Curvature and topology dependency of the cosmological spectra

TL;DR

This paper explores how the geometry and topology of the universe influence the spectral indices of cosmological perturbations, revealing that curvature and topology significantly affect the spectra, especially on sub-horizon scales.

Contribution

It provides explicit equations for the evolution of curvature perturbations in universes with arbitrary spatial curvature and analyzes the impact of topology on spectral indices.

Findings

Spectra depend on curvature and topology at sub-horizon scales.

Time-average of isocurvature index is near zero for flat universe.

Curvature influences the evolution of super-curvature modes.

Abstract

In this article we investigate dependency of the adiabatic and entropy spectral indices of the cosmological perturbations on the geometry and topology of the background universe. Our discussion includes the post-inflationary universe i.e. radiation-dust mixture era. For this purpose, we first extract an explicit equation describing evolution of the comoving curvature perturbation in the FLRW universe with arbitrary spatial sectional curvature. We may percept when $K\neq 0$, curvature scale would be as significant as the perturbations scales to recognize the behavior of the spectral indices. We also focus on the entropy perturbation in order to extract behavior of the isocurvature spectral index in terms of the curvature index and time. Our analysis shows that spectra of curvature and entropy perturbations in sub-horizon scales could be function of topology. Moreover, an accurate analysis makes clear that time-average of isocurvature index in case $K=0$ is about zero,so that imprint of entropy perturbation in time duration may be negligible. We also consider evolution of the cosmological indices for super-curvature modes in the case $K=-1$. In the most results dependency to curvature, initial conditions and scale modes are thoroughly vivid.