Wigner-Eckart theorem in cosmology: Bispectra for total-angular-momentum waves

TL;DR

This paper introduces a formalism using the Wigner-Eckart theorem and TAM waves to simplify the calculation of three-point correlation functions in cosmology, connecting harmonic space bispectra with Fourier-space bispectra.

Contribution

It develops a new approach employing TAM waves and the Wigner-Eckart theorem to relate cosmological bispectra to angular momentum quantum numbers, simplifying spherical sky projections.

Findings

Derived explicit expressions for TAM bispectra in terms of Fourier bispectra.

Showed how Wigner-Eckart theorem constrains the expectation values of TAM wave products.

Provided a framework for easier calculations of cosmological three-point functions.

Abstract

Total-angular-momentum (TAM) waves provide a set of basis functions for scalar, vector, and tensor fields that can be used in place of plane waves and that reflect the rotational symmetry of the spherical sky. Here we discuss three-point correlation functions, or bispectra in harmonic space, for scalar, vector, and tensor fields in terms of TAM waves. The Wigner-Eckart theorem dictates that the expectation value, assuming statistical isotropy, of the product of three TAM waves is the product of a Clebsch-Gordan coefficient (or Wigner-3j symbol) times a function only of the total-angular-momentum quantum numbers. Here we show how this works, and we provide explicit expressions relating the bispectra for TAM waves in terms of the more commonly used Fourier-space bispectra. This formalism will be useful to simplify calculations of projections of three-dimensional bispectra onto the spherical sky.