Fast Spin ±2 Spherical Harmonics Transforms

An exact fast algorithm is developed for the direct spin-weighted spherical harmonics transforms of bandlimited 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(L4) 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(L2 log2 L). Finally, we discuss the application of these generic developments for the efficient computation of the cosmic microwave background (CMB) invariant angular power spectra (TT , EE , BB, and TE) 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. Subject headings: cosmology: cosmic microwave background — methods: data analysis — methods: numerical