The Density Perturbation Power Spectrum to Second-Order Corrections in the Slow-Roll Expansion

TL;DR

This work advances the precision of inflationary perturbation theory by deriving the curvature perturbation power spectrum to second order in the slow-roll expansion. It uses the Mukhanov-Sasaki action and a Green's-function perturbative method around the de Sitter limit, expressing the result in terms of slow-roll parameters and horizon-crossing quantities, and then reframes the spectrum in terms of inflaton potential derivatives. The authors validate their second-order results against exact solutions for power-law inflation and inflation near a maximum, showing exact agreement in these limits. They also provide potential-based expressions for both the spectrum and the spectral index, enabling direct confrontation with observational data.

Abstract

We set up a formalism that can be used to calculate the power spectrum of the curvature perturbations produced during inflation up to arbitrary order in the slow-roll expansion, and explicitly calculate the power spectrum and spectral index up to second-order corrections.