Fast Estimators for Redshift-Space Clustering

TL;DR

This work tackles the challenge of radial redshift-space distortions in galaxy surveys by introducing local estimators that generalize the power spectrum and bispectrum to non-homogeneous statistics. It derives FFT-based multipole estimators for the power spectrum ($P_ ext{ell}$ with $\ell=0,2,4$) and the bispectrum ($B^{(\ell)}$ with $\ell=0,2,4$), including an efficient hexadecapole variant that reduces computational costs via line-of-sight splitting. The authors provide explicit implementations for galaxy surveys, including shot-noise corrections and practical normalization, and demonstrate favorable performance on realistic mocks, achieving full $P$ and $B$ multipoles with modest FFT counts and runtimes. This fast, scalable framework enables accurate covariance estimation from thousands of mocks and enhances constraints on gravity, inflation, and galaxy bias for current and upcoming large-scale structure surveys.

Abstract

Redshift-space distortions in galaxy surveys happen along the radial direction, breaking statistical translation invariance. We construct estimators for radial distortions that, using only Fast Fourier Transforms (FFTs) of the overdensity field multipoles for a given survey geometry, compute the power spectrum monopole, quadrupole and hexadecapole, and generalize such estimators to the bispectrum. Using realistic mock catalogs we compare the signal to noise of two estimators for the power spectrum hexadecapole that require different number of FFTs and measure the bispectrum monopole, quadrupole and hexadecapole. The resulting algorithm is very efficient, e.g. for the BOSS survey requires about three minutes for $\ell=0,2,4$ power spectra for scales up to $k=0.3~h/$Mpc and about fifteen additional minutes for $\ell=0,2,4$ bispectra for all scales and triangle shapes up to $k=0.2~h/$Mpc on a single core. The speed of these estimators is essential as it makes possible to compute covariance matrices from large number of realizations of mock catalogs with realistic survey characteristics, and paves the way for improved constrains of gravity on cosmological scales, inflation and galaxy bias.

Figures (3)

• Figure 1: The bias ratio $b_4 \equiv \langle \widehat{P}_{4b}\rangle/\langle\widehat{P}_{4}\rangle$ between the two estimators of the hexadecapole (symbols with error bars), and the ratio of their cosmic variance $\sigma_4^2\equiv \langle \Delta\widehat{P}_{4b}^2\rangle/\langle\Delta\widehat{P}_{4}^2\rangle$ as a function of $k$ for the Las Damas LRG ($M_g < -21.8$) DR7 mock catalogs.
• Figure 2: Sames as Fig. \ref{['P4LD']} for the PTHalos CMASS DR11 mock catalogs.
• Figure 3: Reduced bispectrum multipoles $Q_{123}^{(\ell)}$ ($\ell=0,2,4$ from top to bottom) for galaxies in the LasDamas LRG ($M_g<-21.8$) DR7 mock catalogs. The triangles correspond to $k_1=0.047 \, h \, {\rm Mpc}^{-1}$, $k_2=2\, k_1$ as a function of the angle $\theta$ between ${\bf k}_1$ and ${\bf k}_2$.