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. With this method, a given multipole of order l requires only 2l+1 FFTs rather than the (l+1)(l+2)= 2 FFTs of the Cartesian approach. For the hexadecapole (l = 4), this translates to 40% fewer FFTs, with increased savings for higher l. The reduction in wall-clock time enables the calculation of finely-binned wedges in P(k; mu ), obtained by computing multipoles up to a large l(max) and combining them. This transformation has a number of advantages. We demonstrate that by using non-uniform bins in mu, we can isolate plane-of-sky (angular) systematics to a narrow bin at mu similar or equal to 0 while eliminating the contamination from all other bins. We also show that the covariance matrix of clustering wedges binned uniformly in mu becomes ill-conditioned when combining multipoles up to large values of l(max), but that the problem can be avoided with non-uniform binning. As an example, we present results using l(max) = 16, for which our procedure requires a factor of 3.4 fewer FFTs than the Cartesian method, while removing the first mu bin leads only to a 7% increase in statistical error on integral sigma 8,as compared to a 54% increase with l(max) = 4.