Fast and Exact Spin-s Spherical Harmonic Transforms

TL;DR

This paper introduces a fast, exact algorithm for spin-s spherical harmonic transforms that efficiently computes multiple transforms simultaneously, with applications in cosmology and gravitational lensing.

Contribution

The paper presents a novel algorithm that computes multiple spin transforms at once using a single set of special functions, improving efficiency over previous methods.

Findings

Computes transforms with O(L^3) complexity.

Allows simultaneous computation of multiple spin transforms.

Achieves high accuracy with modest computational resources.

Abstract

We demonstrate a fast spin-s spherical harmonic transform algorithm, which is flexible and exact for band-limited functions. In contrast to previous work, where spin transforms are computed independently, our algorithm permits the computation of several distinct spin transforms simultaneously. Specifically, only one set of special functions is computed for transforms of quantities with any spin, namely the Wigner d-matrices evaluated at {\pi}/2, which may be computed with efficient recursions. For any spin the computation scales as O(L^3) where L is the band-limit of the function. Our publicly available numerical implementation permits very high accuracy at modest computational cost. We discuss applications to the Cosmic Microwave Background (CMB) and gravitational lensing.