Description
In the stochastic $\delta N$ formalism, the statistics of the primordial density perturbations can be mapped to the first-passage distribution of a stochastic process. In this talk, I will present a general framework to evaluate the rare-event tail of this distribution, based on an instanton approximation to a path integral representation of the transition probability. I will show how this stochastic description can be derived from a more fundamental formulation via the Schwinger-Keldysh path integral, where integrating out short-wavelength modes yields an influence functional that encodes the noise statistics underlying the stochastic approach. Finally, I will apply this method to a number of cases, highlighting its connections with, and advantages over, the existing methodologies.
