Evolution of the spectral index after inflation

TL;DR

This paper analyzes how the spectral indices of curvature and isocurvature perturbations evolve over time after inflation, revealing dependence on initial conditions and revising the Sachs-Wolfe effect calculation.

Contribution

It derives an explicit equation for the evolution of the comoving curvature perturbation and revises the Sachs-Wolfe effect factor based on this analysis.

Findings

Adiabatic spectral index depends on initial conditions and scale.

Adiabatic spectral index approaches a constant after recombination.

Revised the Sachs-Wolfe fudge factor from 1/3 to 0.4.

Abstract

In this article we investigate the time evolution of the adiabatic(curvature) and isocurvature (entropy) spectral indices after end of inflation for all cosmological scales and two different initial conditions. For this purpose,we first extract an explicit equation for the time evolution of the comoving curvature perturbation (which may be known as the generalized Mukhanov-Sasaki equation). It shall be manifested that the evolution of adibatic spectral index severely depends on the intial conditions and just for the super-Hubble scales and adiabatic initial conditions is constat as be expected.Moreover,it shall be clear that the adiabatic spectral index after recombination approach to a constant value for the isocurvature perturbations.Finally,we re-investgate the Sachs-Wolfe effect and show that the fudge factor 1/3 in the adiabatic ordinary Sachs-Wolfe formula must be replaced by 0.4.