Number Density of Peaks in a Chi-Squared Field

TL;DR

This paper analyzes the statistical properties of stationary points in chi-squared fields formed by summing squares of Gaussian fields, providing exact formulas, numerical methods, and exploring dependencies on various parameters.

Contribution

It introduces a new framework for calculating the number density of stationary points in chi-squared fields, extending previous Gaussian peak statistics to these more complex fields.

Findings

Derived exact integral expressions for stationary point density.

Analyzed how stationary point density depends on field amplitude and parameters.

Provided numerical tools for evaluating the integral in general cases.

Abstract

We investigate the statistics of stationary points in the sum of squares of $N$ Gaussian random fields, which we call a "chi-squared" field. The behavior of such a field at a point is investigated, with particular attention paid to the formation of topological defects. An integral to compute the number density of stationary points at a given field amplitude is constructed. We compute exact expressions for the integral in various limits and provide code to evaluate it numerically in the general case. We investigate the dependence of the number density of stationary points on the field amplitude, number of fields, and power spectrum of the individual Gaussian random fields. This work parallels the work of Bardeen, Bond, Kaiser and Szalay, who investigated the statistics of peaks of Gaussian random fields. A number of results for integrating over matrices are presented in appendices.