...

I just heard a report on an NPR station -- it was probably BBC news -- in which freelance mathematician Jeff Weeks (who wrote the book "The Shape of Space") was being interviewed after he had announced that the spatial universe is finite. There was not enough time allotted for him to give even a brief layperson's sketch of his reasoning.

But it seemed to have something to do with the fluctuations in the background radiation from the Big Bang having bounded wavelengths.

Jeff Weeks has been talking about this "shape of space" stuff for quite a while. The original idea was to use the data from the WMAP -- a very high-resolution picture of the cosmic background radiation -- to search for "circles in the sky", a potentially visible trace of the self-intersections of the sphere bounding the visible universe. This sphere is centered at the earth, and its diameter is the distance light has travelled since "decoupling", the point in the cooling of the early universe when light started travelling at the speed of, er, light. If the universe is multiply connected and small enough that the sphere self-intersects, we should be able to notice that we're seeing the circle of self-intersection looking in two different directions. When the data from the WMAP came in, the initial analysis didn't find any circles :-(. But it did notice something very odd, roughly that the autocorrelation function fell off for large angles (over around 60 degrees). Funster Thomas Colthurst probably remembers where to find the initial papers on this; I'm sure they're on the physics arXiv. The dipole can't be measured directly due to interference, but the quadropole is much smaller than it ought to be. Today's story, which made the cover of Nature, is that Weeks and his colleagues have observed that the observed harmonics of the Poincare Dodecahedron are a really good match for this initial data. Shh, don't tell anyone: I put a copy of the letter to Nature at http://people.brandeis.edu/~kleber/nature.pdf But the circles-in-the-sky people have already rebutted this claim, saying they're really sure the circles that the dodecahedron would leave are simply not there. Boy, I'd be delighted if the universe were a Poicare Dodecahedron! (Though I'd been hoping for a nonorientable space: if Weeks is right, then you could go in a straight line for 74 billion light-years and get back to where you started, just rotated by 2pi/5. I'd been hoping you'd get back to where you started mirror imaged.) --Michael Kleber kleber@brandeis.edu

...

But the circles-in-the-sky people have already rebutted this claim, saying they're really sure the circles that the dodecahedron would leave are simply not there.

Where can I read about this rebuttal?

I've only seen this in news reports; I don't think there is a primary source yet. Examples are http://sciencenow.sciencemag.org/cgi/content/full/2003/1008/3 http://www.nytimes.com/2003/10/09/science/09COSM.html?pagewanted=2&ei=5062&e... One of the reports (I don't recall which) said that Weeks et al have been in intense discussion with Cornish et al this week. Hey, it's an experimental science, what do you expect?...

The current evidence is all based on the absolute values of the various spherical harmonics; would there be any predictions about the actual vectors as well if the universe is actually a Poincare dodechedral space?

I do recall an article about some anisotropy in the WMAP autocorrelation data that indicated some preferred direction in the universe -- pointing towards Leo, I think. (Thomas?) But I have no idea how that would relate to the current (highly symmetric) manifold possibility. --Michael Kleber kleber@brandeis.edu

On Friday, October 10, 2003, at 09:00 AM, John Conway wrote:

Well, yes, I'd be delighted too. But what puzzles me about this idea is that that kind of universe, being a quotient of a sphere, has positive curvature, which contradicts the observation that the universe is very very flat. [...]

The WMAP initial data has given the best measurement in existence of just how flat the universe is. Omega_0, the density, is exactly 1 in a flat universe, >1 for spherical and <1 for hyperbolic. The initial best match WMAP data puts Omega_0 at 1.02 +- .02. Further data to appear within the next six months ought to narrow down the interval to about a tenth of that size, I think. The dodecahedral space hypothesis is based on matching the harmonics, and with the best match there, its positive curvature predicts an Omega_0 of 1.013. (So the upcoming improved Omega_0 data will either bolster or shoot down this idea.)

[...] So I think the idea is nonsense.

I've heard a cosmologist say, approximately, that anyone who thought anything about the universe, and hasn't updated that thinking in light of the WMAP data, should now be considered a historical relic. Confidence intervals for cosmological constants have changed from "within an order of magniture" to "+- 2%" in the past year. O brave new world... --Michael Kleber kleber@brandeis.edu

For the MAA, I've been putting various columns together. You can see the current list of them at http://www.maa.org/news/mathgames.html . My latest column is on Square Packings, with an emphasis on Mrs. Perkins's Quilt. The column is at http://www.maa.org/editorial/mathgames/mathgames_12_01_03.html . One of the fun parts of the article was discussing the "state of the art" with Richard Guy. He improved 4 solutions with a bit of hand analysis. I haven't seen http://www.squaring.net/ mentioned here, yet. It's an excellent resource. --Ed Pegg Jr, www.mathpuzzle.com

When this idea is applied to hyperbolic space, we get a finite universe with negative curvature. Such a possibility seems not to have been noticed by early cosmologists; I remember from a long time ago that the universe was supposed to be either finite and of positive curvature, or infinite and of zero or negative curvature.

Even once the idea did come up cosmologists have muddled the manifold part. I think there was a paper at some point called "Open Compact Models." --Michael Kleber kleber@brandeis.edu