The work that I’ve been doing with my student is featured on the cover of this week’s New Scientist. Unfortunately, a subscription is necessary to read the full article online, but if you do manage to find it on the web or the newsstand, you’ll find a much better explanation of the physics than I can manage here, as well as my *koan*-like utterances such as “if you look over here, you’re also looking over there”. There are more illuminating quotes from my friends and colleagues Glenn Starkman, Janna Levin and Dick Bond (all of whom I worked with at CITA in the 90s, coincidentally).

We’re exploring the overall *topology* of space, separate from its *geometry*. Geometry is described by the local curvature of space: what happens to straight lines like rays of light — do parallel rays intersect, do triangles have 180 degrees? But topology describes the way different parts of that geometry are connected to one another. Could I keep going in one direction and end up back where I started — even if space is flat, or much sooner than I would have thought by calculating the circumference of a sphere? The only way this can happen is if space has four-or-more-dimensional “handles” or “holes” (like a coffee mug or a donut). We can only picture this sort of topology by actually curving those surfaces, but mathematically we can describe topology and geometry completely independently, and there’s no reason to assume that the Universe shouldn’t allow both of them to be complicated and interesting. My student, Anastasia Niarchou, and I have made predictions about the patterns that might show up in the Cosmic Microwave Background in these weird “multi-connected” universes. This figure shows the kinds of patterns that you might see in the sky:

The first four are examples of these multi-connected universes, the final one is the standard, simply-connected case. We’ve then carefully compared these predictions with data from the WMAP satellite, using the Bayesian methodology that I never shut up about. Unfortunately, we have determined that the Universe doesn’t have one of a small set of particularly interesting topologies — but there are still plenty more to explore.

** Update:** From the comment below, it seems I wasn’t clear about what I meant by asking if I could “keep going in one direction and end up back where I started”. In a so-called “closed” universe (with

*k*=-1, as noted in the comment) shaped like a sphere sitting in four dimensions, one can indeed go straight on and end up back where you started. This sort of Universe is, however, still simply-connected, and wasn’t what I was talking about. Even in a Universe that is locally curved like a sphere, it’s possible to have multiply-connected topology, so that you end up back again much sooner, or from a different direction, than you would have thought (from measuring the apparent circumference of the sphere). You can picture this in a three-dimensional cartoon by picturing a globe and trying to “tile” it with identical curved pieces. Except for making them all long and then (like peeling an orange along lines of longitude), this is actually a hard problem, and indeed it can only be done in a small number of ways. Each of those ways corresponds to the

*whole*universe: when you leave one edge of the tile, you re-enter another one. In our three-dimensional space, this corresponds to leaving one face of a polyhedron and re-entering somewhere else. Very hard to picture, even for those of us who play with it every day. I fear this discussion may have confused the issue even further. If so, go read the article in New Scientist!

## 2 responses to “The Shape of the Universe”

The only way this can happen is if space has four-or-more-dimensional “handles” or “holes” (like a coffee mug or a donut).hmmmmm… You can derive an expanding finite accelerating spherical k=-1 model from the physics that I previously mentioned:

http://www.lns.cornell.edu/spr/2006-02/msg0073320.html

Correct me if I am wrong, but in this case, the field equations of general relativity can be satisfied with vacuum energy density proportional to the cosmological constant.

Sorry, it was,

“the only way this can happen”…that set me off, and I’m probably the only person in the world that would have even considered my example.