Statistics of Fourier Modes in Non‐Gaussian Fields

Abstract

A theory for statistical distributions of Fourier modes in non-Gaussian fields is established in this paper. In astrophysics and cosmological physics, Fourier analyses of random fields are useful and widely used. While distributions of the Fourier modes in Gaussian fields are well-known, there was no well-established theory for those in non-Gaussian fields. In non-Gaussian random fields, polyspectra, higher order counterparts of the power spectrum, are usually considered to characterize statistical information contained in the fields. The distribution functions of Fourier modes are directly considered in this paper and their explicit relations to the polyspectra are given. Under the condition that any of the polyspectra does not diverge, the distribution function is expanded by dimensionless combinations of polyspectra and a total volume in which the Fourier transforms are performed. The expression up to second order is generally given, and higher order results are also derived in some cases. A concept of the N-point distribution function of Fourier modes is introduced and explicitly calculated. Among them, the one-point distribution function is completely given in a closed form up to arbitrary order. As an application, statistics of Fourier phases are explored in detail. A number of aspects regarding statistical properties of phases are found. It is clarified, for the first time, how phase correlations arise in general non-Gaussian fields. Some of our analytic results are tested against numerical realizations of non-Gaussian fields, showing good agreements.