Curvature and topology dependency of the cosmological spectra

Abstract

In this article we investigate dependency of two important cosmological random fields defined on spatial slices of the FLRW universe and their spectra 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 realize when K≠0, curvature scale may 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 the adiabatic and entropy spectral indices of the cosmological perturbations (spectra of curvature and entropy perturbations) in sub-horizon scales could be function of topology. It may be significant because reveals imprints of geometry on the statistical information deduced by observations. 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 case K=-1. In the most results dependency to curvature, initial conditions and scale modes are thoroughly vivid. Keywords: Adiabatic and entropy spectral indices, Entropy perturbation, Comoving curvature perturbation, Spatial sectional curvature, Spatial topology