The coupling of topology and inflation in Noncommutative Cosmology

TL;DR

This paper demonstrates that in a noncommutative cosmology model based on the spectral action, the universe's topology influences inflation potentials and observable power spectra, linking geometry, topology, and inflation dynamics.

Contribution

It introduces a novel connection between cosmic topology and inflation potentials derived from the spectral action in noncommutative geometry, highlighting topology's impact on inflationary predictions.

Findings

Inflation potentials vary with different cosmic topologies.

Power spectra can distinguish between different topologies.

Slow-roll parameters are insensitive to topology within the same class.

Abstract

We show that, in a model of modified gravity based on the spectral action functional, there is a nontrivial coupling between cosmic topology and inflation, in the sense that the shape of the possible slow-roll inflation potentials obtained in the model from the nonperturbative form of the spectral action are sensitive not only to the geometry (flat or positively curved) of the universe, but also to the different possible non-simply connected topologies. We show this by explicitly computing the nonperturbative spectral action for some candidate flat cosmic topologies given by Bieberbach manifolds and showing that the resulting inflation potential differs from that of the flat torus by a multiplicative factor, similarly to what happens in the case of the spectral action of the spherical forms in relation to the case of the 3-sphere. We then show that, while the slow-roll parameters differ between the spherical and flat manifolds but do not distinguish different topologies within each class, the power spectra detect the different scalings of the slow-roll potential and therefore distinguish between the various topologies, both in the spherical and in the flat case.