Revisiting Cosmologies in Teleparallelism

TL;DR

This paper explores the most general cosmological equations in theories of gravity based on non-metricity and torsion, revealing how different extensions modify or reproduce General Relativity solutions.

Contribution

It systematically derives the field equations for non-linear non-metricity and torsion theories, showing the dynamical role of connections in $f( ext{Q})$ cosmology and fixed connections in $f( ext{T})$ cosmology.

Findings

$f( ext{Q})$ cosmology includes GR solutions and new ones.

In $f( ext{Q})$ theories, connections can be dynamical.

In $f( ext{T})$ theories, connections are non-dynamical.

Abstract

We discuss the most general field equations for cosmological spacetimes for theories of gravity based on non-linear extensions of the non-metricity scalar and the torsion scalar. Our approach is based on a systematic symmetry-reduction of the metric-affine geometry which underlies these theories. While for the simplest conceivable case the connection disappears from the field equations and one obtains the Friedmann equations of General Relativity, we show that in $f(\mathbb{Q})$ cosmology the connection generically modifies the metric field equations and that some of the connection components become dynamical. We show that $f(\mathbb{Q})$ cosmology contains the exact General Relativity solutions and also exact solutions which go beyond. In $f(\mathbb{T})$~cosmology, however, the connection is completely fixed and not dynamical.