We continue our series of excerpts (and discussion) from the outstanding survey paper by George F. R. Ellis, Issues in the Philosophy of Cosmology.
4.3.1 Small universes
A Small Universe: a universe which closes up on itself spatially for topological reasons, and does so on such a small scale that we have seen right round the universe since the time of decoupling. Then we can see all the matter that exists, with multiple images of many objects occurring. This possibility is observationally testable by examining source statistics, by observation of low power in the large angle CBR anisotropies, and by detecting identical temperature variation on various circles in the CBR sky. There are weak hints in the observed CBR anisotropies (the lack of power on large angular scales) that this could actually be the case, but this is not solidly confirmed. Checking if the universe is a small universe or not is an important task; the nature of our observational relationship to the universe is fundamentally different if it is true.
In 1900, Karl Schwarzschild (1873-1916) was perhaps the first to suggest the idea of a small universe topology that would lead to multiple images of the same object at different points in the past. Though most cosmologists favor the idea of a very large universe with a simple topology, the possibility of a more complex small universe topology is still not out of the question. The universe might be measurably finite in some or all directions.
The smaller a finite topological region of space, the easier it should be to discover multiple images of the same object at different ages (except for CMB features which will all be the same age). The distribution of distant sources might show “patterns” that are related to more nearby sources. A comprehensive survey of sources at redshifts between about z=2 to z=6 is still needed before any conclusions can be drawn.
Another approach, of course, is to look at patterns in the CMB temperature (intensity) and polarization. Analyses of the most recent release of Planck satellite data, however, shows no evidence of a compact topology smaller than our visual horizon.
References
Ellis, G. F. R. 2006, Issues in the Philosophy of Cosmology, Philosophy of Physics (Handbook of the Philosophy of Science), Ed. J. Butterfield and J. Earman (Elsevier, 2006), 1183-1285.
[http://arxiv.org/abs/astro-ph/0602280]
Luminet, J.-P. 2016, arXiv:1601.03884v2 [astro-ph.CO]