Tensor perturbations during inflation in a spatially closed Universe

TL;DR

This paper investigates how spatial curvature in a closed universe affects tensor perturbations during inflation, revealing suppressed power at large scales and scale-dependent corrections to the tensor-to-scalar ratio, with minimal impact on observable B-mode polarization.

Contribution

It extends previous work on scalar perturbations to include tensor modes, showing curvature-induced modifications to the power spectra and consistency relations during inflation.

Findings

Tensor power spectrum shows suppression at long wavelengths.

Observable B-mode polarization remains nearly unchanged.

Scale-dependent corrections to tensor-to-scalar ratio are identified.

Abstract

In a recent paper [17], we studied the evolution of the background geometry and scalar perturbations in an inflationary, spatially closed Friedmann-Lema\^itre-Robertson-Walker (FLRW) model having constant positive spatial curvature and spatial topology $\mathbb S^3$. Due to the spatial curvature, the early phase of slow-roll inflation is modified, leading to suppression of power in the scalar power spectrum at large angular scales. In this paper, we extend the analysis to include tensor perturbations. We find that --- similarly to the scalar perturbations --- the tensor power spectrum also shows power suppression for long wavelength modes. The correction to the tensor spectrum is limited to the very long wavelength modes, therefore the resulting observable CMB B-mode polarization spectrum remains practically the same as in the standard scenario with flat spatial sections. However, since both the tensor and scalar power spectra are modified, there are scale dependent corrections to the tensor-to-scalar ratio that lead to violation of the standard slow-roll consistency relation.