5. CLUSTERING OF PROTOGALAXIES

It seems appropriate to provide a brief summary of the method used to generate the clustering lengths and inferred dark matter halo masses given in the Tables. The spatial correlation function (r) = (r / r₀)^- is inferred by fitting a power-law to either the observed spatial or angular correlation function of the sample. If only angular positions are observed, the redshift distribution N(z) must be measured spectroscopically and used to invert the Limber equation as described in Giavalisco et al. (1998). The Landy-Szalay estimator is typically used to estimate the angular or spatial correlation function of the datapoints and to correct for the so-called "integral constraint" caused by measuring the mean density of the population from the observed survey volume (Landy & Szalay 1993). The LBG, LAE, and DRG samples are large enough to use the correlation length r₀ measured from the auto-correlation function to determine the bias factor e.g. b_LBG, following

(2)

where _DM(r) is the dark matter autocorrelation function predicted by the CDM cosmology. The SMG and DLA samples are small, so their cross-correlation with the more numerous LBGs is used to determine their bias factor, e.g.

(3)

The bias factor of each family of protogalaxies determines its typical dark matter halo mass following the method of Mo & White (1996), whose application to the cross-correlation function was first suggested by Gawiser et al. (2001). This method also allows one to predict the number abundance of dark matter halos with mass above the given threshold mass and to compare this with the observed number density of the population to infer the average halo occupation number.

Table 1. The z = 3 universe. References for entries are given in the text, with a few entries still to be determined from MUSYC, ALMA, and JWST. Typical systematic uncertainties are a factor of two.

TLA	Space density	SFR per object	Stellar mass per object	Clustering length (r0)
	[h703 Mpc-3]	[M yr-1]	[M]	[h70-1 Mpc]
LBG	2 × 10-3	30	1010	6 ± 1
LAE	3 × 10-4	6	MUSYC	4 ± 1
DRG	3 × 10-4	200	2 × 1011	9 ± 2
SMG	2 × 10-6	1000	MUSYC	16 ± 7
DLA	ALMA	1-50	JWST	4 ± 2

Table 2. Cosmological quantities. References for entries are given in the text, with a few entries still to be determined from MUSYC and JWST. Typical systematic uncertainties are a factor of three.

TLA	SFR density	Stellar mass density	Dark matter halo mass
	[M yr-1 h703 Mpc-3]	[M h703 Mpc-3]	[M]
LBG	0.1	107	3 × 1011
LAE	0.002	MUSYC	1011
DRG	0.06	6 × 107	3 × 1012
SMG	0.02	MUSYC	1013
DLA	0.03	JWST	1011