (phi), is non-minimally coupled to the torsion scalar T. First, we derive the field equations in this framework. Then, we study the cosmological evolution in a spatially flat, homogeneous, and isotropic universe described by the FRW metric, containing radiation and non-relativistic matter with energy densities rhoᵣ and rhoₘ, respectively. We analyze the system as an autonomous dynamical model of dark energy. The cosmological behavior depends on the coupling function sigma = f' (phi) /sqrt (f (phi) ) and the potential parameter lambda = V' (phi) * f (phi) / V (phi) * f' (phi). A constant coupling sigma leads to a quadratic form f (phi) proportional to (phi + c) ², while a constant lambda results in a power-law potential V (phi) proportional to f (phi) ˡambda. These forms are supported by mathematical and physical considerations. We perform phase space analysis and show that lambda much greater than 1 is a necessary condition to obtain a radiation-dominated era with effective equation of state wₑff approximately 1/3 and a matter-dominated era with wₑff approximately 0. Moreover, small sigma² ensures radiation and matter dominance in the respective eras. Finally, we derive the necessary conditions for a viable cosmological trajectory in this setting.