Isotropic N-Point Basis Functions and Their Properties

Abstract

Isotropic functions of positions ${\hat{\mathbf{r}}}_1,{\hat{\mathbf{r}}}_2,\dots ,{\hat{\mathbf{r}}}_N$, i.e. functions invariant under simultaneous rotations of all the coordinates, are conveniently formed using spherical harmonics and Clebsch-Gordan coefficients. An orthonormal basis of such functions provides a formalism suitable for analyzing isotropic distributions such as those that arise in cosmology, for instance in the clustering of galaxies as revealed by large-scale structure surveys. The algebraic properties of the basis functions are conveniently expressed in terms of 6-$j$ and 9-$j$ symbols. The calculation of relations among the basis functions is facilitated by "Yutsis" diagrams for the addition and recoupling of angular momenta.

arXiv comment57 pages, submitted to Journal of Mathematical Physics, comments welcome

References in corpus (27)

• Non-Gaussianity from Inflation: Theory and Observations
• The Angular Trispectrum of the CMB
• Measuring Statistical isotropy of the CMB anisotropy
• A Spherical Harmonic Analysis of Redshift Space
• The Angular Trispectra of CMB Temperature and Polarization
• The Microwave Background Bispectrum, Paper I: Basic Formalism
• General CMB and Primordial Trispectrum Estimation
• Computing the Three-Point Correlation Function of Galaxies in $\mathcal{O}(N^2)$ Time
• Beyond the plane-parallel approximation for redshift surveys
• Spherical Harmonic Analysis of the PSCz Galaxy Catalogue: Redshift distortions and the real-space power spectrum
• Wide Angle Redshift Distortions Revisited
• The Angular Bispectrum of The Cosmic Microwave Background
• A complete FFT-based decomposition formalism for the redshift-space bispectrum
• Galaxy Bias and Halo-Occupation Numbers from Large-Scale Clustering
• Accelerating the 2-point and 3-point galaxy correlation functions using Fourier transforms
• Perturbation theory approach to predict the covariance matrices of the galaxy power spectrum and bispectrum in redshift space
• Total Angular Momentum Waves for Scalar, Vector, and Tensor Fields
• A Practical Computational Method for the Anisotropic Redshift-Space 3-Point Correlation Function
• Spherical Harmonic Analyses of Intensity Mapping Power Spectra
• Odd-Parity Bipolar Spherical Harmonics
• Towards a self-consistent analysis of the anisotropic galaxy two- and three-point correlation functions on large scales: application to mock galaxy catalogues
• Testing gaussianity, homogeneity and isotropy with the cosmic microwave background
• The CMB Bispectrum, Trispectrum, non-Gaussianity, and the Cramer-Rao Bound
• Projected bispectrum in spherical harmonics and its application to angular galaxy catalogues
• Bipolar Harmonic encoding of CMB correlation patterns
• General and consistent statistics for cosmological observations
• Non-Gaussianity and extragalactic foregrounds to the Cosmic Microwave Background

Cited by in corpus (17)

• Information Content of Higher-Order Galaxy Correlation Functions
• Analytic Gaussian Covariance Matrices for Galaxy $N$-Point Correlation Functions
• A First Detection of the Connected 4-Point Correlation Function of Galaxies Using the BOSS CMASS Sample
• Efficient Computation of $N$-point Correlation Functions in $D$ Dimensions
• No evidence for parity violation in BOSS
• Cosmological Probes of Structure Growth and Tests of Gravity
• Anatomy of Parity-violating Trispectra in Galaxy Surveys
• Probing vector chirality in the early Universe
• Breaking parity: the case of the trispectrum from chiral scalar-tensor theories of gravity
• Improving the Line of Sight for the Anisotropic 3-Point Correlation Function of Galaxies: Centroid and Unit-Vector-Average Methods Scaling as $\mathcal{O}(N^2)$
• A Model for the Redshift-Space Galaxy 4-Point Correlation Function
• Mixing Bispectrum Multipoles Under Geometric Distortions
• A general formalism for machine-learning models based on multipolar-spherical harmonics
• Parity-odd Four-Point Correlation Function from DESI Data Release 1 Luminous Red Galaxy Sample
• An Optimal In-Situ Multipole Algorithm for the Isotropic Three-Point Correlation Functions
• Analytical Template for the 4-Point Correlation Function Covariance Beyond the Gaussian Random Field $\mathrm{I}$: 1-Loop Corrections involving Second-Order Densities
• Analytical Template for the 4-Point Correlation Function Covariance Beyond the Gaussian Random Field $\mathrm{II}$: 1-Loop Corrections with Third-Order Densities