Halo Model at Its Best: Constraints on Conditional Luminosity Functions from Measured Galaxy Statistics

Using the conditional luminosity function (CLF) — the luminosity distribution of galaxies in a dark matter halo — as the fundamental building block, we present an empirical model for the galaxy distribution. The model predictions are compared with the published luminosity function and clustering statistics from Sloan Digital Sky Survey (SDSS) at low redshifts, and galaxy correlation functions from COMBO-17 survey at a redshift of 0.6, DEEP2 survey at a redshift of unity, and Great Observatories Deep Origins Survey (GOODS) at a redshift around 3. The comparison with statistical measurements allows us to constrain certain parameters related to analytical descriptions that describe the relation between a dark matter halo and its central galaxy luminosity, its satellite galaxy luminosity, and the fraction of earlyand late-type galaxies of that halo. With the SDSS r-band LF at Mr < −17, the log-normal scatter in the central galaxy luminosity at a given halo mass in the central galaxy–halo mass, Lc(M), relation is constrained to be 0.17 +0.02 −0.01, with 1 σ errors here and below. For the same galaxy sample, we find no evidence for a low-mass cut off in the appearance of a single central galaxy in dark matter halos, with the 68% confidence level upper limit on the minimum mass of dark matter halos to host a central galaxy is 2× 10 h M⊙. On the other hand, the appearance of satellites with luminosities Mr < −17 at z < 0.1, using a total luminosity-halo mass relation of the form Lc(M)(M/Msat) β s , is constrained with SDSS to be at a halo mass of Msat = (1.2 +2.9 −1.1) × 10 13 h M⊙ with a power-law slope βs of (0.56 +0.19 −0.17). At z ∼ 0.6, COMBO-17 data allows these parameters for MB < −18 galaxies to be constrained as (3.3 +4.9 −3.0) × 10 13 h M⊙ and (0.62 −0.27), respectively. DEEP2 and GOODS measurements only allow an upper limit on the power-law slope of total luminosity at a given minimum halo mass for the appearance of satellites. The single parameter well constrained by clustering measurements is the average of total satellite galaxy luminosity corresponding to the dark matter halo distribution probed by the galaxy sample. For SDSS, 〈Lsat〉 = (2.1 +0.8 −0.4) × 10 10 h−2 L⊙, while for GOODS at z ∼ 3, 〈Lsat〉 < 2 × 10 11 h L⊙. Certain quantities derivable from our model such as the fraction of satellite galaxies at a given galaxy luminosity are not strongly constrained with SDSS data. In addition to constraints on central and satellite CLFs, we also determine model parameters of the analytical relations that describe the fraction of earlyand late-type galaxies in dark matter halos. We use our CLFs to establish probability distribution of halo mass in which galaxies of a given luminosity could be found either at halo centers or as satellites. Finally, to help establish further properties of the galaxy distribution, we propose the measurement of cross-clustering between galaxies divided into two distinctly different luminosity bins. Our analysis show how CLFs provide a stronger foundation to built up analytical models of the galaxy distribution when compared to models based on halo occupation number alone.