Stability of scalar perturbations in scalar-torsion f(T,f) gravity theories in the presence of a matter fluid

We study the viability conditions for the absence of ghost, gradient and tachyonic instabilities, in scalar-torsion f(T,f) gravity theories in the presence of a general barotropic perfect fluid. To describe the matter sector, we use the Sorkin-Schutz action and then calculate the second order action for scalar perturbations. For the study of ghost and gradient instabilities, we found that the gravity sector keeps decoupled from the matter sector and then applied the viability conditions for each one separately. Particularly, we verified that this theory is free from ghost and gradient instabilities, obtaining the standard results for matter, and for the gravity sector we checked that the corresponding speed of propagation satisfies c_s,g2=1. On the other hand, in the case of tachyonic instability, we obtained the general expressions for the mass eigenvalues and then evaluated them in the scaling matter fixed points of a concrete model of dark energy. Thus, we found a space of parameters where it is possible to have a stable configuration respecting the constraints from the CMB measurements and the BBN constraints for early dark energy. Finally, we have numerically corroborated these results by solving the cosmological equations for a realistic cosmological evolution with phase space trajectories undergoing scaling matter regimes, and then showing that the system presents a stable configuration throughout cosmic evolution.