5.3. The Large-Scale Distribution of Clusters and the Mean Density of Matter in the Universe
Hubble's (1934) study of the distribution of galaxies revealed by 60- and 100-inch telescope photographs showed that to the accuracy of the observations the galaxies, except for a clustering tendency, appeared to be distributed uniformly in depth in all directions. The distribution of the rich clusters has similarly been investigated by the writer (Abell 1958). The homogeneous sample selected from the Abell catalog yields the frequency distribution N(z), of clusters with (estimated) redshift, z. There is, as we have seen, a small-scale clustering tendency of the clusters themselves, and N(z) cannot be determined (from these data) accurately enough to choose between different cosmological models. There is, however, no evidence for departure from uniformity over regions of space that are large compared with the scale of the second-order clustering. Neither the Hubble nor the Abell data are accurate or complete enough to ensure that the cosmological principle is precisely realized throughout the observable region of space, but both are at least compatible with the assumption of large-scale homogeneity.
An important cosmological datum is the mean density of matter in space. The available velocity-dispersion data permit us to make meaningful estimates of the possible range of the density of the Universe - estimates which include any unseen matter within the clusters that can produce dynamical effects.
The mean density of a rich cluster gives us an upper limit. If the Corona Borealis cluster (number 2065) has a mass of 3 × 1015 (probably an upper limit), it has a mean density of the order of 10-27 g cm-3. Because cluster 2065 is an unusually rich and compact one, typical rich clusters probably have mean densities an order of magnitude lower. A more realistic upper limit is the mean density of a typical second-order cluster. For a mass of 1016 and a radius of 40 Mpc, such a system would have a mean density of the order 10-30 g cm-3.
A lower limit to the density of the Universe is found by assuming that all of its mass is contained in clusters as rich as those in the writer's catalog. After correcting for the fact that the statistical sample does not cover the entire sky, we estimate that about 4000 such clusters probably exist within a distance of 1.2 × 109 pc (corresponding to the distance of distance class 6 clusters). If 1014 is a lower limit to the mass of a rich cluster, there are at least 8 × 1050 grams of matter within a distance of 1.2 × 109 pc, corresponding to a mean density of about 4 × 10-33. This estimate is probably too low for the mean density of the Universe by at least an order of magnitude.
Thus, we estimate that the mean density of that matter in the Universe whose gravitational influence produces observable kinematical effects lies in the range of 10-32 to 10-30 g cm-3. The best guess, to order of magnitude, is 10-31 g cm-3. A Hubble constant, H = 50 km s-1 Mpc-1, is assumed for these estimates; of course, the density estimate is proportional to H2. The corresponding value of the deceleration parameter, q0 (which is independent of H), is 10-2.
Note added 1974 January 18. - The manuscript for this chapter was submitted early in 1966. Because of unavoidable delays in the publication of this volume, the chapter has become outdated in many respects. When the author received the manuscript with copy editing for final printing, he attempted to incorporate some recent references. Time did not permit a complete revision, however, and much of the chapter still has the flavor of an 8-year-old review. In particular, Sections 3.6 and 4.3 should be read with cognizance that much recent work is not reflected therein.