Fast spin ±2 spherical harmonics transforms and application in cosmology

A fast and exact algorithm is developed for the spin ±2 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(L2log22L), 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 ±2 functions considered. The algorithm is presented as an alternative to existing fast algorithms with an asymptotic complexity of order O(L3)O(L3) 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.