Galaxy Bias and Halo-Occupation Numbers from Large-Scale Clustering

TL;DR

The study demonstrates that higher-order galaxy clustering statistics, notably the bispectrum and trispectrum, contain substantial information about galaxy bias at large scales, enabling robust constraints on the mean halo occupation distribution (HOD) without relying on detailed intra-halo galaxy modeling. By deriving and employing practical estimators for the trispectrum alongside the bispectrum, the authors show that including trispectrum information reduces uncertainties on linear and quadratic bias parameters and helps constrain the mass dependence of the HOD mean, providing a valuable consistency check against small-scale, two-point analyses. A likelihood framework with both ideal and SDSS-like survey geometries indicates that trispectrum data can improve bias parameter constraints by roughly 20–40% depending on geometry and modeling, and can map these constraints into HOD parameters such as $M_1$ and $eta$ with meaningful precision. Overall, the work highlights a feasible, large-scale pathway to test and refine the galaxy-halo connection, potentially validating or refining small-scale modeling assumptions while reducing degeneracies with cosmology. The approach hinges on the halo model and perturbation theory, and calls for improved trispectrum estimators and validation against simulations to fully realize its potential.

Abstract

We show that current surveys have at least as much signal to noise in higher-order statistics as in the power spectrum at weakly nonlinear scales. We discuss how one can use this information to determine the mean of the galaxy halo occupation distribution (HOD) using only large-scale information, through galaxy bias parameters determined from the galaxy bispectrum and trispectrum. After introducing an averaged, reasonably fast to evaluate, trispectrum estimator, we show that the expected errors on linear and quadratic bias parameters can be reduced by at least 20-40%. Also, the inclusion of the trispectrum information, which is sensitive to "three-dimensionality" of structures, helps significantly in constraining the mass dependence of the HOD mean. Our approach depends only on adequate modeling of the abundance and large-scale clustering of halos and thus is independent of details of how galaxies are distributed within halos. This provides a consistency check on the traditional approach of using two-point statistics down to small scales, which necessarily makes more assumptions. We present a detailed forecast of how well our approach can be carried out in the case of the SDSS.

Figures (2)

• Figure 1: Slices 50 Mpc/$h$ thick of a mock galaxy distribution obtained from an HOD fit in a $\Lambda$CDM model to the $M_r<-20$ galaxy two-point function in SDSS (left) and a Rayleigh-Lèvy flight (right). Despite their obvious differences, these two distributions have the same two-point statistics, the differences seen are entirely due to those in higher-order correlations, see Fig. \ref{['PkQBQT']}.
• Figure 2: The distributions in Fig. \ref{['HODRLwalkdist']} have the same power spectrum (left) but can be easily distinguished by their bispectrum and trispectrum (right). Square symbols correspond to the HOD galaxies, triangles to the Rayleigh-Lèvy flight. The bispectrum ($Q_B$) and trispectrum ($Q_T$) are for all shapes of triangles and "quads" (see section \ref{['BTestim']}) in the range $0.04 \, h \, {\rm Mpc}^{-1} \leq k \leq 0.4 \, h \, {\rm Mpc}^{-1}$, here binned into $N_T=170$ and $N_Q=203$ configurations, respectively. The variations seen in $Q_B$ and $Q_T$ in the HOD galaxies are due to the dependence of higher-order correlations on the shape of the configuration, a reflection of the filamentary structure seen in the left panel in Fig. \ref{['HODRLwalkdist']}.