Your idea of a donut shaped universe is intriguing, Homer. I may have to steal it.-Stephen Hawking, The Simpsons.
The mathematical structure resembling a donut is called a torus. It's the shape that is generated if you take a rectangle, attach two opposite sides together so that it forms a cylinder, and than attaching the two circular ends of the cylinder together.
A torus is the geometry of the game Asteroids and many others, where going off the screen on one side makes you appear on the others. Solving physics problems on a toroidal geometry can be pretty useful, because you don't have to deal with boundary conditions (Onsager's solution to the square-lattice Ising model is such an example). Conceptually, this cosmic looparound is comforting, avoiding both the edges of the universe and the fact that there are none.
|It looks like a rectangle but it's a donut.|
Our universe apparently has three spatial dimensions, so if it were donut shaped it would have to be a 3-torus, which is beyond my ability to visualize as its enclosed volume is some kind of hyperdonut in four dimensional space. But what the physical implications of the universe being a 3-donut, and can we look for them?
Even before getting into cosmology, one might want to consider special relativity in a torus. Because two inertial observes can move with respect to one another and cross each others' path multiple times, the so-called twin paradox cannot be resolved by requiring that one of the twins has to change directions in order to compare the elapsed time. The resolution is that in a toroidal spacetime, there is a preferred reference frame, which is the one that makes a given side of the torus appear shortest. So, special relativity in the land of donuts is not the same as the version we are familiar with. This challenges our Copernican sensibilities, because there will be some place and frame from which the universe appears smallest.
|Homer Simpson, you are accused of breaking half the Lorentz symmetry of the planet of the donuts.|
Considering a cosmological torus, light that is emitted and travels far enough would reach the point where it was emitted, so if we looked far enough we might see another Earth. However, we live in a universe that used to be a hot opaque plasma, so light from that epoch reaches us in the form of the cosmic microwave background, which is seen in all directions, and we can't see beyond that. So even if we can't see another Milky Way, we could detect the donutness of the universe. Consider Earth in a toroidal universe. Light is emitted from some distant point in all directions. Instead of only one of those light rays reaching Earth, many of them take a different path, each arriving at Earth in a different direction. When we look up at the sky, we would see multiple versions of the same image, in a circle whose angular size depended on the relative size of the torus and distance to the source. This took me a while to figure out, so I drew a crappy MS Paint drawing to illustrate it.
|Yellow, red, and green all go from the star thingy to Earth in different directions, arriving at different angles in the sky.|
As was the case in the photon decay paper, the cosmic microwave background is the most distant light source we have, so this has the best chance of being duplicated by the topology of the universe. To see if this is the case we can look at measurements of the temperature anisotropy of the universe, such as those taken by the WMAP satellite. That's what these guys did, and by their non-observation of obvious cosmic circles (the analysis was considerably more detailed), they placed the bounds of the size of any potential torus at about 78 billion lightyears. I am not sure whether this should be compared with the radius (46 billion lightyears) or diameter (92 billion) of the observable universe. There are a number of independent analyses of this, and to my surprise they do not really rule out the donut universe, although they do not support it either.
|If the size of the universe exceeds the size of the torus, the "intersection" will appear at multiple points on the sky.|
So, the jury is still out on the Simpson-Hawking donut universe. Did Hawking himself every discuss this? In a 1992 paper on chronology projection, he did brush over it slightly:
"For example, if the initial surface is a three-torus, the Cauchy horizon will also be a three-torus, and the generators can be nonrational curves that do not close up on themselves. However, this kind of behavior is unstable."
|However, this kind of behavior is unstable.|