$\delta N$ formalism with gradient interactions

The standard $\delta N$ formalism is a cornerstone technique for calculating nonlinear curvature perturbations on super-Hubble scales. However, its validity relies heavily on the separate universe assumption, in which spatial gradients are neglected. This approximation is known to break down in scenarios that are critical for primordial black hole formation, such as transitions to an ultra-slow-roll phase, where gradient interactions induce a significant non-conservation of the comoving curvature perturbation. In this paper, we introduce a framework that systematically incorporates gradient corrections into the $\delta N$ formalism at a desired order by adding an effective source term to the background Klein–Gordon equation. This approach allows for a fully 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 $f_{\mathrm{N}\mathrm{L}}^{\mathrm{e}\mathrm{q}}$, we demonstrate that our method captures essential physical features missed by the standard $\delta N$ approach, offering a simple yet rigorous pathway to determine the nonlinear evolution expected from full cosmological perturbation theory.