LAUSR

We use the Hamilton-Jacobi (H-J) formulation of stochastic inflation to describe the evolution of the inflaton during a period of Ultra-Slow Roll (USR), taking into account the field's velocity and its gravitational backreaction. We demonstrate how this formalism allows one to modify existing slow-roll (SR) formulae to be fully valid outside of the SR regime. We then compute the mass fraction, β, of Primordial Black Holes (PBHs) formed by a plateau in the inflationary potential. By fully accounting for the inflaton velocity as it enters the plateau, we find that PBHs are generically overproduced before the inflaton's velocity reaches zero, ruling out a period of free diffusion or even stochastic noise domination on the inflaton dynamics. We also examine a local inflection point and similarly conclude that PBHs are overproduced before entering a quantum diffusion dominated regime. We therefore surmise that the evolution of the inflaton is always predominantly classical with diffusion effects always subdominant. Both the plateau and the inflection point are characterized by a very sharp transition between the under- and over-production regimes. This can be seen either as severe fine-tunning on the inflationary production of PBHs, or as a very strong link between the fraction β and the shape of the potential and the plateau's extent.