**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 × 10^{15}
_{}
(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 10^{16}
_{} 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 × 10^{9} pc (corresponding to the
distance of distance class 6 clusters). If 10^{14}
_{}
is a lower limit to the mass of a rich cluster, there are at least
8 × 10^{50} grams of matter within a
distance of 1.2 × 10^{9} 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 *H*^{2}. The corresponding value of the
deceleration
parameter, *q*_{0} (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.