Stochastic effects in ultra slow-roll inflation

In this work we apply the stochastic inflation formalism to backgrounds that deviate from standard slow-roll in a way that can lead to the production of primordial black holes. We analyse the effects of sudden transitions on the stochastic noise amplitude and its potential impact on the production of primordial black holes in single field inflation. We justify the claim that primordial black holes can be responsible for a significant fraction of the dark matter abundance today and estimate the required enhancement in the power spectrum. We introduce the Hamiltonian formalism and the coarse-graining of the quantum field and its momentum, allowing us to obtain a quantitative measure of the role of quantum diffusion in the production of primordial black holes. We mainly focus our analysis on a Starobinsky potential given that it is rich enough to allow for the dynamics of the scalar field during inflation to include an ultra slow-roll phase induced by a transition from a relatively large slow-roll parameter to a hierarchically smaller one. This has the effect of making the field perturbations undergo sudden transitions and rise from its ground state to an excited state. We present two procedures that we employed when calculating the power spectrum: a numerical and an analytical one. Both methods show that stochastic effects are negligible at small scales where their amplitude is time dependent and vanishes at late times, and that for scales leaving the horizon after the ultra slow-roll phase the de Sitter estimate of $H^2/(4\pi^2)$ is approximately correct. We therefore demonstrate that the estimates in the literature are incomplete and that a revaluation of the role of stochastic effects on primordial black hole production is in order.