Bianchi Type Cosmological Models in $f(T)$ Tele-parallel Gravity

TL;DR

This paper investigates Bianchi type cosmological models within $f(T)$ tele-parallel gravity, deriving conditions for symmetry compatibility and highlighting differences between Bianchi types A and B in relation to the field equations.

Contribution

It provides a systematic analysis of spatially homogeneous Bianchi models in $f(T)$ gravity, identifying which symmetries are compatible with the theory's field equations.

Findings

Bianchi type A models are compatible with $f(T)$ gravity field equations.

Bianchi type B models are generally incompatible unless reduced to GR-like theories.

The study clarifies symmetry conditions necessary for consistent $f(T)$ cosmological solutions.

Abstract

Symmetry assumptions on the geometrical framework have provided successful mechanisms to develop physically meaningful solutions to many problems. In tele-parallel gravity, invariance of the frame and spin-connection under a group of motions defines an affine symmetry group. Here, we assume there exists a three-dimensional group of affine symmetries acting simply transitively on a spatial hypersurface and that this group of symmetry actions defines our affine frame symmetry group. We determine the general form of the co-frame and spin connection for each spatially homogeneous Bianchi type. We then construct the corresponding field equations for $f(T)$ tele-parallel gravity. We show that if the symmetry group is of Bianchi type A ($I$, $II$, $VI_0$, $VII_0$, $VIII$ or $IX$) then there exists a co-frame/spin connection pair that is consistent with the antisymmetric part of the field equations of $f(T)$ tele-parallel gravity. For those geometries having a Bianchi type B symmetry group ($IV$, $V$, $VI_h$, $VII_h$), we find that in general these geometries are inconsistent with the antisymmetric part of the $f(T)$ tele-parallel gravity field equations unless the theory reduces to an analog of General Relativity with a cosmological constant.