Triple-Spherical Bessel Function Integrals with Exponential and Gaussian Damping: Towards an Analytic N-Point Correlation Function Covariance Model

TL;DR

This paper develops new analytic methods for evaluating integrals of spherical Bessel functions with exponential and Gaussian damping, aiding models of physical observables like the galaxy power spectrum.

Contribution

It introduces a generalized recursion-based method and a novel non-recursive approach for these integrals, providing closed-form solutions involving special functions.

Findings

Closed-form solutions using Legendre, Gamma, and hypergeometric functions.

Simplified evaluation of integrals relevant to physical observables.

Enhanced analytical tools for covariance modeling in physics.

Abstract

Spherical Bessel functions appear commonly in many areas of physics wherein there is both translation and rotation invariance, and often integrals over products of several arise. Thus, analytic evaluation of such integrals with different weighting functions (which appear as toy models of a given physical observable, such as the galaxy power spectrum) is useful. Here we present a generalization of a recursion-based method for evaluating such integrals. It gives relatively simple closed-form results in terms of Legendre functions (for the exponentially-damped case) and Gamma, incomplete Gamma functions, and hypergeometric functions (for the Gaussian-damped case). We also present a new, non-recursive method to evaluate integrals of products of spherical Bessel functions with Gaussian damping in terms of incomplete Gamma functions and hypergeometric functions.