Non-linear perturbations in multiple-field inflation

TL;DR

The paper presents a non-linear, long-wavelength framework for perturbations in multi-field inflation using gradient-based, time-slicing invariant variables $\zeta^A_i$ and ${\mathcal{Q}}^A_i$, coupled to a closing set of constraints. It derives fully non-linear evolution equations with a field-space mass matrix $\Omega^A{}_B$, introduces a covariant field-basis and the corresponding slow-roll-like parameters, and adds stochastic source terms to model horizon-crossing influx from short-wavelength modes; the linear limit recovers standard gauge-invariant perturbation theory, while the non-linear system enables straightforward perturbative (to second order) and numerical analyses without slow-roll assumptions. A clear connection to the $\delta N$ formalism is discussed, together with a practical numerical scheme that reconstructs local backgrounds from constraints and evolves the stochastic system on a grid. The approach provides a tractable, gauge-consistent method to predict non-Gaussian features in multi-field inflation and to study the impact of super-horizon evolution on observational signatures, with explicit guidance on field-basis formulations and stochastic implementation. Overall, the work establishes a versatile framework for accurate, non-perturbative exploration of inflationary perturbations beyond linear theory and slow-roll approximations.

Abstract

We develop a non-linear framework for describing long-wavelength perturbations in multiple-field inflation. The basic variables describing inhomogeneities are defined in a non-perturbative manner, are invariant under changes of time slicing on large scales and include both matter and metric perturbations. They are combinations of spatial gradients generalising the gauge-invariant variables of linear theory. Dynamical equations are derived and supplemented with stochastic source terms which provide the long-wavelength initial conditions determined from short-wavelength modes. Solutions can be readily obtained via numerical simulations or analytic perturbative expansions. The latter are much simpler than the usual second-order perturbation theory. Applications are given in a companion paper.