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

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.

arXiv comment29 pages

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• A Model for the Redshift-Space Galaxy 4-Point Correlation Function
• Analytical Template for the 4-Point Correlation Function Covariance Beyond the Gaussian Random Field $\mathrm{II}$: 1-Loop Corrections with Third-Order Densities
• Analytical Template for the 4-Point Correlation Function Covariance Beyond the Gaussian Random Field $\mathrm{I}$: 1-Loop Corrections involving Second-Order Densities