Ultra-Slow-Roll Inflation on the Lattice II: Nonperturbative Curvature Perturbation

Abstract: Building on the recent lattice simulations of ultra-slow-roll (USR) dynamics presented in arXiv:2410.23942, we investigate the role of the nonlinear relation between the inflaton field configuration and the curvature perturbation $\zeta$, the key observable after inflation. Using a nonperturbative $\delta N$ approach applied to the lattice output, we generate fully nonlinear three-dimensional maps of $\zeta$. This calculation captures both the non-Gaussianity arising from the nonlinear mapping between $\phi$ and $\zeta$, and the intrinsic non-Gaussianity generated around Hubble crossing by the nonlinear field dynamics, which is neglected in stochastic approaches. We find that the nonlinear mapping has a profound impact on the statistics, significantly enhancing the positive tail of the $\zeta$ probability distribution, with important implications for observable quantities. A central part of this work is the comparison with the standard perturbative treatment based on a gauge transformation, which allows us to quantify when and how the perturbative picture breaks down as fluctuations grow large. Together with arXiv:2410.23942, this work sets the basis for robust, nonperturbative predictions of primordial black hole production and scalar-induced gravitational wave emission from inflation using lattice simulations.