Beyond scalar, vector and tensor harmonics in maximally symmetric three-dimensional spaces

TL;DR

This paper develops a comprehensive formalism for constructing scalar, vector, and tensor harmonics on maximally symmetric three-dimensional spaces, introducing spin-weighted spherical harmonics and a generalized helicity basis for detailed decomposition.

Contribution

It introduces a unified approach using spin-weighted harmonics and helicity basis, enabling recursive construction of higher tensor harmonics and clarifying their properties in curved spaces.

Findings

Provides explicit expressions for harmonics in curved spaces

Develops recursive relations for higher tensor harmonics

Shows that normal modes in curved spaces cannot be factorized as previously thought

Abstract

We present a comprehensive construction of scalar, vector and tensor harmonics on maximally symmetric three-dimensional spaces. Our formalism relies on the introduction of spin-weighted spherical harmonics and a generalized helicity basis which, together, are ideal tools to decompose harmonics into their radial and angular dependencies. We provide a thorough and self-contained set of expressions and relations for these harmonics. Being general, our formalism also allows to build harmonics of higher tensor type by recursion among radial functions, and we collect the complete set of recursive relations which can be used. While the formalism is readily adapted to computation of CMB transfer functions, we also collect explicit forms of the radial harmonics which are needed for other cosmological observables. Finally, we show that in curved spaces, normal modes cannot be factorized into a local angular dependence and a unit norm function encoding the orbital dependence of the harmonics, contrary to previous statements in the literature.