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 , Great Observatories Deep Origins Survey ( GOODS ) at a redshift around 3 , and Subaru/XMM-Newton Deep Field data at a redshift of 4 .
The comparison with statistical measurements allows us to constrain certain parameters related to analytical descriptions on the relation between a dark matter halo and its central galaxy luminosity , its satellite galaxy luminosity , and the fraction of early- and late-type galaxies of that halo .
With the SDSS r-band LF at M _ { r } < -17 , the log-normal scatter in the central galaxy luminosity at a given halo mass in the central galaxy–halo mass , L _ { c } ( M ) , relation is constrained to be 0.17 ^ { +0.02 } _ { -0.01 } , with 1 \sigma 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 , with luminosity M _ { r } < -17 , is 2 \times 10 ^ { 10 } h ^ { -1 } M _ { \hbox { $ \odot$ } } .
On the other hand , the appearance of satellites with luminosities M _ { r } < -17 at z < 0.1 , using a total luminosity-halo mass relation of the form L _ { c } ( M ) ( M / M _ { sat } ) ^ { \beta } _ { s } , is constrained with SDSS to be at a halo mass of M _ { sat } = ( 1.2 _ { -1.1 } ^ { +2.9 } ) \times 10 ^ { 13 } h ^ { -1 } M _ { \hbox { $ \odot$ } } with a power-law slope \beta _ { s } of ( 0.56 ^ { +0.19 } _ { -0.17 } ) .
At z \sim 0.6 , COMBO-17 data allows these parameters for M _ { B } < -18 galaxies to be constrained as ( 3.3 _ { -3.0 } ^ { +4.9 } ) \times 10 ^ { 13 } h ^ { -1 } M _ { \hbox { $ \odot$ } } and ( 0.62 ^ { +0.33 } _ { -0.27 } ) , respectively .
At z \sim 4 , Subaru measurements constrain these parameters for M _ { B } < -18.5 galaxies as ( 4.12 _ { -4.08 } ^ { +5.90 } ) \times 10 ^ { 12 } h ^ { -1 } M _ { \hbox { $ \odot$ } } and ( 0.55 ^ { +0.32 } _ { -0.35 } ) , respectively .
The redshift evolution associated with these parameters can be described as a combination of the evolution associated with the halo mass function and the luminosity–halo mass relation .
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 , \langle L _ { sat } \rangle = ( 2.1 ^ { +0.8 } _ { -0.4 } ) \times 10 ^ { 10 } h ^ { -2 } L _ { \hbox { $ \odot$ } } , while for GOODS at z \sim 3 , \langle L _ { sat } \rangle < 2 \times 10 ^ { 11 } h ^ { -2 } L _ { \hbox { $ \odot$ } } .
For SDSS , the fraction of galaxies that appear as satellites is 0.13 ^ { +0.03 } _ { -0.03 } , 0.11 ^ { +0.05 } _ { -0.02 } , 0.11 ^ { +0.12 } _ { -0.03 } , and 0.12 ^ { +0.33 } _ { -0.05 } for galaxies with luminosities in the r ^ { \prime } -band between -22 to -21 , -21 to -20 , -20 to -19 , and -19 to -18 , respectively .
In addition to constraints on central and satellite CLFs , we also determine model parameters of the analytical relations that describe the fraction of early- and 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 the halo occupation number alone .