Annu. Rev. Astron. Astrophys. 2000. 38: 667-715
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5.6. Clustering of high-redshift galaxies

Until relatively recently, evolution of the spatial two-point correlation function xi has typically been modeled as a simple power-law evolution in redshift

Equation 3 (3)

where r and r0 are expressed in comoving coordinates (e.g. [Groth & Peebles 1977]). For epsilon = 0, the formula above corresponds to stable clustering (fixed in proper coordinates), while for epsilon = gamma - 3 it corresponds to a clustering pattern that simply expands with the background cosmology. Although these two cases may bound the problem at relatively low redshift, the situation becomes much more complex as galaxies surveys probe to high redshifts. In particular, the complex merging and fading histories of galaxies make it unlikely that such a simple formula could hold, and the differences in sample selection at high and low-z make it unclear whether the analysis is comparing the same physical entities. In practice it is likely that galaxies are biased tracers of the underlying dark-matter distribution, with a bias factor b that is non-linear, scale-dependent, type-dependent, and stochastic [Magliocchetti et al. 1999, Dekel & Lahav 1999].

In hierarchical models, the correlation function xi(r) of halos on linear scales is given by the statistics of peaks in a Gaussian random field [Bardeen et al. 1986]. Peaks at a higher density threshold delta rho / rho have a higher correlation amplitude, and their bias factor b, relative to the overall mass distribution is completely specified by the number-density of peaks (e.g [Mao & Mo 1998]).

The comparison of the clustering of high-redshift galaxies in the HDF and in ground-based surveys provides a strong test of whether there is a one-to-one correspondence between galaxies and peaks in the underlying density field. The bias factor observed in Lyman-break samples [Steidel et al. 1998, Adelberger et al. 1998, Giavalisco et al. 1998] is in remarkable agreement with the expectations from such a one-to-one correspondence. As the luminosity of Lyman-break galaxies decreases, the number density increases, and the clustering amplitude decreases [Giavalisco et al. 2000]. These trends are all in agreement with a scenario in which lower-luminosity Lyman-break galaxies inhabit lower-mass halos. Thus, at redshifts z ~ 3, the connection between galaxies and dark matter halos may in fact be quite simple and in good agreement with theoretical expectations.

The picture becomes less clear for the samples described in Section 4.12. In particular, the interpretation of the apparent evolution in the correlation length r0 or the bias factor b from the [Magliocchetti & Maddox 1999] and [Arnouts et al. 1999] studies involves a sophisticated treatment of the merging of galaxies and galaxy halos over cosmic time. Fairly detailed attempts at this have been made using either semi-analytic models [Baugh et al. 1999, Kauffmann et al. 1999] or more generic arguments (e.g. [Matarrese et al. 1997, Moscardini et al. 1998, Magliocchetti et al. 1999], following on earlier work by [Mo & Fukugita 1996], [Mo & White 1996], [Jain1997], [Ogawa et al. 1997] and others). These studies all assume that galaxy mass is directly related to halo mass, which may not be true in reality at lower z, but which is an assumption worth testing. A generic prediction of the models is that the effective bias factor (the value of b averaged over the mass function) should increase with increasing redshift. The correlation length r0 is relatively independent of redshift (to within factor of ~ 3), in contrast to the factor of ~ 10 decline from z = 0 to z = 4 predicted for non-evolving bias. The results shown by [Arnouts et al. 1999] are qualitatively consistent with this behavior. However, it is worth re-emphasizing that the HDF is a very small field. The standard definition for the bias factor b is the ratio of the root mean square density fluctuations of galaxies relative to mass on a scale of 8h-1 Mpc. Fluctuations on this angular scale clearly have not been measured in the HDF, and the interpretation rests on (a) the assumption of a powerlaw index gamma = 1.8 for the angular correlation function (which is not in fact predicted by the models) [Moscardini et al. 1998], and (b) the fit to the integral constraint. Clearly much larger areas are needed before secure results can be obtained.

[Roukema & Valls-Gabaud 1997] and [Roukema et al. 1999] point out that much of the measured clustering signal in the HDF comes from scales 25 - 250 kpc, which is within the size of a typical L* galaxy halo at z = 0. The connection of the observed correlation function to hierarchical models thus depends quite strongly on what happens when multiple galaxies inhabit the same halo. Do they co-exist for a long time (e.g. as in present-day galaxy groups and clusters), or rapidly merge together to form a larger galaxy (in which case one should consider a "halo exclusion radius" in modeling the correlation function)? As the halo exclusion radius increases, the predicted correlation amplitude on scales of 5" decreases relative to the standard cosmological predictions. The slope of omega(theta) also differs from gamma = 1.8 for theta ltapprox 20". [Roukema et al. 1999] find that the HDF data for 1.5 < zphot < 2.5 are best fit with stable clustering, a halo-exclusion radius of rhalo = 200h-1 kpc, and a low-density universe.

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