Ultra-slow-roll inflation with quantum diffusion

TL;DR

This paper analyzes how quantum diffusion influences ultra-slow-roll inflation, extending stochastic formalism to derive the probability distribution of primordial density fluctuations, with implications for primordial black hole formation.

Contribution

It extends the stochastic-$ abla ext{delta} ext{N}$ formalism to ultra-slow-roll inflation and develops analytical and numerical methods to characterize quantum diffusion effects.

Findings

Derived the full probability distribution function for density fluctuations.

Identified exponential tails in the distribution in the diffusion limit.

Demonstrated potential for primordial black hole production during ultra-slow-roll phases.

Abstract

We consider the effect of quantum diffusion on the dynamics of the inflaton during a period of ultra-slow-roll inflation. We extend the stochastic-$\delta\mathcal{N}$ formalism to the ultra-slow-roll regime and show how this system can be solved analytically in both the classical-drift and quantum-diffusion dominated limits. By deriving the characteristic function, we are able to construct the full probability distribution function for the primordial density field. In the diffusion-dominated limit, we recover an exponential tail for the probability distribution, as found previously in slow-roll inflation. To complement these analytical techniques, we present numerical results found both by very large numbers of simulations of the Langevin equations, and through a new, more efficient approach based on iterative Volterra integrals. We illustrate these techniques with two examples of potentials that exhibit an ultra-slow-roll phase leading to the possible production of primordial black holes.