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.

29 pages

References in corpus (10)

• Primordial non-Gaussianity and the CMB bispectrum
• FFT-PT: Reducing the two-loop large-scale structure power spectrum to low-dimensional radial integrals
• 2-FAST: Fast and accurate computation of projected two-point functions
• A First Detection of the Connected 4-Point Correlation Function of Galaxies Using the BOSS CMASS Sample
• Rotation method for accelerating multiple-spherical Bessel function integrals against a numerical source function
• Fast Calculation of Nonlinear Redshift-space Galaxy Power Spectrum Including Selection Bias
• The Ewald sums for singly, doubly and triply periodic electrostatic systems
• On a General Method for Resolving Integrals of Multiple Spherical Bessel Functions Against Power Laws into Distributions
• SARABANDE: 3/4 Point Correlation Functions with Fast Fourier Transforms
• Approximations for modeling light scattering by spheres with uncertainty in physical parameters

Cited by in corpus (3)

• 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