2.4.2. Correlation functions
An alternative to the morphological approach described above is to use statistics. Statistical analyses of galaxy catalogues have been considered by various authors (Rubin, 1954; Limber, 1954; Neyman, Scott and Shane, 1954; Totsuji and Kihara, 1969) though much of the recent interest has been due to the results of Peebles and coworkers. Peebles et al. have applied the low-order correlation functions as a measure of galaxy clustering to the Zwicky, Lick and Jagellonian galaxy catalogues. This work has been reviewed extensively by Fall (1979a) and Peebles (1980a), to which we refer the interested reader for details; here we shall briefly discuss the main results.
The two-point correlation function (r) is defined so that the joint probability of finding galaxies in the volume elements V1, V2 separated by a distance r is
where n is the mean space density of galaxies. Hence (r) measures deviations of the pattern of galaxy clustering from a Poisson distribution. The data available to Peebles et al. has consisted of galaxy coordinates in projection. Hence in order to estimate the form of (r) they have measured the angular two-point function w(). w() is defined in an exactly analogous way to (r) and is related to via an integral equation first derived by Limber (1954). Higher order correlation functions may also be defined, e.g. the three-point function :
where P is the joint probability of finding galaxies in each of the three volume elements V1, V2, V3 with separations r12, r23, r13. Similarly, the four-point and higher order correlations may be defined, though so far useful results have not been obtained for correlation functions of higher order than the four-point function.
The main result (Peebles, 1974a) is that the spatial two-point correlation function has an approximately power law from over a wide range of scales,
The normalization factor, r0, quoted here comes from Peebles' analysis of the Kirshner et al. (1978) redshift sample (Peebles, 1979) and the formal error may be an underestimate because of sampling problems. In a revised analysis of the Lick catalogue, Groth and Peebles (1977) report the existence of a sharp break from the power law (2.27) at a value of approximately
Thus there appears to be negligible clustering on scales > rbreak. This result is rather tentative as the analysis may be subject to systematic errors, though a similar feature has been found from deeper galaxy samples (Shanks et al., 1980). Gott and Turner (1979) have extended the correlation function analysis for the Zwicky catalogue down to small separations using the accurate positions of binary pairs in Turner's (1976) catalogue. They find the new data to be consistent with the extrapolation of Eq. (2.27) to very small scales, r 5h-1 kpc.
In addition to these results, Peebles and Groth (1975) have discovered that the three-point correlation function has the following simple analytic form,
Equation (2.28) is found to be a good representation of the data over the range of scales 0.1h-1 Mpc r 2h-1 Mpc. Fry and Peebles (1978) have estimated the four-point function which is consistent with the generalization of (2.28) to a sum of triple products of . Several other results have been obtained by Peebles and coworkers, such as cross-correlations between galaxies and other extragalactic objects but these will not be discussed here. One result worth mentioning, however, concerns the two-point correlation function of rich clusters of galaxies cc. Hauser and Peebles (1973) find that rich clusters are more strongly correlated than galaxies, with cc 10 (r). (5) Thus rich clusters and galaxies cannot both be good tracers of the mass distribution.
The relations between the two-point and higher order correlation functions suggest that galaxies are grouped in a nested clustering hierarchy. If we look at the clustering pattern with a resolution r, typical clumps of galaxies will have an overdensity with respect to the background / ~ (r). The mean number of neighbours per unit volume at a distance r from a randomly chosen galaxy will scale as n(r) , thus the model predicts that the three-point correlation function should scale according to Eq. (2.28) with 2 (Peebles and Groth, 1975).
Clumps with a high overdensity should be bound and stable in which case one can apply the virial theorem. The mass of a typical clump is M ~ r3, hence the internal velocities should scale with clump size as
A more rigorous version of this argument has been given by Peebles (1976a, b). The one-dimensional mean square relative peculiar velocity <v212> between pairs of separation r12 is related to the sum of the accelerations over triplets of galaxies,
if the velocity distribution is isotropic and the clustering is bound and stable. Using Eqs. (2.28) and (2.27), Eq. (2.30) may be written
This relation is sometimes called the "cosmic" virial theorem and agrees with the simple application of the virial theorem which led to Eq. (2.29). Thus a measure of the relative peculiar velocities between galaxy pairs provides an estimate of the cosmological density parameter provided that the galaxy correlation functions accurately measure the mass distribution. The scaling of <v212> with pair separation provides a check of this point, for Eq. (2.32) predicts that <v212> should be very nearly independent of scale. Fall (1975) has derived a relation between the mean-square peculiar velocities of single galaxies and the two-point correlation function which, in principle, can be used to estimate . Unfortunately, Fall's method is sensitive to the shape of (r) at large separations.
Several applications of the cosmic virial theorem may be found in the literature (Peebles, 1976b; Davis, Geller and Huchra, 1978; Peebles, 1979, 1980b). The most extensive results come from the Center for Astrophysics redshift survey (Davis and Peebles, 1983) and the deep redshift survey of Bean et al. 1983. These surveys suggest
with a weak dependence on pair separation consistent with Eq. (2.32). Applying the cosmic virial theorem using Q = 1.3 (Eq. 2.28) gives,
Thus, the relative peculiar velocities suggest a low mean matter density. Comparing with Eq. (2.24) we find a mean mass-to-light ratio of ~ 50h(M/L) which is similar to the results obtained from application of the virial theorem to groups but somewhat less than the results from the cores of rich clusters (e.g. Rood et al. 1972).
The results do not necessarily exclude a high density universe for it is possible that most of the dark material is in a broadly distributed component that is not highly clustered on small scales. The present redshift samples are too small to check this point.
5 See the recent extensive discussion of this point by Bond and Szalay (1983). Back.