Effective noise in stochastic description of inflation

TL;DR

This paper analyzes how the stochastic noise term in inflationary dynamics should be characterized for reliable simulations. The authors compute the correlator of the noise field for general smoothing windows and show a universal scaling with the smoothing length, clarifying the impact of window choice versus a sharp cutoff. They demonstrate that the asymptotic noise correlation decays as $C(r,t;\epsilon) \sim e^{-2Ht}(\epsilon H r)^{-4}$ for large separations, aligning with the derivative correlators in de Sitter space and revealing that previously reported strong correlations were artefacts of a sharp cutoff. They also evaluate a sine-wave approximation used in some simulations and show it introduces nonphysical long-range correlations, recommending Gaussian random-field noise with a known correlator for accurate large-scale behavior.

Abstract

Stochastic description of inflationary spacetimes emulates the growth of vacuum fluctuations by an effective stochastic ``noise field'' which drives the dynamics of the volume-smoothed inflaton. We investigate statistical properties of this field and find its correlator to be a function of distance measured in units of the smoothing length. Our results apply for a wide class of smoothing window functions and are different from previous calculations by Starobinsky and others who used a sharp momentum cutoff. We also discuss the applicability of some approximate noise descriptions to simulations of stochastic inflation.

Figures (1)

• Figure 1: Correlator $C\left( \rho ,t\right)$ of the effective noise field as function of $\rho$ computed at equal times ($t=0$). The distance was measured in smoothing scale units ($\rho \equiv \epsilon Hr$).