Primordial black hole compaction function from stochastic fluctuations in ultra-slow-roll inflation

We study the formation of primordial black holes (PBH) with ultra-slow-roll inflation when stochastic effects are important. We use the $\mathrm{\Delta }N$ formalism and simplify the stochastic equations with an analytical constant-roll approximation. Considering a viable inflation model, we find the spatial profile of the PBH compaction function numerically for each stochastic patch, without assumptions about Gaussianity or the radial profile. The stochastic effects that lead to an exponential tail for the density distribution also make the compaction function very spiky, unlike assumed in the literature. Naively using collapse thresholds found for smooth profiles, the PBH abundance is enhanced by up to a factor of ${10}^9$, and the PBH mass distribution is spread over three orders of magnitude in mass. The results point to a need to redo numerical simulations of PBH formation with spiky profiles.