Nonlinear Lattice Framework for Inflation: Bridging stochastic inflation and the $\delta{N}$ formalism

TL;DR

This paper introduces a nonlinear lattice framework for single-field inflation that captures inhomogeneous expansion, curvature effects, and nonlinear perturbations efficiently, bridging existing methods.

Contribution

It presents a novel nonlinear lattice approach that extends the $ ext{delta}N$ formalism and stochastic inflation, validated in slow-roll and ultra-slow-roll regimes.

Findings

Captured the separation of curvature perturbation estimators during USR.

Demonstrated growth and stabilization of non-Gaussianity.

Showed transient effects when inflaton velocity is very small.

Abstract

Understanding when inflationary perturbations become genuinely nonlinear near the horizon crossing requires methods that go beyond both linear perturbation theory and the gradient expansion. In this work, we introduce a nonlinear lattice framework for single-field inflation based on a shear-free, locally Friedmann-Lema\^itre-Robertson-Walker geometry. This approach captures inhomogeneous local expansion rates, curvature contributions to the local Friedmann equation, and proper-volume weighting at a fraction of the computational cost of full numerical relativity. We construct fully nonlinear $\delta N$ observables on uniform-density slices, together with other practical time-dependent estimators for the curvature perturbations. After validating the framework in a standard slow-roll regime, we apply it to Starobinsky's linear-potential model featuring an intermittent ultra-slow-roll (USR) phase and a sharp potential transition. During this non-attractor USR regime, the lattice captures the separation of curvature perturbation estimators, the growth and subsequent stabilisation of non-Gaussianity, and a transient weakening of the shear-free approximation when the inflaton velocity becomes very small. Our framework provides a practical intermediate approach between rigid background lattice simulations and full numerical relativity, offering a nonlinear bridge between lattice methods, the $\delta N$ formalism, and the stochastic inflation formalism.