$δN$ formalism with gradient interactions

Abstract: The standard $δN$ formalism, a cornerstone for calculating nonlinear curvature perturbations on super-Hubble scales, relies on the separate universe assumption, in which spatial gradients are neglected. However, this approximation breaks down in scenarios critical for primordial black hole formation, such as transitions to an ultra-slow-roll phase, where gradient interactions induce significant non-conservation of the comoving curvature perturbation. In this Letter, we introduce a framework that incorporates gradient corrections into the $δN$ formalism at a desired order by adding an effective source term to the background Klein--Gordon equation. This approach allows for a nonlinear treatment of curvature perturbations at the end of inflation considering initial conditions at the time of horizon exit. By computing the equilateral non-Gaussianity parameter $f_{\mathrm{NL}}^{\mathrm{eq}}$, we demonstrate that our method captures essential features missed by the standard $δN$, offering a simple yet rigorous pathway to determine nonlinear evolution expected from cosmological perturbation theory.