2.3.1 Statistics
Statistics appeared in cosmology before the nature of the particles to
be used was established. In the first paper
(Sect. 3.1.1) which
describes the universe in terms of the general theory of relativity
(Einstein 1917)
it is assumed that the particles filling the universe
show Boltzmann distribution. Einstein's particles are stars.
Charlier (1922)
calculated the number of collisions between galaxies
from Maxwell's equation in his hierarchical universe which is based on
Lambert's (1761)
principles but starts with galaxies instead of
planetary (or even Jovian) systems.
The universe described by
Milne (1933)
starts from statistics.
``We. . . .impose the condition that this spatio-velocity distribution
formula shall actually represent, statistically, a concourse of
material objects; this is done by a modification of Boltzmann's gas
equation.
. . .since we expect the clots or sub-systems to be formed in the
spatial positions of the singularities in the original swarm, we were
let to infer an exact correlation between position and velocity. . .;
this being contrasted with the merely statistical correlation. . .It is
clear that once we introduce an exact correlation between velocity and
position, our system is not a statistical one but a hydrodynamical
one.''
Milne concludes:
``Relativity - statistics - hydrodynamics - dynamics -
gravitation - such was our course of investigation.''
Several points seem noteworhty concerning the earliest applications of
statistics to cosmology:
- they include implicitly or explicitly both the distributions of
positions and motions
- no assumption about the distributions are made by Milne; the
assumptions made by the other authors are simple statistical or
hierarchical distributions.
Another statistic and probabilistic aspect entered extragalactic
astronomy with
Wirtz (1918)
in his search for parameter correlation.
Covariance functions, now generally called correlation functions,
became the standard tools.
Positional distributions became of interest as a consequence of the
increasing body of data. The early application of probabilistic
methods is well exemplified by the work of Zwicky
(1937c,
Sect. 2.2):
``By a bold extrapolation of well-known results of ordinary statistical
mechanics we adopt the following working hypothesis. . .:
1. The system of extragalactic nebulae throughout the known parts of
the universe forms a statistically stationary system.
2. Every constellation of nebulae is to be endowed with a probability
weight f(
) which is
a function of the total energy of this
constellation. Quantitatively the probability P of the occurrence of a
certain configuration of nebulae is assumed to be of the type
Here V is the volume occupied by the configuration or cluster
considered, V0 is the volume to be allotted, on the
average, to any
individual nebula in the known parts of the universe, and is the
total energy of the cluster in question, while 
will
probably be
found to be proportional to the average kinetic energy of individual
nebulae. The function A (V / V0) can be determined a
priori. On the other
hand, F( / ) presumably
will be found to be a monotonously decreasing
function in / , analogous
in type to a Boltzmann factor
Assuming the principles stated in the preceding to be correct, we may
draw the following hypothetical conclusions:
a) The clustering of nebulae is favoured by high values of f and is
partially checked by low values of the a priori probability a.
b) If, as would appear to be certain, nebulae are not all of the same
mass, nebulae of high mass are favored in the process of clustering,
since they contribute most to produce high values of the weight
function f.
c) As a consequence of b, we should expect that the frequency with
which different types of nebulae occur will not be the same among
field nebulae and among cluster nebulae. In other words, clustering is
a process which tends to segregate certain types of nebulae from the
remaining types. This may contribute toward the correct interpretation
of the well-known fact that cluster nebulae are preponderantly of the
globular and elliptical types, whereas field nebulae are mostly
spirals. . . It is not necessary as yet to call on evolutionary
processes to explain why the representation of nebular types in
clusters differs from that in the general field. . .
d) If cluster nebulae, on the average, are really more massive than
field nebulae, the conclusion suggests itself that globular nebulae
may, somewhat unexpectedly, be
among the most massive systems. It will be of great interest to check
this inference by a search for gravitational lens effects among
globular nebulae.''
Compare also the work of Holmberg in
Sect. 2.3.2.
Another step in analyzing galaxy distributions is the adaption of the
correlation function from parameter space to real space
through the
definition of n-point correlation functions for two and three
dimensions: the angular and the spatial correlation functions.
The concept of clustering (Sect. 2.1)
basic to some of the formal
statistical approaches used since the 1950s is presented by
Carpenter (1938):
``It is concluded that the density restrictions and the mass
restrictions in metagalactic space are real and represent a
fundamental property of the distribution of matter in space. It
follows as a corollary and from the interpretation of
[Fig. 18] that
there is no basic and essential distinction between the large, rich
clusters and the small, loose groups. Rather, the objects commonly
recognized as physical clusterings are merely
the extremes of a nonuniform though not random distribution which is
limited by density as well as by population. From this point of view,
the term `supergalaxy' is of questionable propriety, since it implies
a distinctive and coherent organic structure inherently of a higher
order than the individual galaxies themselves.''
|
Figure 18. Density-diameter relation of
clusters of galaxies
(Carpenter 1938).
``The densities and diameters of 42 clusters of galaxies. The abscissae
indicate the diameter,  , of
the clusters, expressed in megaparsecs.
The ordinates indicate the logarithm of the mean density, p, of each
cluster expressed as the number of galaxies per cubic megaparsec. The
source of the data are distinguished as follows: Harvard, open
circles; Mount Wilson complete counts, disks; Mount Wilson incomplete
counts, barred disks; Lundmark cross; Carpenter, rectangle. The heavy
curve is the envelope pmax = 6040 -3/2. The four lighter
curves represent
clusters having memberships N = 1, 10, 100 and 1000 galaxies.''
|
Upon the publication of the first data from the Lick catalogue,
Neyman and Scott
(1952)
and Neyman et
al. (1953)
introduced the angular pair
correlation functions and Limber
(1953,
1954)
the angular and the
spatial pair correlation function and the integral equation relating
the two functions. This initiated the first period of probabilistic
studies of the distribution of galaxies. Extensive discussions were
given by
Neyman et at. (1956).
Results of the first applications to the Lick Survey by Neyman et at.
and Limber are displayed in Figs. 19 and
20.
The full bloom of statistical studies of galaxy clustering using
correlation functions began to develop almost two decades after the
initial phase. Essential steps leading to it were the publication of
the complete Lick catalogue in 1967, its new reduction in 1977
(Seldner et al.),
the impact of Peebles' books
(1971,
1980)
and the availability of powerful computing devices.
The importance of Limber's equation was realized when more galaxy
redshifts became available.
Totsuji and Kihara
(1969)
published an
approximate solution for small angles, and the equation was extended
to the general relativistic case by
Groth and Peebles
(1977).
More detail is given in an extensive review article covering the 1970s
(Fall 1979).
Further statistical techniques are mentioned by
Ott (1988).
Fractals
are a new approach to test hierarchical clustering. It was introduced
by Mandlebrot and first applied by him to astronomical problems in
1975.
M, and 
-
F