Fast Spin ±2 Spherical Harmonics Transforms

An exact fast algorithm is developed for the direct spin-weighted spherical harmonics transforms of band-limited spin ±2 functions on the sphere. First, we define spin functions on the sphere and their decomposition in an orthonormal basis of spin-weighted spherical harmonics. Second, we discuss the a priori O(L 4) asymptotic complexity of the spin ±2 spherical harmonics transforms, 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. We derive an explicit expression for the spin ±2 spherical harmonics as linear combinations of standard scalar spherical harmonics. An exact algorithm is developed for the spin ±2 spherical harmonics transforms, based on the Driscoll and Healy fast scalar spherical harmonics transform. The associated asymptotic complexity is of order O(L 2 log 2 2 L). Finally, we discuss the application of these generic developments for the efficient computation of the cosmic microwave background (CMB) invariant angular power spectra (T T , EE, BB, and T E) from the observable temperature T and the linear polarization Stokes parameters Q and U. In this perspective, times for the exact computation of spin ±2 spherical harmonics transforms from megapixels all-sky maps of the present WMAP or the future Planck Surveyor satellite missions are typically reduced from days to seconds on a single standard computer. This renders the calculation easily affordable.