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 𝒫 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 𝒩 and derive an approximate expression for the mean number of e-folds 〈𝒩〉. 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.

Numerical stochastic inflation constrained by frozen noise

“…The choice of final boundary conditions in this paper and in [67,78] differs from that of many other studies, which employ the first passage time formalism [41,49,52,57,60,64,73,74,79,[81][82][83][84]. There the stochastic kicks continue all the way to the end of the computation, which terminates when the field first crosses the final hypersurface at φ = φ final .…”

Numerical stochastic inflation constrained by frozen noise

“…The choice of final boundary conditions in this paper and in [67,78] differs from that of many other studies, which employ the first passage time formalism [41,49,52,57,60,64,73,74,79,[81][82][83][84]. There the stochastic kicks continue all the way to the end of the computation, which terminates when the field first crosses the final hypersurface at φ = φ final .…”

Primordial black holes and stochastic inflation beyond slow roll. Part I. Noise matrix elements

“…There has been a profusion of work on SI in the recent literature in the context of PBH formation [56][57][58][59][60][61][62][63][64][65][66][67][68][69][70][71][72][73][74], and it has been shown that SI generically predicts a (highly non-Gaussian) exponential tail in the PDF [56,59]. However, since PBH formation usually requires slow-roll violation, it is important to develop the SI formalism beyond slow-roll [57,[60][61][62]66].…”

“…This assumption has been questioned, especially because the sharp cut-off window function may not lead to well-behaved coarse-grained field correlators in the physical space. See refs [73,88,89]. for more discussion.…”