It seems clear that there is no 'repetition' as we look out into the cosmos, on a scale that is less than say a hundred million light years (i.e. the diameter of our local supercluster of galaxies). But on scales larger than that it is difficult to be so sure. One has to think rather carefully about what our image would actually look like. Not only would it be 'us' as seen from an unfamiliar point-of-view, but it would also be 'us' as we were several hundred million years ago. There would be no shortcut for the light coming to us from the distant image.
Is there any possibility of seeing two images of the same object which have the same 'date-stamp', i.e. such that the light which reaches us left both images at the same epoch? Yes, if the two images are equidistant from us. Imagine an object that is just half-way between me and my image. Then the object will have an image that is halfway between me and the second image of myself that I see by looking behind me - and the two images of my object will be equidistant from me.
Now, the cosmological (constant-curvature) PDS is constructed by identifying opposite faces of a non-Euclidean dodecahedron. Where am I in the dodecahedron? The at first sight surprising answer is that I might be anywhere - and it really doesn't matter. The space of the PDS is absolutely homogeneous; the curvature is the same everywhere, there are no special points. Admittedly space is not quite so isotropic - or 'the same in all directions' - as it was when space was topologically trivial. At any point there are now special directions: the six pairs of directions in which I see images of myself (or could do so, if I could see far enough).
The point is, that the edges and vertices of the dodecahedron are artifices of the construction by which we conceived of the PDS. Now we have got there, we can discard the original dodecahedron: it is no more than a map of the actual space. Other maps are possible, and there is nothing to stop me using a map which is centred on me.
To get from the space in which I find myself to the map, I need to measure out a regular dodecahedron, centred on myself, of such a size (and orientation) that any two opposite pentagonal faces are images of one another, in other words, identical.
Then, any object which happens to be located on one of the faces of this mapping-dodecahedron, will have a second image on the opposite face, and both images will be equidistant from me. I will get two different views of the same object, at the same moment of its existence.
Given that the scale of this dodecahedron is likely to be of the order of hundreds of millions if not billions of light-years, there is a danger that it will be too big for me to see any repeated images at all. This will be the case if a sphere drawn with me as centre, and radius the distance travelled by light since the origin of the universe (approximately 14 billion light years), is small enough to fit inside the dodecahedron without touching any of its faces.
There is also the possibility that this sphere is just about the same size as the dodecahedron, so that it intersects each of its twelve faces in a circle (see the picture below). This is the possibility that Roukema and his colleagues investigated. It is only objects lying within these 12 circles in the sky that can have double images (in opposite directions), and they must be far enough away that both the objects and their images lie within the visibility sphere (also called the cosmological horizon) - the sphere centred on me whose radius is the age of the universe times the speed of light. (The point is that if the object is too close to me, its image will be too far away to be seen by me.)
The visiblity sphere just beginning to burst through
the cosmic dodecahedron
In this scenario, only objects that are nearly as old as the universe are going to have double images. But there's not much we can see out there - with the notable exception of the primaeval plasma which emits the microwave background radiation, and at which (with the help of the WMAP sattellite) we have been gazing so intently during the last few years.
In other words, if the visibility sphere is only just big enough to intersect the dodecahedron, the only thing of which we are likely to be able to see a double image is the primaeval plasma. The glowing plasma is very nearly as old as the universe itself, so to find the double images we need to look in the direction of the 12 circles, above-mentioned, in which the visibility sphere intersects the dodecahedron. If Roukema's hypothesis is correct, when we look at one of these circles and its opposite number (which is also its topological image), we will be looking at the same bits of the primaeval plasma, and they should look the same so far as their tiny temperature variations are concerned.
Testing the theory is a question of looking at all possible opposing circles and testing them for correlations (not forgetting the 36° twist!). The unkowns are the orientation of the dodecahedron, and the size of the circles of intersection (which depends on the exact relationship between the size of the dodecahedron and the age of the universe).
In the latest (January 2008) paper by Roukema et al. (link) the configuration of the dodecahedron which yields the best correlations turns out to be such that the face-centres are at the following positions (specified by galactic longitude l and latitude b; I have also converted to Right Ascension and Declination - using Nasa'sCoordinate Converter - so that you know where to look in the sky; and have added the nearest constellation in each case):
Face 1 l=117° b=020° RA=22h17 Decl=81° Cepheus
Face 2 l=184° b=062° RA=10h47 Decl=37° Leo Minor
Face 3 l=046° b=049° RA=15h57 Decl=28° Corona B.
Face 4 l=060° b=-013° RA=20h30 Decl=17° Dolphin
Face 5 l=125° b=-042° RA=01h00 Decl=21° Andromeda
Face 6 l=176° b=-004° RA=05h20 Decl=30° Taurus
Note that the five faces 2..6 are arranged symmetrically around face 1. The remaining six face-positions are antipodal to the six given ones. (You may find it easier to think in terms of vertices of an icosahedron instead of face-centres of a dodecahedron. The angular distance between adjacent vertices is 63.43°, or tan-1(2).)
The circles that give the best correlation have an angular radius of 11°. Even if the data were entirely random, there would no doubt be a dodecahedron, and a circle radius that yielded a maximum for the correlation. The question is, whether the quality of the optimum correlation is such as to make one believe that there really is a 'signal' - indicating the PDS topology - in all those masses of data from WMAP, rather than just noise. Much of the latest paper from Roukema et al. is devoted to trying to convince us that there really is a signal. I haven't yet understood the argument. But to quote from the paper's conclusion:
Do we really live in a Poincaré Dodecahedral Space? Further constraints either for or against the model are certainly still needed, but the evidence in favour of a PDS-like signal in the WMAP data does seem to be cumulating.
'Bye for now!
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