A non-perturbative framework for N-point functions of locally non-Gaussian fields

TL;DR

This paper develops a non-perturbative toolbox for N-point functions of locally non-Gaussian fields by starting from a Gaussian field ζ_G and a local mapping ζ=F(ζ_G). It rewrites n-point statistics using a Gaussian path integral and the Kibble–Slepian decomposition, introducing a set of one-point coefficients C_s and a map G_n that yields non-Gaussian n-point functions from Gaussian two-point data, enabling a semi-perturbative resummation that does not require F to be analytic. The authors provide explicit constructions for two-, three-, and four-point functions (power spectra, bispectra, trispectra) and present a diagrammatic interpretation with momentum-space Feynman-like rules, where convolution powers of the Gaussian spectrum enter. As a case study, they analyze exponentially tailed locally non-Gaussian fields, obtaining exact analytic results in the strong NG limit and showing NG can flatten or even suppress the power spectrum and generate infrared k^3 tails. The framework offers a versatile, non-perturbative approach for early-Universe phenomenology, including scalar-induced GWs and PBH formation, and highlights the potential for testing perturbative validity in strongly non-Gaussian regimes.

Abstract

We present a non-perturbative approach to correlation functions and polyspectra of locally non-Gaussian fields and develop a simple semi-perturbative framework that does not rely on the local expansion. As an example, we apply it to locally non-Gaussian fields possessing exponential tails and derive some exact analytic results in the strongly non-Gaussian limit.

Figures (6)

• Figure 1: Diagrams contributing to the correlation function and the power spectrum. The line and the vertices carry the same multiplicity $n \equiv \nu_{12}=s_1=s_2 \in \mathbb{N}$ and the same momentum.
• Figure 2: Diagrams contributing to the 3-point function up to permutations of the vertices. Momenta (or positions) at the vertices and lines are not shown.
• Figure 3: Diagrams contributing to the 4-point function up to permutations of the vertices.
• Figure 4: Left panel: The first 5 coefficients $\bar{\mathcal{C}}_n$ for USR-like models estimated as the average \ref{['eq:Cn_USR']} (solid) numerically and from the first 4 terms in the perturbative expansion \ref{['eq:Cn_USR_pert']} (dashed). The solid line depicts the dispersion $\sqrt{\xi(0)/\xi_0}$ of the non-Gaussian field. Right panel: The map $G_2$ for USR-like models for different $\bar{\beta}$ shown in the figure. The solid lines show the exact numerical computation and the dashed lines the expansion in $\mathcal{C}_n$ containing the first 5 terms.
• Figure 5: Left panel: The deformation of the 2-point function $\mathcal{P}_{\rm G,1}(k)$ in Eq. \ref{['eq:PG_templates']} (black line) due to varying amount of local NG (colored lines). Right panel: The deformation of the Gaussian power spectrum (black line) due to varying amount of local NG (colored lines). The normalization of the $\bar{\beta} \to \infty$ power spectrum is arbitrary.