Ultra-Slow-Roll Inflation on the Lattice I: Backreaction and Nonlinear Effects

Abstract

Violating the slow-roll regime during the final stages of inflation can significantly enhance curvature perturbations, a scenario often invoked in models producing primordial black holes and small-scale scalar induced gravitational waves. When perturbations are enhanced, one approaches the regime in which tree-level computations are insufficient, and nonlinear corrections may become relevant. In this work, we conduct lattice simulations of ultra-slow-roll (USR) dynamics to investigate the significance of nonlinear effects, both in terms of backreaction on the background and in the evolution of perturbations. Our systematic study of various USR potentials reveals that nonlinear corrections are significant when the tree-level curvature power spectrum peaks at ${\mathcal{P}}_{\zeta }^{\max }=\mathcal{O}(10^{-3})-\mathcal{O}(10^{-2})$, with 5%-20% corrections. Larger enhancements yield even greater differences. We find a simple universal relation between simulation and tree-level quantities $\dot{\varphi }={\dot{\varphi }}_{\mathrm{tree}}\left(1+\sqrt{{\mathcal{P}}_{\zeta ,\mathrm{tree}}^{\max }}\right)$ at the end of the USR phase, which is valid in all cases we consider. Additionally, we explore how nonlinear interactions during the USR phase affect the clustering and non-Gaussianity of scalar fluctuations, crucial for understanding the phenomenological consequences of USR, such as scalar-induced gravitational waves and primordial black holes. Our findings demonstrate the necessity of going beyond leading order perturbation theory results, through higher-order or non-perturbative computations, to make robust predictions for inflation models exhibiting a USR phase.

17 pages, 11 figures; animations are available at https://github.com/caravangelo/USR-on-the-lattice; minor modifications matching the version published in PRD

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