Smooth coarse-graining and colored noise dynamics in stochastic inflation

Abstract

We consider stochastic inflation coarse-grained using a general class of exponential filters. Such a coarse-graining prescription gives rise to inflaton-Langevin equations sourced by colored noise that is correlated in e-fold time. The dynamics are studied first in slow-roll for simple potentials using first-order perturbative, semi-analytical calculations which are later compared to numerical simulations. Subsequent calculations are performed using an exponentially correlated noise which appears as a leading order correction to the full slow-roll noise correlation functions of the type 〈ξ(N)ξ(N')〉(n) ∼ (cosh[n(N-N'+1])-1. We find that the power spectrum of curvature perturbations \U0001d4ab ζ is suppressed at small e-folds, with the suppression controlled by n. Furthermore, we use the leading order, exponentially correlated noise and perform a first passage time analysis to compute the statistics of the stochastic e-fold distribution \U0001d4a9 and derive an approximate expression for the mean number of e-folds 〈\U0001d4a9〉. Comparing analytical results with numerical simulations of the inflaton dynamics, we show that the leading order noise correlation function can be used as a very good approximation of the exact noise, the latter being more difficult to simulate.