An optimal FFT-based anisotropic power spectrum estimator

Abstract

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 $\ell$ requires only $2\ell+1$ FFTs rather than the $(\ell+1)(\ell+2)/2$ FFTs of the Cartesian approach. For the hexadecapole ($\ell = 4$), this translates to $40\%$ fewer FFTs, with increased savings for higher $\ell$. 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 $\ell_{\rm max}$ 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 $\mu \simeq 0$ 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 $\ell_{\rm max}$, but that the problem can be avoided with non-uniform binning. As an example, we present results using $\ell_{\rm max}=16$, 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 $f \sigma_8$, as compared to a 54% increase with $\ell_{\rm max}=4$.