Halo Occupation Distribution Modeling of Clustering of Luminous Red Galaxies

TL;DR

This work uses a flexible halo occupation distribution (HOD) framework to interpret SDSS clustering of luminous red galaxies (LRGs) and their cross-correlation with $L_*$ galaxies. By decomposing LRGs into central and satellite populations and anchoring the $L_*$ occupancy to low-redshift results with a high-mass extension, the authors extract the halo-mass scales, satellite fractions, and the central-luminosity–halo-mass relation $L_c \propto M^p$. They find LRGs reside in halos of $M \sim$ a few $\times 10^{13}$ to $10^{14} h^{-1}M_\odot$ with a small satellite fraction and a central luminosity–mass slope $p \sim 0.5$–0.66; the $L_*$ occupation in massive halos shows a high-mass slope $\alpha_h \approx 1.5$ and a concentration consistent with dark matter. The scale-dependent cross-clustering is explained by the evolving balance of one-halo central–satellite, satellite–satellite, and two-halo pairs, providing insights for galaxy-formation theories and cosmological analyses using LRGs.

Abstract

We perform Halo Occupation Distribution (HOD) modeling to interpret small-scale and intermediate-scale clustering of 35,000 luminous early-type galaxies and their cross-correlation with a reference imaging sample of normal L* galaxies in the Sloan Digital Sky Survey. The modeling results show that most of these luminous red galaxies (LRGs) are central galaxies residing in massive halos of typical mass M ~ a few times 10^13 to 10^14 Msun/h, while a few percent of them have to be satellites within halos in order to produce the strong auto-correlations exhibited on smaller scales. The mean luminosity Lc of central LRGs increases with the host halo mass, with a rough scaling relation of Lc \propto M^0.5. The halo mass required to host on average one satellite LRG above a luminosity threshold is found to be about 10 times higher than that required to host a central LRG above the same threshold. We find that in massive halos the distribution of L* galaxies roughly follows that of the dark matter and their mean occupation number scales with halo mass as M^1.5. The HOD modeling results also allows for an intuitive understanding of the scale-dependent luminosity dependence of the cross-correlation between LRGs and L_* galaxies. Constraints on the LRG HOD provide tests to models of formation and evolution of massive galaxies, and they are also useful for cosmological parameter investigations. In one of the appendices, we provide LRG HOD parameters with dependence on cosmology inferred from modeling the two-point auto-correlation functions of LRGs.

Figures (9)

