*The Permissible Type of an Expanding Universe*

These conclusions in themselves are important factors in the study of
expanding models of the universe,
but the implications of the empirical data can be pushed
still farther. The nature of the expansion and the
nature of the spatial curvature are now determined, but
there still remains a third arbitrary element in the
general theory, namely, the cosmological constant.
This constant occurs in formulae connecting both of
the other elements with the contents of the universe.
Now we know the nebulae well enough, but we do not
know what lies between the nebulae. Consequently,
we cannot directly determine the numerical value of
the cosmological constant. Nevertheless, with the help
of two presumably necessary assumptions, we can make
a reliable estimate of the order of the constant. The
assumptions are that neither the mean density in the
universe, nor the mean pressure, is less than zero -
that the universe is not less than empty, either of
matter or of radiation. Then, with the aid of the known
elements of expansion and curvature, we can set upper
and lower limits between which the constant must lie.
Fortunately, the limits are so close together that we
may state the approximate value of the constant with
confidence. It is about 4.5 × 10^{-18} (years)^{-2}.

The significant features are the facts that the constant is positive, and is slightly larger than a certain critical value - the value it would have in a particular, unstable model called the Einstein universe. These facts assign the actual world to a single class known as monotonic, or `ever-expanding', universes. The other elements are sufficient to identify, within the class, a unique model known as the ever-expanding universe of the first kind.

The model expands without reversal. The radius increases from zero to infinity. Past time is finite, future time is infinite. Comparatively recently, perhaps a thousand million years ago, the model started to expand from a small compact mass. The expansion was very rapid at first, but it has steadily slowed down to the rate we measure today. At each moment the model is homogeneous, the contents are uniformly distributed, but as time goes on the mean density diminishes, the-average distance between neighbouring nebulae increases. Eventually a state of complete isolation will be reached.

The disturbing features of this picture of the universe is the very small scale both in time and in space. With existing telescopes we explore at least a third of the `age' and about a quarter of the volume. And there is a third questionable feature. The general formulae indicate a definite relation between spatial curvature, the cosmological constant, and the contents of space. The universe suggested by the surveys is very massive. The mean density is of the order of a thousand times that which can be accounted for by the nebulae we observe. The excess matter, if it exists, must be scattered between the luminous nebulae.

We suppose that such internebular material would probably be in the form of dust or ionized gas. In that case the medium would absorb light, and would be readily detected. But we find no trace of space absorption; space is sensibly transparent. Therefore, the matter, if it exists, must be in some unlikely form such as chunks or non-ionized gas. Even then, the necessary quantity is barely permissible. For, on other evidence, we can set an upper limit to the possible density, regardless of the form of the material. We find no trace of internebular material, but our investigations have been pushed only to a certain limit. At the moment, that limit is just about the density corresponding to the curvature. Thus the theory might be valid provided the universe were packed with matter to the very threshold of perception. Nevertheless, the ever-expanding model of the first kind seems rather dubious. It cannot be ruled out by the observations, but it suggests a forced interpretation of the data.

The disturbing features are all introduced by the recession factors, by the assumption that red shifts are velocity-shifts. The departure from a linear law of red-shifts, the departure from uniform distribution, the curvature necessary to restore homogeneity, the excess material demanded by the curvature, each of these is merely the recession factor in another form. These elements identify a unique model among the array of possible expanding worlds, and, in this model, the restriction in the time-scale, the limitation of the spatial dimensions, the amount of unobserved material, is each equivalent to the recession factor.

On the other hand, if the recession factor is dropped, if red-shifts are not primarily velocity-shifts, the picture is simple and plausible. There is no evidence of expansion and no restriction of the time-scale, no trace of spatial curvature, and no limitation of spatial dimensions. Moreover, there is no problem of internebular material. The observable region is thoroughly homogeneous; it is too small a sample to indicate the nature of the universe at large. The universe might even be an expanding model, provided the rate of expansion, which pure theory does not specify, is inappreciable. For that matter, the universe might even be contracting.