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.