Complete classification of cosmological teleparallel geometries

TL;DR

This paper provides a comprehensive classification of all homogeneous and isotropic teleparallel geometries, extending to include spatial reflections, and derives their properties and cosmological field equations.

Contribution

It offers the first complete classification of cosmologically symmetric teleparallel geometries using multiple methods, confirming their equivalence and exploring their properties.

Findings

All classified geometries are consistent across different construction methods.

The geometries' torsion tensors and transformation behaviors are explicitly characterized.

The most general cosmological field equations for teleparallel gravity are derived.

Abstract

We consider the notion of cosmological symmetry, i.e., spatial homogeneity and isotropy, in the field of teleparallel gravity and geometry, and provide a complete classification of all homogeneous and isotropic teleparallel geometries. We explicitly construct these geometries by independently employing three different methods, and prove that all of them lead to the same class of geometries. Further, we derive their properties, such as the torsion tensor and its irreducible decomposition, as well as the transformation behavior under change of the time coordinate, and derive the most general cosmological field equations for a number of teleparallel gravity theories. In addition to homogeneity and isotropy, we extend the notion of cosmological symmetry to also include spatial reflections, and find that this further restricts the possible teleparallel geometries. This work answers an important question in teleparallel cosmology, in which so far only particular examples of cosmologically symmetric solutions had been known, but it was unknown whether further solutions can be constructed.