Smooth coarse-graining and colored noise dynamics in stochastic inflation

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 $\big\langle \xi(N)\xi(N') \big\rangle_{(n)}\sim\left( \cosh\left[ n(N-N')+1 \right] \right)^{-1}$. We find that the power spectrum of curvature perturbations $\mathcal{P}_{\zeta}$ 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 $e$-fold distribution $\mathcal{N}$ and derive an approximate expression for the mean number of $e$-folds $\big\langle \mathcal{N} \big\rangle$. 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.