Spin connection and cosmological perturbations in f (T) gravity

In this paper we study small scalar cosmological perturbations in teleparallel gravity in linear order. We discuss problems which appear in the standard approach to f (T) gravity, and find that these problems may be solved within covariant formulation of teleparallel gravity, which take into account spin connection. We calculate spin connection which symmetrizes the equation for perturbation and split diagonal and nondiagonal parts of the equation of motion. We demonstrate that there is a solution for spin connection, which leads to zero gravitational slip; however, in this case one additional equation appears, so the system may become overdetermined. After that, we show that a more general solution exists, which is incompatible with zero slip, but allows one to write down the equations of motion for cosmological perturbation in a self-consistent way without additional equations to be satisfied.