Numerically modeling stochastic inflation in slow-roll and beyond

TL;DR

This paper develops a numerical approach to model stochastic inflation, capturing quantum fluctuations' effects on inflaton dynamics and power spectrum, including scenarios like ultra slow-roll inflation relevant for primordial black holes.

Contribution

It introduces a comprehensive numerical method for solving stochastic inflationary equations, extending analysis to ultra slow-roll scenarios and primordial black hole formation.

Findings

Numerical solutions match traditional power spectrum calculations.

Method applicable to ultra slow-roll inflation.

Insights into primordial black hole formation potential.

Abstract

We present a complete numerical treatment of inflationary dynamics under the influence of stochastic corrections from sub-Hubble modes. We discuss how to exactly model the stochastic noise terms arising from the sub-Hubble quantum modes that give rise to the coarse-grained inflaton dynamics in the form of stochastic differential equations. The stochastic differential equations are solved event-by-event on a discrete time grid. We then compute the power spectrum of curvature perturbations that can be compared with the power spectrum computed in the traditional fashion using the Mukhanov-Sasaki equation by canonically quantizing the inflaton fluctuations. Our numerical procedure helps us to easily extend the formalism to ultra slow-roll inflation and study the possibility of primordial black hole formation.