An optimal FFT-based anisotropic power spectrum estimator

Measurements of line-of-sight dependent clustering via the galaxy power spectrum's multipole moments constitute a powerful tool for testing theoretical models in large-scale structure. Recent work shows that this measurement, including a moving line-of-sight, can be accelerated using Fast Fourier Transforms (FFTs) by decomposing the Legendre polynomials into products of Cartesian vectors. Here, we present a faster, optimal means of using FFTs for this measurement. We avoid redundancy present in the Cartesian decomposition by using a spherical harmonic decomposition of the Legendre polynomials. Consequently, our method is substantially faster: a given multipole of order requires only 2+1 FFTs rather than the (+1) (+2) /2 FFTs of the Cartesian approach. For the hexadecapole (= 4), this translates to 40\% fewer FFTs, with increased savings for higher. The reduction in wall-clock time enables the calculation of finely-binned wedges in P (k, ), obtained by computing multipoles up to a large ₌₀ₗ and combining them. This transformation has a number of advantages. We demonstrate that by using non-uniform bins in, we can isolate plane-of-sky (angular) systematics to a narrow bin at 0 while eliminating the contamination from all other bins. We also show that the covariance matrix of clustering wedges binned uniformly in becomes ill-conditioned when combining multipoles up to large values of ₌₀ₗ, but that the problem can be avoided with non-uniform binning. As an example, we present results using ₌₀ₗ=16, for which our procedure requires a factor of 3. 4 fewer FFTs than the Cartesian method, while removing the first bin leads only to a 7% increase in statistical error on f ₈, as compared to a 54% increase with ₌₀ₗ=4.