Deviations from Gaussian White Noise in Stochastic Inflation

TL;DR

This paper investigates how relaxing common assumptions in stochastic inflation affects the noise's character, showing that deviations from ideal conditions can produce colored, non-stationary, and non-Gaussian noise.

Contribution

It demonstrates how deviations from de Sitter background, window functions, and initial states modify the noise properties in stochastic inflation, including the emergence of colored and non-Gaussian noise.

Findings

Deviations from de Sitter background maintain white noise with time-dependent amplitude.

Non-Heaviside window functions or non-Bunch-Davies initial states can produce colored noise.

Non-Gaussian noise arises when the initial state deviates from Bunch-Davies vacuum.

Abstract

Stochastic inflation is widely used as a framework to study scalar field perturbations on an inflationary spacetime in a classical manner. In Starobinsky's seminal work and most of the subsequent literature, stochastic inflation is driven by a white noise. This is a consequence of a number of assumptions about the background metric, the window function, and the initial state. Given that noise is the central object in this approach, it is worthwhile to investigate how the noise is modified upon relaxing some of these assumptions. We show that while deviation from an exact de Sitter background maintains the white character of the noise (only with a time-dependent amplitude), deviation from the Heaviside window function or the Bunch-Davies initial state can produce colored noise. We calculate the power spectrum and the memory of the noise for a toy model with a piecewise linear window function. We also show that, in order to produce a colored noise, the deviation from the Bunch-Davies vacuum should essentially be a sum of two-particle states. The resulting noise is non-stationary and we find its instantaneous power spectrum in a concrete example. Furthermore, while deviations from de Sitter background and sharp cutoff do not affect Gaussianity, changing the initial state yields a non-Gaussian noise.