## Cosmic Topology April 16, 2009

Posted by keithkchan in Cosmology.

Thanks to my colleagues, in particular Mr David, who urged me to update this blog. So I decide not to watch Spiderman cartoon tonight, and write something on this blog. By the way, Spiderman is also a geek. Although he studied at an imaginary university (Empire State University) in Mahattan, we believe that the writer in fact imply NYU. Also he rests on the trademark of NYU, the characteristic NYU flag frequently. Of course, I like Spiderman before coming to NYU.

What do I want to talk about? Well let me write up the paper I presented in the journal club this week. The paper I chose is arXiv:astro-ph/0402324, whose title is Cosmic Topology: a Brief Overview. Well this sounds rather off the main stream of cosmology. Yes, that’s true. The main motivation for me to learn something about these stuffs is that I wonder if the current observed cosmic acceleration can arise from some nontrivial global topology of the universe. It turns out that this is rather unlikely (should I just say impossible?).

In standard cosmology, we usually assume that the spatial topology of the universe is either the Euclidean space $E^3$, 3-sphere ($S^3$), or a 3-hyperbolic space ($H^3$). However, as realized by Karl Scharzschild, in fact the spatial topology can be $M/\Gamma$, where $M$ is one of the homogeneous and isotropic space and $\Gamma$ is some symmetry transformation that leaves the metric invariant. For example, instead of the whole infinite Euclidean space, we may live in a cube with the opposite face identified, that is a 3-torus. The point is that general relativity is a local metrical theory, it does not allow us to determine the spatial topology. My goal to realize cosmic acceleration by nontrivial topology evaporates and my interest in this subject has dropped by a factor of 3. Incidentally, there are a few papers in the literature trying to argue that spatial topology is the cause of cosmic acceleration. I have read one, and my impression is that it is totally nonsense.

Then can we really tell the spatial topology of the universe by observations. Yes, it is possible. The most striking consequence of the nontrivial topology is that we can see multiple images of the same object in the sky. Unfortunately there are great difficulties in direct detection. The images of the object, such galaxies, are seen from different angles, at different stages of their life, and some of the images may be blocked by other objects. Thus it is almost impossible to tell by direct detection. Instead we can look for some signatures of nontrivial global topology in statistics. I am not going to the details of the statistics, but I just outline the main ideas. One of the way is to compute the distance between the galaxies in a catalogue, and plot the distribution as a function of their separations. Because the separation between the copies of the object is some characteristic length scale, it will show up as a sharp spike in the distribution. One can also look for identical circles in the CMB temperature map, which is so far the best way to probe nontrivial topology. It has ruled out the famous (or infamous) dodecahedral universe that some people proposed to explain the lower-than-expected power in the quadrupole and octopole in the CMB map. Incidentally many people made fun of this model.

Now I should point out that if the particle horizon is much smaller than some “periodic scale” of the universe, even if we live in a universe with nontrivial topology, we have no way to tell. So far, there is no observational evidence that the universe has nontrivial topology. So we can apply Occam’s razor to cut out the unnecessary complications and stick to the standard cosmology happily. On the other hand, we should be open-minded, exhausted all the possibilities, and don’t make fun of those people who have good reasons to do non-standard cosmology.