On second-order superhorizon perturbations in multifield inflationary models

TL;DR

This work extends inflationary perturbation theory to second order in models with multiple scalar fields and nontrivial field-space metrics, focusing on superhorizon scales. By changing coordinates in field space, it separates adiabatic and isocurvature modes and constructs second-order gauge-invariant variables, notably a second-order curvature perturbation $\mathcal{R}_{(2)}$, to capture nonlinear effects. The authors derive the second-order Einstein equations, separating gravity and matter sectors, and show that isocurvature perturbations can source the gravitational potential unless the background trajectory is a geodesic in field space; nonlinear evolution is handled via quadratic source terms from first-order perturbations. The framework offers a principled procedure to study nonlinear and non-Gaussian signatures in multifield inflation, including a two-field diagonal-metric example and a path to quantify $\,\chi^2$-type non-Gaussianity on superhorizon scales, with plans for future elaborations.

Abstract

We present a method for the study of second-order superhorizon perturbations in multi field inflationary models with non trivial kinetic terms. We utilise a change of coordinates in field space to separate isocurvature and adiabatic perturbations generalizing previous results. We also construct second order gauge invariant variables related to them. It is found that with an arbitrary metric in field space the isocurvature perturbation sources the gravitational potential on long wavelengths even for ``straight'' trajectories. The potential decouples from the isocurvature perturbations if the background fields' trajectory is a geodesic in field space. Taking nonlinear effects into account shows that, in general, the two types of perturbations couple to each other. This is an outline of a possible procedure to study nonlinear and non-Gaussian effects during multifield inflation.

Figures (1)

• Figure 1: The new basis in N-dimensional field space. Here, $\sigma$ is a coordinate that varies along the background trajectory. It defines a basis vector ${\mathbf{\hat{e}}}_{\sigma}$ tangent to the curve. There are $N-1$ coordinates $s_i$ which are constant along the curve and define $N-1$ coordinate directions normal to it. The basis vectors along these directions ${\mathbf{\hat{e}}}_{s_i}$ are taken to be orthonormal and define the $N-1$ isocurvature perturbations.