Fast spin $\pm 2$ spherical harmonics transforms and application in cosmology

Journal of Computational Physics

Abstract: A fast and exact algorithm is developed for the spin \(\pm2\) spherical harmonics transforms on equi-angular pixelizations on the sphere. It is based on the Driscoll and Healy fast scalar spherical harmonics transform. The theoretical exactness of the transform relies on a sampling theorem. The associated asymptotic complexity is of order \(O\big(L^2, (\log_2 L)^{2}\big)\), where \(2L\) stands for the square-root of the number of sampling points on the sphere, also setting a band limit L for the spin \(\pm2\) functions considered. The algorithm is presented as an alternative to existing fast algorithms with an asymptotic complexity of order \(O(L^3)\) on other pixelizations. We also illustrate these generic developments through their application in cosmology, for the analysis of the cosmic microwave background (CMB) polarization data.