Eigenvalue formulation of Stochastic Inflation and application to large perturbation generating inflationary features

Abstract

Stochastic inflation is a powerful technique for calculating the probability distribution function (PDF) of large inflationary perturbations, which may collapse to form Primordial Black Holes. The PDF, $P({\cal N})$, of the stochastic number of e-folds, ${\cal N}$, satisfies an adjoint Fokker-Planck Equation. We develop a new self-contained eigenvalue technique which can be used to determine $P({\cal N})$. First we apply this method to the simple case of quantum diffusion along a flat potential without any classical drift. We recover the expression for the PDF that has previously been found using characteristic functions, with an exponential tail, and a power-law behaviour, $P({\cal N}) \propto {\cal N}^{-3/2}$, in the intermediate regime between the peak and the tail of the PDF. Finally we apply the method to constant drift inflation, in the narrow- and broad-well limits. In the narrow-well limit, there is an analytic solution and the PDF is similar to the drift-free case, with a mildly suppressed tail. In the broad-well limit, determining the full set of eigenvalues and eigenfunctions requires a piecewise construction of the spectrum, and the broad-well PDF is qualitatively different, with an enhanced peak and a strongly suppressed tail.