Nonlinear perturbations of cosmological scalar fields

TL;DR

This paper develops a covariant, fully nonlinear perturbation framework for multiple canonical scalar fields, focusing on the two-field case and introducing adiabatic and entropy covectors $\sigma_a$ and $s_a$ to decompose the dynamics. It derives exact second-order, covariant evolution equations for these components, and shows that on large scales they reduce to a closed, local system with a decaying nonlocal term, linking to the standard linear theory for $\zeta_a$, $\mathcal{R}_a$, and the Sasaki–Mukhanov variable $Q_{SM}$. The authors map the covariant results to coordinate-based perturbation theory, recover the known linear two-field results, and extend to second order by obtaining gauge-invariant entropy perturbations $\delta s^{(2)}$ and evolution equations for curvature perturbations $\zeta^{(2)}$ and $\mathcal{R}^{(2)}$ sourced by entropy. The framework clarifies the nonlinear generation of adiabatic and isocurvature modes, provides a robust basis for non-Gaussianity studies in multi-field inflation, and yields practical large-scale equations for numerical or analytic work.

Abstract

We present a covariant formalism for studying nonlinear perturbations of scalar fields. In particular, we consider the case of two scalar fields and introduce the notion of adiabatic and isocurvature covectors. We obtain differential equations governing the evolution of these two covectors, as well as the evolution equation for the covector associated with the curvature perturbation. The form of these equations is very close to the analogous equations obtained in the linear theory, but our equations are fully nonlinear and exact. As an application of our formalism, we expand these equations at second order in the perturbations. On large scales, we obtain a closed system of coupled scalar equations giving the evolution of the second-order adiabatic and entropy perturbations in terms of the first-order perturbations. These equations in general contain a nonlocal term which, however, rapidly decays in an expanding universe.