Beyond perturbation theory in inflation

Abstract

Inflationary perturbations are approximately Gaussian and deviations from Gaussianity are usually calculated using in-in perturbation theory. This method, however, fails for unlikely events on the tail of the probability distribution: in this regime non-Gaussianities are important and perturbation theory breaks down for |ζ| ≳ |f NL|-1. In this paper we show that this regime is amenable to a semiclassical treatment, ħ → 0. In this limit the wavefunction of the Universe can be calculated in saddle-point, corresponding to a resummation of all the tree-level Witten diagrams. The saddle can be found by solving numerically the classical (Euclidean) non-linear equations of motion, with prescribed boundary conditions. We apply these ideas to a model with an inflaton self-interaction ∝λζ̇4. Numerical and analytical methods show that the tail of the probability distribution of ζ goes as exp(-λ-1/4ζ3/2), with a clear non-perturbative dependence on the coupling. Our results are relevant for the calculation of the abundance of primordial black holes.