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

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-Lemaître-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 $δN$ 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 $δN$ formalism, and the stochastic inflation formalism.

Figures (8)

• Figure 1: An illustration of how lattice differs in handling the modes from the conventional $\delta{N}$ formalism, based on the leading order of the gradient expansion.
• Figure 2: We plot the evolution of the volume-averaged field value (left panel) and the energy components (normalised by the total energy $\langle{\tilde{\rho}_i}\rangle_V = \langle{\rho_i}\rangle_V/\langle{\rho_{\mathrm{tot}}}\rangle_V$) (right panel) for the $m^2\phi^2$ model.
• Figure 3: These plots show the final spectra and PDF of different variables for the $m^2\phi^2$ model.
• Figure 4: We plot the evolution of the averaged energy components normalised by the total energy density and the effective second slow-roll parameters for the Starobinsky's linear model. The dashed line in the right panel indicates the USR limit, $\eta_H=-6$.
• Figure 5: We plot the time evolution of the power spectra for the comoving curvature perturbation $\mathcal{R}^{\mathrm{est}}$ (left panel) and the curvature perturbation on uniform-density slices $\zeta^{\mathrm{est}}$ (middle panel) during the initial SR, intermediate USR phase to the final SR phase, at intervals of $0.25$$e$-fold. The inflaton crosses the branch points $\phi_1$ at around $\mathbb{N} = 0.5$ and $\phi_2$ around $\mathbb{N} = 2$, which determines the duration of the USR phase during which the spectra of these two estimators don't converge. The colored vertical lines indicate the instantaneous comoving Hubble scale ($k = aH$) corresponding to the spectrum of the same colour. For any given snapshot, modes to the left of the respective vertical line are super-Hubble, while those to the right are sub-Hubble. At the onset of the simulation, the majority of the resolved spectrum is sub-Hubble, which is required to properly impose the adiabatic vacuum initial conditions for the field fluctuations. By the end of the simulation (solid black line), the entire dynamical range of the lattice has crossed the horizon, leaving all modes deeply super-Hubble. The rightmost figure shows the final spectrum, along with spectra obtained using the $\delta{N}$ formalism.