# Parity violating Friedmann Universes

## TL;DR

This paper explores parity-violating solutions in Einstein-Cartan cosmology with a variable cosmological constant, revealing new degrees of freedom and modifications to standard Friedmann universe models.

## Contribution

It introduces and analyzes parity-violating solutions within Einstein-Cartan theory, highlighting their distinct gauge structures and the role of the Pontryagin term without spinorial matter.

## Key findings

- Parity-violating solutions have non-zero Weyl curvature in FRW models.
- Parity even and odd solutions belong to different branches with distinct degrees of freedom.
- The parity-violating branch introduces a new degree of freedom not present in standard relativity.

## Abstract

We revisit models, within the Einstein-Cartan theory, where the cosmological constant $\Lambda$ is promoted to a variable, at the cost of allowing for torsion even in the absence of spinors. We remark that some standard notions about FRW Universes collapse in these theories, most notably spatial homogeneity and isotropy may now co-exist with violations of parity invariance. The parity violating solutions have non-vanishing Weyl curvature even within FRW models. The presence of parity violating torsion opens up the space of possible such theories with relevant FRW modifications: in particular the Pontryagin term can play an important role even in the absence of spinorial matter. We present a number of parity violating solutions with and without matter. The former are the non-self dual vacuum solutions long suspected to exist. The latter lead to tracking and non-tracking solutions with a number of observational problems, unless we invoke the Pontryagin term. An examination of the Hamiltonian structure of the theory reveals that the parity even and the parity violating solutions belong to two distinct branches of the theory, with different gauge symmetries (constraints) and different numbers of degrees of freedom. The parity even branch is nothing but standard relativity with a cosmological constant which has become pure gauge under conformal invariance if matter is absent, or a slave of matter (and so not an independent degree of freedom) if non-conformally invariant matter is present. In contrast, the parity violating branch contains a genuinely new degree of freedom.