Time-reversed Stochastic Inflation

Cosmic inflation may exhibit stochastic periods during which quantum fluctuations dominate over the semi-classical evolution. Extracting observables in these regimes is a notoriously difficult program as quantum randomness makes them fully probabilistic. However, among all the possible quantum histories, the ones which are relevant for Cosmology are conditioned by the requirement that stochastic inflation ended. From an observational point of view, it would be more convenient to model stochastic periods as starting from the time at which they ended and evolving backwards in times. We present a time-reversed approach to stochastic inflation, based on a reverse Fokker-Planck equation, which allows us to derive non-perturbatively the probability distribution of the field values at a given time before the end of the quantum regime. As a motivated example, we solve the flat semi-infinite potential and derive a new and exact formula for the probability distribution of the quantum-generated curvature fluctuations. It is normalisable while exhibiting tails slowly decaying as a Levy distribution. Our reverse-time stochastic formalism could be applied to any inflationary potentials and quantum diffusion eras, including the ones that can lead to the formation of primordial black holes.