Non-Gaussian corrections to the probability distribution of the curvature perturbation from inflation

TL;DR

The paper provides a direct, non-perturbative bridge between inflationary $n$-point correlation functions of the curvature perturbation and the probability distribution of spatial fluctuations. Using a Schrödinger–like wavefunctional and Schwinger–Keldysh generating functionals, it derives the Gaussian baseline for the smoothed perturbation $\epsilon$ and then computes the leading slow-roll correction from the bispectrum. It also extends the analysis to the local spectrum $\varrho(k)$, producing functional PDFs and showing regulator independence in the infinite-cutoff limit. The framework enables systematic inclusion of higher-order non-Gaussianities and has potential implications for structure formation and primordial black hole abundances, illustrating how subtle inflationary signatures propagate into observable statistics.

Abstract

We show how to obtain the probability density function for the amplitude of the curvature perturbation, R, produced during an epoch of slow-roll, single-field inflation, working directly from n-point correlation functions of R. These n-point functions are the usual output of quantum field theory calculations, and as a result we bypass approximate statistical arguments based on the central limit theorem. Our method can be extended to deal with arbitrary forms of non-Gaussianity, appearing at any order in the n-point hierarchy. We compute the probability density for the total smoothed perturbation within a Hubble volume, ε, and for the spectrum of ε. When only the two-point function is retained, exact Gaussian statistics are recovered. When the three-point function is taken into account, we compute explicitly the leading slow-roll correction to the Gaussian result.