The curvature perturbation at second order

TL;DR

This paper derives a second-order relation between scalar-field fluctuations and curvature perturbations, facilitating precise calculations in multi-field inflation models and demonstrating the equivalence of different perturbation approaches.

Contribution

It provides an explicit second-order formula linking scalar-field fluctuations to curvature perturbations, simplifying calculations in multi-field inflation.

Findings

Equivalent expressions from perturbation theory and the separate universe approach for superhorizon scales

Elimination of nonlocal terms from second-order perturbation expressions

Facilitation of observable quantity calculations at second order

Abstract

We give an explicit relation, up to second-order terms, between scalar-field fluctuations defined on spatially-flat slices and the curvature perturbation on uniform-density slices. This expression is a necessary ingredient for calculating observable quantities at second-order and beyond in multiple-field inflation. We show that traditional cosmological perturbation theory and the `separate universe' approach yield equivalent expressions for superhorizon wavenumbers, and in particular that all nonlocal terms can be eliminated from the perturbation-theory expressions.