Stochastic inflation beyond slow roll: noise modelling and importance sampling

TL;DR

This paper advances the simulation of rare inflationary fluctuations by extending stochastic methods and importance sampling, enabling efficient modeling of quantum noise effects beyond slow-roll conditions.

Contribution

It introduces an improved numerical approach combining importance sampling with stochastic inflation modeling, allowing rapid simulation of rare events in the early universe.

Findings

Stochastic noise can be accurately modeled with a Bessel-function ansatz.

Importance sampling reduces simulation time from supercomputers to minutes.

The method captures non-linear stochastic evolution on super-Hubble scales.

Abstract

We simulate the distribution of very rare, large excursions in the primordial density field produced in models of inflation in the very early universe which include a strong enhancement of the power spectrum. The stochastic $\delta \mathcal{N}$ formalism is used to identify the probability distribution for the primordial curvature perturbation with the first-passage-time distribution, $P(\delta \mathcal{N})$, and we compare our stochastic results with those obtained in the classical $\delta \mathcal{N}$ approach. We extend the PyFPT numerical code to simulate the full 2D phase space, and apply importance sampling which allows very rare fluctuations to be simulated in $\mathcal{O}(10)$ minutes on a single CPU, where previous direct simulations required supercomputers. We demonstrate that the stochastic noise due to quantum fluctuations after a sudden transition to ultra-slow roll can be accurately modelled using an analytical Bessel-function ansatz to identify the homogeneous growing mode. The stochastic noise found in this way is a function of the field value only. This enables us to coarse grain the inflation field at the Hubble scale and include non-linear, stochastic evolution on all super-Hubble length scales.