Measuring line-of-sight dependent Fourier-space clustering using FFTs

TL;DR

The paper tackles the challenge of measuring line-of-sight (LOS) dependent clustering in Fourier space when the LOS varies across a galaxy survey. It adopts the Yamamoto approximation, replacing the pair LOS with the LOS to one galaxy and implements LOS-weighted power-spectrum moments via multiple FFTs, achieving a computational scaling of $N_k \log N$ and a large speed-up over direct pair counting. The method decomposes LOS-weighted terms into FFT-friendly building blocks $A_n({\bf k})$, $B_{ij}({\bf k})$, and $C_{ijl}({\bf k})$, enabling accurate recovery of the monopole, quadrupole, and hexadecapole with 1, 7, and 22 FFTs respectively, and extends to wedges through Legendre expansions of LOS dependence. Performance tests on CMASS mocks show excellent agreement with traditional summation approaches while delivering ~1000x speed-ups, making it practical for upcoming large surveys like DESI and Euclid and adaptable to higher-order statistics such as the bispectrum.

Abstract

Observed galaxy clustering exhibits local transverse statistical isotropy around the line-of-sight (LOS). The variation of the LOS across a galaxy survey complicates the measurement of the observed clustering as a function of the angle to the LOS, as fast Fourier transforms (FFTs) based on Cartesian grids, cannot individually allow for this. Recent advances in methodology for calculating LOS-dependent clustering in Fourier space include the realization that power spectrum LOS-dependent moments can be constructed from sums over galaxies, based on approximating the LOS to each pair of galaxies by the LOS to one of them. We show that we can implement this method using multiple FFTs, each measuring the LOS-weighted clustering along different axes. The N log(N) nature of FFTs means that the computational speed-up is a factor of >1000 compared with summing over galaxies. This development should be beneficial for future projects such as DESI and Euclid which will provide an order of magnitude more galaxies than current surveys.

Figures (2)

• Figure 1: Top panel: power spectrum multipoles: monopole (blue lines), quadrupole (red lines) and hexadecapole (green lines), obtained from the average of 50 realization of pthalos mocks corresponding to the BOSS DR11 CMASS NGC survey geometry. The solid lines display the computation of Eq. (\ref{['eq:P_yama']}) using the FFT-based method using $1024^3$ grid cells. The dashed and dotted lines display the computation of the Yamamoto estimator using the sum-grid (with $512^3$ cells) and sum-gal methods, respectively. In both these cases an orthonormal base of $512^3$$k$-vectors has been used. The bottom panels show the corresponding sum-gal and sum-grid multipoles divided by the FFT-based multipoles to highlight differences among these implementations.
• Figure 2: Top panel: power spectrum 'Wedges': perpendicular-to-the-LOS power spectrum monopole, $P_\perp$ (blue lines) and parallel-to-the-LOS power spectrum monopole (red lines) obtained from the average of 50 realization of pthalos mocks corresponding to the BOSS DR11 CMASS NGC survey geometry. The solid lines display the approximation presented by Eq. (\ref{['P_perp']}-\ref{['P_para']}) using the monopole, quadrupole and hexadecapole computed by using the FFT-based method described in §\ref{['sec:algorithm']} placing the particles in $1024^3$ grid cells. The dashed lines display the computation of the "Wedges" using sum-gal and Eq. (\ref{['P_perp_def']}-\ref{['P_para_def']}), so the sum is exact. In this case an orthonormal base of $512^3$$k$-vector have been used. The bottom panels show the fractional differences between the sum-gal and the FFT-based method, for $P_\perp$ and $P_\parallel$ as labeled.