Beyond scalar, vector and tensor harmonics in maximally symmetric three-dimensional spaces
arXiv:1909.13687 · doi:10.1103/PhysRevD.100.123535
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.
39 pages, 2 figures
References in corpus (5)
- Relativistic cosmology and large-scale structure
- Non-linear multipole interactions and gravitational-wave octupole modes for inspiralling compact binaries to third-and-a-half post-Newtonian order
- Total Angular Momentum Waves for Scalar, Vector, and Tensor Fields
- Scalar, Vector and Tensor Harmonics on the Three-Sphere
- Emission-angle and polarization-rotation effects in the lensed CMB