Extended $\delta N$ formalism: Nonspatially Flat Separate Universe Approach

TL;DR

This paper extends the $ abla N$ formalism to include gradient corrections by incorporating homogeneous spatial curvature, enabling accurate modeling of large-scale curvature perturbations beyond slow-roll inflation.

Contribution

It introduces a novel extension of the $ abla N$ formalism that accounts for gradient effects via curvature contributions, improving its applicability.

Findings

Successfully describes large-scale evolution of $$ from horizon exit in ultra-slow-roll inflation.

Captures gradient corrections that are significant beyond slow-roll models.

Provides insights into non-Gaussianities in the extended formalism.

Abstract

The $\delta N$ formalism is a powerful approach to compute non-linearly the large-scale evolution of the comoving curvature perturbation $\zeta$. It assumes a set of FLRW patches that evolve independently, but in doing so, all the gradient terms are discarded, which are not negligibly small in models beyond slow-roll. In this Letter, we extend the formalism to capture these gradient corrections by encoding them in a homogeneous-spatial-curvature contribution assigned to each FLRW patch. For a concrete example, we apply this formalism to the ultra-slow-roll inflation, and find that it can correctly describe the large-scale evolution of the comoving curvature perturbation from the horizon exit. We also briefly discuss non-Gaussianities in this context.