Annu. Rev. Astron. Astrophys. 2000. 38: 667-715
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4.12. Galaxy Clustering

The measurements of clustering using the HDF have varied in the catalog and object selection, in the angular scales considered, in the use or non-use of photometric redshifts, and in the attention to object masking and sensitivity variations. In an early paper on the HDF-N, [Colley et al. 1996] explored whether the galaxy counts in the HDF are "whole numbers," i.e. whether fragments of galaxies were incorrectly being counted as individual galaxies. From an analysis of the angular correlation function of objects with color-redshifts z > 2.4 they conclude that many of these objects are HII regions within a larger underlying galaxy. This analysis showed a strong clustering signal on scales 0".2 < theta < 10". [Colley et al. 1997] explored various hypotheses for the neighboring galaxies and favored a scenario in which the faint compact sources in the HDF are giant star-forming regions within small Magellanic irregulars.

It is not clear whether the Colley et al. (1996, 1997) results apply to other catalogs, given the different cataloging algorithms used. In particular, programs such as SExtractor [Bertin & Arnouts 1996] and FOCAS [Tyson & Jarvis 1979] use sophisticated and (fortunately or unfortunately) highly tunable algorithms for merging or splitting objects within a hierarchy of isophotal thresholds. The DAOFIND algorithm used by [Colley et al. 1996] provides no such post-detection processing. The extent to which this affects the results can only be determined by an object-by-object comparison of the catalogs, which has not (yet) been done in any detail. [Ferguson 1998b] (Figs. 1-4) presents a qualitative comparison of several catalogs (not including that of Colley et al.) which shows significant differences in how objects with overlapping isophotes are counted. The [Colley et al. 1997] analysis focused specifically on 695 galaxies with color-redshifts z > 2.4. This number of galaxies is considerably larger than the 69 identified by [Madau et al. 1996], using very conservative color selection criteria, or the 187 identified by [Dickinson 1998] with somewhat less conservative criteria. For these smaller samples, we suspect that the "overcounting" problem is not as severe as Colley et al. contend.

Problems separating overlapping objects, although important for clustering studies on small angular scales, do not have much effect on the overall galaxy number-magnitude relation [Ferguson 1998b] or angular clustering measurements on scales larger than ~ 2".

Other studies of clustering in the HDF have been restricted to separations larger than 2". The analysis has focused primarily on the angular correlation function omega(theta), which gives the excess probability deltaP, with respect to a random Poisson distribution of n sources, of finding two sources in solid angles delta Omega1, delta Omega2 separated by angle theta:

Equation 1 (1)

The angular correlation function is normally modeled as a power law

Equation 2 (2)

Because of the small angular size of the field, omega(theta) is suppressed if the integral of the correlation function over the survey area is forced to be zero. Most authors account for this "integral constraint" by including another parameter and fitting for omega(theta) = A(theta0 theta1-gamma - C, with a fixed value of gamma = 1.8 and with A and C as free parameters.

[Villumsen et al. 1997] measured the overall angular correlation function for galaxies brighter than R = 29 and found an amplitude A(theta = 1") decreasing with increasing apparent magnitude, roughly consistent with the extrapolation of previous ground-based results. The measured amplitude was roughly the same for the full sample and for a subsample of red galaxies (considered typically to be at higher redshift). The motivation for this color cut was to try to isolate the effects of magnification bias due to weak lensing by cosmological large-scale structure [Moessner et al. 1998], but the predicted effect is small and could not be detected in the HDF (and will ultimately be difficult to disentangle from the evolutionary effects discussed in Section 5.6).

The remaining HDF studies have explicitly used photometric redshifts. [Connolly et al. 1999] considered scales 3" < theta < 220" and galaxies brighter than I814 = 27. Within intervals Deltaz = 0.4 the amplitude A(theta = 10") ~ 0.13 shows little sign of evolution out to zphot = 1.6 (the highest considered by Connolly et al.). [Roukema et al. 1999] analyzed a U-band selected sample, isolated to lie within the range 1.5 < zphot < 2.5 and angular separations 2" < theta < 40". The results are consistent with those of [Connolly et al. 1999], although [Roukema et al. 1999] point out that galaxy masking and treatment of the integral constraint can have a non-negligible effect on the result.

Measurements of omega(theta) for samples extending out to zphot > 4 have been carried out by [Magliocchetti & Maddox1999], [Arnouts et al. 1999] and [Miralles & Pello 1998]. The minimum angular separations considered for the three studies were 9", 5", and 10", respectively: i.e. basically disjoint from the angular scales considered by [Colley et al. 1997]. All three studies detect increased clustering at z gtapprox 2, although direct comparison is difficult because of the different redshift binnings. The interpretation shared by all three studies is that the HDF shows strong clustering for galaxies with z gtapprox 3, in qualitative agreement with Lyman-break galaxy studies from ground-based samples [Giavalisco et al. 1998, Adelberger et al. 1998]. Although this may be the correct interpretation, the significant differences in the photometric redshift distributions and the derived clustering parameters from the three studies leave a considerable uncertainty about the exact value of the clustering amplitude. The two most comprehensive studies differ by more than a factor of 3 in A(theta = 10") at z = 3 [Arnouts et al. 1999, Magliocchetti & Maddox 1999]. Both studies use photometric redshifts to magnitudes I814 = 28, which is fainter than the detection limits in the F300W and F450W bands for even flat spectrum galaxies. A large part of the disagreement may thus be due to scatter in zphot. The analysis also hinges critically on the assumed power-law index gamma = 1.8 and the necessity to fit the integral constraint. Thus, although it seems reasonably secure that a positive clustering signal has been measured in the HDF at high zphot, it will require much larger data sets to constrain the exact nature of this clustering and determine its relation to clustering at lower redshift.

At brighter magnitudes, spectroscopic surveys show clear evidence for clustering in redshift space [Cohen et al. 1996, Adelberger et al. 1998, Cohen 1999], with pronounced peaks even at redshifts z > 1. These structures, and others within the flanking fields, do not show evidence for centrally concentrated structures, and are probably analagous to walls and filaments observed locally.

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