Annu. Rev. Astron. Astrophys. 1997. 35: 389-443
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3.5. Evolutionary Constraints from Galaxy Correlation Functions

The analysis of positional data in deep galaxy catalogs in terms of 2-D angular correlations or 3-D redshift space correlations is primarily useful as a probe of the evolution of clustering with look-back time (Phillipps et al 1978). The role of correlation functions in constraining galaxy evolution, as stressed by Koo & Szalay (1984), has, to a large extent, been overtaken by the redshift surveys, which give a much clearer and less ambiguous indicator of luminosity changes. Nonetheless, a bewildering amount of 2-D data has been analyzed in recent years, mostly from panoramic photographic and CCD-based surveys in various bands, and the results arising from this work have featured prominently in the debate on faint blue galaxies.

The angular correlation function, w(theta), of faint galaxies is linked to its spatial equivalent, xi(r), by terms that depend on the redshift distribution (see Peebles 1994 for definitions). The connecting relationship (Limber 1953, Phillipps et al 1978 takes into account both the angular diameter distance, dA = dL / (1 + z)2, and dilution from uncorrelated pairs distributed along the line of sight. In practice, as xi(r) is locally type-dependent Davis & Geller 1976, w(theta) is weighted by the type-dependent redshift distribution N(z, j) determined by the k-correction and evolutionary behavior of each type. A further factor is the likely evolution in spatial clustering conventionally parameterized in proper space as

Equation 6 (6)

where r0 (= 5 h-1 Mpc; Peebles 1980) is the current scale length of galaxy clustering and epsilon = 0, corresponds to clustering fixed in proper coordinates. For a correlation function xi(r) = (r / r0)-1.8, clustering fixed in comoving coordinates yields epsilon = -1.2, whereas linear theory with Omega = 1 indicates growth equivalent to epsilon = +0.8 (Peacock 1997).

Broadly speaking, the angular correlation function results for the faint surveys are consistent with a decline in the spatial correlation function with increasing redshift (Roche et al 1993, 1996b, Infante & Pritchet 1995, Brainerd et al 1995, Hudon & Lilly 1996, Villumsen et al 1996), but any quantitative interpretation that would allow statements to be made on luminosity evolution would have to make careful allowance for the change in apparent mix of types with redshift. Because of the morphology-density relation, the k-correction works in the direction of suppressing the visibility of correlated spheroidal galaxies, and therefore an apparent decrease in the amplitude of clustering is inevitable. Using color-selected subsamples, some workers (Efstathiou et al 1991, Neuschaefer et al 1991, Neuschaefer & Windhorst 1995) have addressed this issue and claimed that a large fraction of the faintest blue samples must be weakly correlated unless they are at very high redshift (z > 3). The surprisingly large decrease in angular clustering with apparent magnitude has even led to the suggestion that the bulk of the faintest sources represent a population unrelated to normal galaxies (Efstathiou 1995).

The dramatic evolution claimed from the 2-D data sets contrasts somewhat with those studies based on 3-D data from the redshift surveys. Cole et al (1994b) derived spatial correlation functions from the Autofib redshift survey, and Bernstein et al (1994) analyzed panoramic 2-D data within magnitude limits where the redshift range is likewise known. Neither found significant changes in the clustering scale to z = 0.3 other than can be accounted for if the bluer star-forming galaxies were somewhat less clustered, as seen locally in IRAS-selected samples. Estimates of spatial clustering at higher redshift in the CFRS redshift surveys give rather uncertain, though modest, growth estimates (0 < epsilon < 2) (LeFevre et al 1996b). The results from the deeper Keck LRIS survey (Carlberg et al 1997) also indicate modest evolution (a 60% decrease in r0 from z = 0.6 to 1.1) with a possible strong cross-correlation between high and low luminosity galaxies on 100 h-1 kpc scales.

The apparent conflict between the weak angular clustering of faint sources and the modest decline in the spatial clustering to z = 1 might be understood if an increasing proportion of the detected population is (a) less clustered and (b) itself evolving at a physically reasonable rate. Villumsen et al (1997)) argue that over 20 < R < 29, w(theta) can be explained via the linear growth of a component whose present correlation length is comparable to that of local IRAS galaxies (r0 approx 4 h-1 Mpc). Smaller correlation lengths (approx2 h-1 Mpc) are claimed by Brainerd et al (1995), but the differences may relate to the adopted redshift distributions for the faint populations as well as possible sampling variations in the small fields available to date. Peacock (1997) concludes the evolution in xi(r) seen by LeFevre et al can be simply interpreted via a single population of sources evolving with epsilon approx 1 - corresponding to Omega approx 0.3. That such diverse conclusions can emerge from these basic observational trends gives some indication of the degeneracies involved in analyzing the data.

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