The spectral action and cosmic topology

TL;DR

This paper explores how the spectral action functional, used as a gravity-matter model, varies with different cosmic topologies, affecting inflationary parameters and potentially revealing the universe's shape.

Contribution

It explicitly computes the spectral action for various cosmic topologies and analyzes their impact on inflationary slow-roll parameters.

Findings

Quaternionic and Poincaré spaces yield similar inflation behavior to spherical topology.

Flat tori exhibit different inflationary characteristics.

The study includes analysis of lens spaces as well.

Abstract

The spectral action functional, considered as a model of gravity coupled to matter, provides, in its non-perturbative form, a slow-roll potential for inflation, whose form and corresponding slow-roll parameters can be sensitive to the underlying cosmic topology. We explicitly compute the non-perturbative spectral action for some of the main candidates for cosmic topologies, namely the quaternionic space, the Poincare' dodecahedral space, and the flat tori. We compute the corresponding slow-roll parameters and see we check that the resulting inflation model behaves in the same way as for a simply-connected spherical topology in the case of the quaternionic space and the Poincare' homology sphere, while it behaves differently in the case of the flat tori. We add an appendix with a discussion of the case of lens spaces.