Spin connection and cosmological perturbations in $f(T)$ gravity

TL;DR

This paper investigates cosmological perturbations in $f(T)$ gravity, addressing issues in the standard approach by incorporating spin connection within a covariant framework, leading to more consistent equations of motion.

Contribution

It introduces a covariant formulation of $f(T)$ gravity with spin connection, resolving problems in standard perturbation analysis and deriving self-consistent equations.

Findings

Minimal spin connection solution leads to zero slip but over-determined system.

More general spin connection solutions allow non-zero slip and consistent equations.

The covariant approach improves the theoretical consistency of cosmological perturbation analysis.

Abstract

In this paper we study cosmological perturbations in teleparallel gravity. We discuss problems which appear in 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 symmetrize equation for perturbation and split diagonal and non-diagonal part of equation of motion. We demonstrate that there is minimal solution for spin connection, which lead to zero slip, however, in this case one additional equation appears, so the system may become over-determined. After that, we show that a more general solution exists, which is incompatible with zero slip, but allows to write down the equations of motion for cosmological perturbation in a self-consistent way without additional equations to be satisfied.