Nonlinear Lattice Framework for Inflation: Bridging stochastic inflation and the $\delta{N}$ formalism
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 as perturbations transition beyond the linear regime.