Beyond Perturbation Theory in Inflation

TL;DR

This paper introduces a semiclassical approach to analyze the tail of the probability distribution of inflationary perturbations, overcoming the breakdown of perturbation theory in rare, non-Gaussian events, with implications for primordial black hole abundance.

Contribution

It develops a semiclassical method to study non-Gaussian tails in inflationary perturbations, providing a non-perturbative framework beyond traditional in-in perturbation theory.

Findings

The tail of the distribution follows an exponential decay with a non-perturbative dependence on coupling.

Numerical and analytical results show the probability distribution scales as exp(-λ^{-1/4} ζ^{3/2}).

The method applies saddle-point solutions to non-linear Euclidean equations of motion for rare event analysis.

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 $|\zeta| \gtrsim |f_{\rm \scriptscriptstyle NL}|^{-1}$. In this paper we show that this regime is amenable to a semiclassical treatment, $\hbar \to 0$. 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 $\propto \lambda \dot\zeta^4$. Numerical and analytical methods show that the tail of the probability distribution of $\zeta$ goes as $\exp(-\lambda^{-1/4}\zeta^{3/2})$, with a clear non-perturbative dependence on the coupling. Our results are relevant for the calculation of the abundance of primordial black holes.