Spin connection and cosmological perturbations in scalar-torsion gravity

In this paper we generalize our previous results on spin connection for the linear scalar cosmological perturbation in $f(T)$ theory to wider class of theories which includes a scalar field $\mathrm{\Phi }$ non-minimally coupled to torsion, and allows $\mathrm{\Phi }$-dependence of the function $f$. The case of $f_{TT}=0$, with a scalar field non-minimally coupled to torsion needs a special treatment. In both considered cases we derive self-consistent equations of motion for cosmological perturbation using a solution for non-trivial spin connection.