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 slowroll 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 early 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 efold 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.

Review on Stochastic Approach to Inflation

“…The main consequence of this is that, under the separate universe condition, we can express all the scalar fluctuations only in terms of the field inhomogeneities not only in the spatially flat gauge, but also in the uniform-N gauge. This can be clearly seen when looking at the linear "separate-universe" gauge invariant MS variable of (68). For non-linear generalizations of this variables see [117,118].…”

“…We know that, as a consequence of the separate universe approach used here, the linearized equation for the field perturbation in the uniform-N gauge will be the same as the equation for the "separate-universe" gauge invariant variable Q sep defined in (68), in other words, Q sep = δφ f = δφ δN under the separate universe assumption. Now, since the linearized equation for Q sep only gives the correct solution for the true gauge-invariant MS variable Q defined in (44) if we set 1 = 0 as checked in Section 4.1.2, the equation we will try to solve here is the linearized equation for the gauge invariant quantity Q over a local background in which all the SR parameters have been set to zero (note that, in order for this approximation to work we need − 3 2 2 − 1 4 2…”

“…However, it was noted in [28] that this choice of window function leads to some problems in the asymptotic of the noise correlator at large spatial distances. This is why some works with smooth window functions that lead to coloured noises have also been proposed [34,68].…”