Fast spin ±2 spherical harmonics transforms and application in cosmology

An exact fast algorithm is developed for the spin-weighted spherical harmonics transforms of band-limited spin ±2 functions on the sphere. First, we recall the notion of 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 ±2Ylm (of integer l and m, with l ≥ 2, |m| ≤ l) as linear combinations of the standard scalar spherical harmonics Ylm and Y(l−1)m. An exact algorithm is developed for the spin ±2 spherical harmonics transforms on equi-angular grids, based on the Driscoll and Healy fast scalar spherical harmonics transform. The theoretical exactness of the transform relies on a sampling theorem existing on equi-angular grids. The associated asymptotic complexity is of order O(L2 log2 L). Alternative algorithms exist on other pixelizations, based on the technique of separation of variables. However, they are not theoretically exact and have an associated asymptotic complexity of order O(L3). Finally, we consider the application of these generic developments in cosmology, for the efficient computation of the cosmic microwave background (CMB) invariant angular power spectra from the observable temperature T and the linear polarization Stokes parameters Q and U (direct transform), or for the simulation of temperature and polarization maps from theoretical spectra (inverse transform). Preprint submitted to Elsevier Science 19th March 2008