• Figure 1: Projected two-point auto-correlation functions and best-fit HODs for the two luminosity-threshold LRG samples. Panel ( a): The measured two-point correlation functions (data points and error bars) and the HOD model fits (solid curves). The two dashed curves for each sample show the envelope of predictions from models with $\Delta\chi^2<4$. The predicted one-halo and two-halo terms (dot-dashed curves) are also shown for the sample with the lower luminosity threshold. Panel ( b): The mean occupation functions (solid curves) of LRGs from the best fits, with contributions from central (dotted) and satellite (dashed) LRGs. For each sample, the two sets of curves are the envelope from models with $\Delta\chi^2<4$ and those for all galaxies are shaded. Panel ( c): The marginalized distribution of the satellite fraction in each sample with the central 68% distribution marked by the two dashed lines. Panel ( d): The probability distribution of halo masses for the LRGs in each sample (solid lines), obtained from the occupation function shown in panel ( b) weighted by the differential halo mass function. The dotted lines show the halo mass probability distribution from just the central galaxies.
• Figure 2: Constraints on HOD parameters from the LRG auto-correlation functions and the LRG-$L_*$ cross-correlation function (see the text for details). Panel ( a): The marginalized distribution of the parameter $\sigma_{M_g}$, which is the width (in magnitude) of central galaxy luminosity distribution at a fixed halo mass. Panel ( b): The marginalized distribution of the parameter $p$, which characterizes the relation between mean central galaxy luminosity and halo mass through $L_c\propto M^p$. In panels ( a) and ( b), the two dashed vertical lines indicate the central 68% of the distribution. Panel ( c): Illustration of the effect of $\sigma_{M_g}$ and $p$ on the mean occupation functions of LRGs. The thick dotted and dashed curves are for the $M_g<-21.2$ LRG sample, with $\sigma_{M_g}$ varied from 0.37 to 0.43 (i.e., a $\pm 1\sigma$ change). The thin dashed and dotted curves are for the $M_g<-21.8$ LRG sample, with $p$ varied from 0.46 to 0.51 (a $\pm 1\sigma$ change in $p$). Panel ( d): The marginalized joint distribution of the concentration parameter $c_0$ (normalized to that at the $z\sim 0.3$ nonlinear mass scale) and the high mass slope $\alpha_h$ of the mean occupation function for the $L_*$ galaxies. The contours show the 68% and 95% confidence levels for two parameters.
• Figure 3: Mass scales of the LRG HODs as a function of threshold luminosity. Shown are the relation between the characteristic minimum mass $M_{\rm min}$ at which 50% of halos host central galaxies above the luminosity threshold and the mass $M_1$ of halos that on average host one satellite galaxy, as a function of the threshold absolute magnitude. The square points are taken from Zheng07 for the SDSS main galaxy sample (corrected to $\sigma_8=0.8$). Open and filled circles are the results for the two LRG samples (note that the $z=0.3$$g$-band luminosity is converted to $z=0.1$$r$-band by adopting an apparent color of $g-r=0.4$). Dotted lines indicate different relations between luminosity of central galaxy and mass of host halo.
• Figure 4: Mean occupation functions and cross-correlation functions of $L_*$ galaxies and LRGs from the HOD modeling. Left panels: the predicted two-point cross-correlation function between LRGs and $L_*$ galaxies separated into contributions from central and satellite LRGs paired with $L_*$ galaxies within same halos and LRGs paired with $L_*$ galaxies from different halos. The scale $a$ is in units of proper $h^{-1}{\rm Mpc}$ to be consistent with that adopted in Eisenstein05a. The quantity $\phi_0=2.267\times 10^{-2} h^3{\rm Mpc}^{-3}$ is the proper number density of $L_*$ galaxies at the mean redshift $z\sim 0.3$. The lower panel shows the fractional difference between model fits (thick for the flexible model and thin for the slope-fixed model) and data (see the text). Right panel: the mean occupation function for the $L_*$ galaxies in massive halos (top solid curves) and that for the luminosity-bin LRGs. (bottom curves). The square window at the low mass end represents the mean occupation function of central $L_*$ galaxies. The envelopes of the mean occupation functions are derived from models with $\Delta\chi^2<4$. Dotted and dashed curves in the LRG mean occupation function are contributions from central and satellite galaxies, respectively. The dot-dashed curve shows the mean occupation function for the $L_*$ galaxies from a model with the slope fixed to be 1.10 in the whole mass range.
• Figure 5: The luminosity dependence of the LRG--$L_*$ cross-correlation functions at different scales. The six panels correspond to (proper) scales $a$= 0.125 to 4$h^{-1}{\rm Mpc}$, respectively, as labeled in each panel. The quantity $\phi_0=2.267\times 10^{-2} h^3{\rm Mpc}^{-3}$ is the proper number density of $L_*$ galaxies at the mean redshift $z\sim 0.3$, and $V=3(2\pi a^2)^{3/2}$ is the effective volume for the window function [see eq. (\ref{['eqn:window']})]. In each panel, the shaded regions are predictions from the modeling results. Since the model predictions are based on clustering information up to only $\sim 4L_*$, the shaded regions below and above $4L_*$ are model interpolations and extrapolations, respectively. The data points with error bars are the measurements in Eisenstein05a. Thick (thin) dotted curves are contributions from central (satellite) LRGs paired with $L_*$ galaxies within common halos (calculated from the best-fit HOD model), and dashed curves, which can be clearly seen in panels ( d)--( f), represent the two-halo pair contribution.