Abstract. We calculate the full probability density function (PDF) of inflationary curvature perturbations, even in the presence of large quantum backreaction. Making use of the stochastic-δN formalism, two complementary methods are developed, one based on solving an ordinary differential equation for the characteristic function of the PDF, and the other based on solving a heat equation for the PDF directly. In the classical limit where quantum diffusion is small, we develop an expansion scheme that not only recovers the standard Gaussian PDF at leading order, but also allows us to calculate the first non-Gaussian corrections to the usual result. In the opposite limit where quantum diffusion is large, we find that the PDF is given by an elliptic theta function, which is fully characterised by the ratio between the squared width and height (in Planck mass units) of the region where stochastic effects dominate. We then apply these results to the calculation of the mass fraction of primordial black holes from inflation, and show that no more than ∼ 1 e-fold can be spent in regions of the potential dominated by quantum diffusion. We explain how this requirement constrains inflationary potentials with two examples.

We study the stochastic formalism of inflation beyond the usual slow-roll approximation. We verify that the assumptions on which the stochastic formalism relies still hold even far from the slow-roll attractor. This includes demonstrating the validity of the separate universe approach to evolving long-wavelength scalar field perturbations beyond slow roll. We also explain that, in general, there is a gauge correction to the amplitude of the stochastic noise. This is because the amplitude is usually calculated in the spatially-flat gauge, while the number of e-folds is used as the time variable (hence one works in the uniform-N gauge) in the Langevin equations. We show that these corrections vanish in the slow-roll limit, but we also explain how to calculate them in general. We compute them in difference cases, including ultra-slow roll and the Starobinsky model that interpolates between slow roll and ultra-slow roll, and find the corrections to be negligible in practice. This confirms the validity of the stochastic formalism for studying quantum backreaction effects in the very early universe beyond slow roll. Keywords: physics of the early universe, inflationRecently, situations in which non-slow-roll stochastic effects are at play have been highlighted [24][25][26][27][28]. For instance, if the inflationary potential features a very flat section close to the end of inflation, large curvature perturbations could be produced that later collapse into primordial black holes. If such a flat portion exists, it may be associated with both large stochastic diffusion [29] and deviations from slow-roll, e.g. along the so-called ultra-slow-roll (or "friction dominated") regime [30,31], which in some cases can be stable [32]. This explains the need for implementing the stochastic inflation programme beyond slow roll, which is the aim of the present work.This paper is organised as follows. In Sec. 2, we quickly review the stochastic inflation formalism and identify the three main requirements for the validity of this approach: the quantum-to-classical transition of super-Hubble fluctuations, the validity of the separate universe approach, and the consistent implementation of gauge corrections. The two latter requirements are the non-trivial ones and we examine them in Secs. 3 and 4 respectively. Although recently questioned [33], we find the separate universe approach to hold beyond slow roll, and we explain how the gauge corrections to the amplitude of the stochastic noise (that vanish in the slow-roll regime if the number of e-folds is used as a time variable) can be derived in general. We then apply this program to three situations of interest: slow roll in Sec. 5, where we recover the usual results, ultra-slow roll in Sec. 6, and the Starobinsky model in Sec. 7, which interpolates between an ultra-slow-roll and a slow-roll phase. In all cases, we find the gauge corrections to be negligible, allowing for the usual stochastic formulation to be employed.

We consider the effect of quantum diffusion on the dynamics of the inflaton during a period of ultra-slow-roll inflation. We extend the stochastic-δ𝒩 formalism to the ultra-slow-roll regime and show how this system can be solved analytically in both the classical-drift and quantum-diffusion dominated limits. By deriving the characteristic function, we are able to construct the full probability distribution function for the primordial density field. In the diffusion-dominated limit, we recover an exponential tail for the probability distribution, as found previously in slow-roll inflation. To complement these analytical techniques, we present numerical results found both by very large numbers of simulations of the Langevin equations, and through a new, more efficient approach based on iterative Volterra integrals. We illustrate these techniques with two examples of potentials that exhibit an ultra-slow-roll phase leading to the possible production of primordial black holes.

It is often claimed that the ultra-slow-roll regime of inflation, where the dynamics of the inflaton field are friction dominated, is a non-attractor and/or transient. In this work we carry out a phase-space analysis of ultra-slow roll in an arbitrary potential, V (φ). We show that while standard slow roll is always a dynamical attractor whenever it is a self-consistent approximation, ultra-slow roll is stable for an inflaton field rolling down a convex potential with M Pl V > |V | (or for a field rolling up a concave potential with M Pl V < −|V |). In particular, when approaching a flat inflection point, ultra-slow roll is always stable and a large number of e-folds may be realised in this regime. However, in ultra-slow roll,φ is not a unique function of φ as it is in slow roll and dependence on initial conditions is retained. We confirm our analytical results with numerical examples.

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