Ian Agol's Research Blog

3/3/03
The Wilkinson Microwave Anisotropy Probe (WMAP) was sent up on June 30, 2001. Its results were finally released on February 11, 2003. I watched the announcement on NASA tv. WMAP's map of the cosmic microwave background is the most accurate to date, and confirms various theories about the origin of the universe. The simplest model for the universe is a Friedman universe, which is a 3-dimensional space-form which changes in size with time, according to Einstein's theory of general relativity. These are the simplest Einstein metrics, and since the universe is very homogeneous, seem to be a good model for our universe. The main question was which sort of space-form is our universe - positively curved, flat, or negatively curved? If it were positively curved, then if we measured the angles of a very large triangle, the sum would be greater than 180, if it is flat, the sum would be 180, and if negatively curved it would be < 180. Depending on the universe's curvature, it could have non-trivial topology as well. If we flew in certain directions, we could end up where we started, similar to circumnavigating the earth.  If so, then when we look out in space, we could see copies of our galaxy, but of course far in the past. Anyway, searches for similar patterns of distribution of galaxies don't seem to have worked. But if we consider another copy of ourselves, they would see the cosmic microwave background as a humongous sphere 13.3 billion light years away, but from a different vantage point from our view. If we are close enough to another copy of ourselves, then our two views of the cosmic microwave background would be spheres intersecting in a common circle, so we should expect the patterns of the CMB to agree on such circles. If we see enough of these, then the we can reconstruct the topology of the universe (see "Measuring the shape of the universe" by Cornish and Weeks and other papers by Weeks). This excited 3-manifold topogists for a bit, since if the universe was negatively curved, then it might be a hyperbolic manifold, and the field of 3-manifold topology might actually be useful! But measurements by BOOMERANG and measurements of supernovas seemed to indicate that the universe is almost flat. Since I had written a paper giving a lower bound on the volume of a closed hyperbolic 3-manifold, I was contacted by a Brazilian astrophysicist Marcelo Reboucas, who wrote a paper with collaborators using these measurements and estimates on hyperbolic manifolds by myself and Przeworski to determine what sort of constant negative curvature manifold could be observed given the constraints on its curvature. It is conjectured that the Weeks manifold is the smallest volume hyperbolic 3-manifold, for which there is much experimental evidence by the program snappea (and the fact that it is the smallest volume arithmetic 3-manifold). It would be quite embarrassing if cosmologists discovered that the universe is a hyperbolic 3-manifold of volume < .9427,  the volume of the Weeks manifold! Since they were using experimental evidence for the physics part of their paper, I'm not sure why they didn't want to use experimental evidence for the mathematical part? Anyway, it looks like the universe is likely to be flat. But it still could be a Euclidean manifold. At the WMAP news conference, they didn't mention anything about this, even though the chief theoretician, David Spergel, had written papers on measuring topology. So I wrote him an e-mail:

On Tue, 11 Feb 2003, Ian Agol wrote:

> Hi,
>       I just watched the announcement of
> the WMAP results on nasa TV, congratuations
> on your accomplishment!
>       I believe there was no mention of
> determining the topology of the universe.
> I was wondering whether your team has
> investigated this yet, or if this is still
> to come?
>
> thanks,
> Ian Agol

Still to come.  We have the paper mostly written and
should get it out within two weeks.

Reading between the lines, if they have already done the research, but didn't announce anything at the news conference, then I'm guessing they haven't discovered that the universe has non-trivial topology. But I'm still looking forward to what the paper has to say.