Primordial black holes and stochastic inflation beyond slow roll: I -noise matrix elements

Primordial Black Holes (PBHs) may form in the early Universe, from the grav-itational collapse of large density perturbations, generated by large quantum fluctuations during inflation. Since PBHs form from rare over-densities, their abundance is sensitive to the tail of the primordial probability distribution function (PDF) of the perturbations. It is therefore important to calculate the full PDF of the perturbations, which can be done non-perturbatively using the 'stochastic inflation' framework. In single field inflation models generating large enough perturbations to produce an interesting abundance of PBHs requires violation of slow roll. It is therefore necessary to extend the stochastic inflation formalism beyond slow roll. A crucial ingredient for this are the stochastic noise matrix elements of the inflaton potential. We carry out analytical and numerical calculations of these matrix elements for a potential with a feature which violates slow roll and produces large, potentially PBH generating, perturbations. We find that the transition to an ultra slow-roll phase results in the momentum induced noise terms becoming larger than the field noise whilst each of them falls exponentially for a few e-folds. The noise terms then start rising with their original order restored, before approaching constant values which depend on the nature of the slow roll parameters in the post transition epoch. This will significantly impact the quantum diffusion of the coarse-grained inflaton field, and hence the PDF of the perturbations and the PBH mass fraction