*The Problem of Distribution in an Expanding
Universe*

Therefore, we must consider the alternative scale of
distance, and formulate the law of distribution on the
assumption that red-shifts are the familiar velocity-shifts, and do
measure the expansion of the universe.
The actual recession necessitates one correction to
apparent luminosities and another to the epoch of the
various surveys. The first correction has already been
discussed. Recession reduces apparent luminosities of
the nebulae by the `recession factors'
1 + *d* /
. When
these effects are removed from the measures, the nebulae appear
brighter, and the distances are less than those
estimated from the uncorrected measures. Now the
latter data, as we have just seen, indicate uniform
distribution. Consequently, the revised distances might
be expected to introduce departures from uniformity,
in the sense that the volumes increase less rapidly than
the numbers of nebulae, or, in other words, that the
distribution increases outwards leaving the observer in
an unwelcomed, favoured position. However, this conclusion does not
necessarily follow, because the surveys
represent different epochs in the history of the expanding universe.

The light which reaches us today left the limits of the various surveys far back in past time. From the limit of the deepest survey, for instance, the light started about 4.00 million years ago. It travelled for about 120 million years before it reached the limit of the next deepest-survey, and another 130 million years before crossing the limit of the shallowest survey. During these immense intervals of time the nebulae at the limits of the different surveys were receding at enormous velocities to still greater distances.

We count a certain number of nebulae and we know
that they were scattered through a certain volume of
space *when the light left the limit of the survey.* But today, many
millions of years later, these same nebulae
are scattered through a much larger volume of space,
and the increase is different for each survey. Evidently,
all the surveys must be reduced to the same epoch
before the law of distribution can be formulated. Moreover, the law will
continually change, for the recession
implies that the distribution thins out with time.

These considerations emphasize the complexity of
the problem of distribution in an expanding universe.
The first step in the solution is the choice of a common
epoch to which the different surveys will be reduced in
order to make the comparison of numbers of nebulae
and volumes of space. As a matter of convenience, the
epoch selected is *now,* the time at which the surveys
were made. Then, knowing the law of red-shifts or, in
other words, the law of expansion, it seems possible to expand the
volumes of all the surveys up to the epoch, now.

At this point the procedure becomes arbitrary. The calculations, in the present stage of knowledge, may be made in various ways, and the choice involves assumptions concerning the nature of the universe. As a simple illustration, does an individual nebula maintain a constant velocity as it recedes into the depths of space, or does its velocity steadily increase with increasing distance? This and other more technical questions must be answered before the reductions can be made with confidence. Thus the problem of reduction to a common epoch forces us to consider cosmological theory and some of the models of the universe which loom in that shadowy realm.

Most of the current models are derived from relativistic cosmology. Moreover, the outstanding exception, Professor Milne's kinematical model, is so outwardly similar, in several of its aspects, to a special case in the relativistic theory, that the observer, faced with a small sample, can scarcely hope to distinguish between them. Therefore, in the brief discussion which follows, the relativistic models alone will be considered. There are, it is said, many compelling reasons for concentrating on the theory, but the observer is not the proper authority to present them in their technical details. Instead, a few of the underlying principles will be mentioned together with the features of the models that may be compared with observations.