Constraining the LRG Halo Occupation Distribution using Counts-in-Cylinders

TL;DR

This paper introduces Counts-In-Cylinders (CiC) as a high-fidelity method to constrain the satellite component of the LRG Halo Occupation Distribution by leveraging higher-order statistics of close LRG pairs. By identifying one-halo pairs with specific transverse and line-of-sight separations and grouping them via a FoF algorithm, CiC yields a measured group multiplicity function that is calibrated against mock catalogs built from SO halo catalogs. The analysis fixes the central-halo occupation parameters while fitting the satellite occupation $ig\langle N_{sat}(M)\big\rangle = \big(M - M_{cut}\over M_1\big)^{\alpha}$ through a maximum-likelihood approach that accounts for offsets and stochasticity in CiC detections. The resulting constraints indicate a low satellite fraction of $f_{sat} \approx 6.4\%$ and a near-linear satellite scaling with halo mass ($\alpha \approx 1.04$), with $M_{cut}$ and $M_1$ tightly bounded, and demonstrate that SO-based mocks reproduce both CiC and projected clustering, while FoF-based catalogs underproduce close pairs at $\sim 1\,h^{-1}$Mpc. The CiC approach thus provides a robust, halo-finder-insensitive route to accurate HOD inferences and will be extended to optimize FOG compression in a companion study.

Abstract

The low number density of the Sloan Digital Sky Survey (SDSS) Luminous Red Galaxies (LRGs) suggests that LRGs occupying the same dark matter halo can be separated from pairs occupying distinct dark matter halos with high fidelity. We present a new technique, Counts-in-Cylinders (CiC), to constrain the parameters of the satellite contribution to the LRG Halo-Occupation Distribution (HOD). For a fiber collision-corrected SDSS spectroscopic LRG subsample at 0.16 < z < 0.36, we find the CiC multiplicity function is fit by a halo model where the average number of satellites in a halo of mass M is <Nsat(M)> = ((M - Mcut)/M1)^alpha with Mcut = 5.0 +1.5/-1.3 (+2.9/-2.6) X 10^13 Msun, M1 = 4.95 +0.37/-0.26 (+0.79/-0.53) X 10^14 Msun, and alpha = 1.035 +0.10/-0.17 (+0.24/-0.31) at the 68% and 95% confidence levels using a WMAP3 cosmology and z=0.2 halo catalog. Our method tightly constrains the fraction of LRGs that are satellite galaxies, 6.36 +0.38/-0.39, and the combination Mcut/10^{14} Msun + alpha = 1.53 +0.08/-0.09 at the 95% confidence level. We also find that mocks based on a halo catalog produced by a spherical overdensity (SO) finder reproduce both the measured CiC multiplicity function and the projected correlation function, while mocks based on a Friends-of-Friends (FoF) halo catalog has a deficit of close pairs at ~1 Mpc/h separations. Because the CiC method relies on higher order statistics of close pairs, it is robust to the choice of halo finder. In a companion paper we will apply this technique to optimize Finger-of-God (FOG) compression to eliminate the 1-halo contribution to the LRG power spectrum.

Figures (9)

• Figure 2: The mean $c_{\parallel}-z$ relation and bands including 95% of objects in each $\Delta z = 0.01$ bin. $c_{\parallel}$ (Eqn. \ref{['cparcuteqn']}) is used as a redshift indicator for objects targeted as LRGs but lacking spectra.
• Figure 3: Contours for $\Delta \ln L = \{-1.15, -3.09, -5.9\}$ in the $M_{1}$ vs. $M_{cut}$ plane after marginalizing over $\alpha$. For a $\chi^2$ distribution, these contours would enclose 1, 2, and 3$\sigma$ confidence regions. The cross indicates the maximum likelihood parameter values.
• Figure 4: Same as Figure \ref{['fig:sdssM1Mcut']}, but for $\alpha$ vs. $M_{cut}$. $M_{cut}/(10^{14} M_{\odot}) + \alpha$ is tightly constrained, while $M_{cut}/(10^{14} M_{\odot}) - \alpha$ is only weakly constrained.
• Figure 5: Same as Figure \ref{['fig:sdssM1Mcut']}, but for $\alpha$ vs. $M_{1}$.
• Figure 6: The dashed curve ranging from 0 to 1 shows the $N_{cen}(M)$ term for the maximum likelihood HOD; it should vary only slightly with HOD since the satellite fraction is well constrained by our model, and we hold $\sigma_{log M}$ fixed. The solid curve shows $\left<N_{sat}(M) N_{cen}(M)\right>$, with the error bars computed by Eqn. \ref{['varHODeqn']}.