Some of my recent work has focused on foliations of spacetime – which are essentially 1- or 2-dimensional parametrizations of 3- or 4-manifolds. I am mostly interested in them because my work often requires 1- and 2-dimensional embedded spaces, and you can use these initial spaces to construct entire foliations of the manifold. If you have a 2-dimensional space (a surface), you can always smooth it out by squishing all the curvature to a single point, called a conical point (you can try this – next time you get an ice cream cone, take the paper off the cone and try to flatten it. You will be able to do it everywhere except for one point, at the tip of the cone). Mathematically, a surface with a conical point looks exactly like a surface intersecting a cosmic string, so this is a long winded way of explaining how my interest in foliations has lead to an interest in cosmic strings!
The latest part of this story is using some of the interesting properties of cosmic strings, I’ve been able to use the lack of observational evidence for them to constrain the nature of the global topology of the universe. This is a pretty interesting idea because there are very few ways that we can study the overall shape of the universe (shape here means topology, so does the surface looks like a plane, a sphere, a donut, or what?). There are lots of ways we can study the local details of the universe (the geometry), because we can look for the gravitational effects from massive objects like stars, galaxies, black holes, etc. However, we have essentially no access to the topological structure, because gravity is actually only a local theory, not a global theory (I could write forever about this, but I’ll just leave it for a later post maybe…). We have zero theoretical understanding of the global topology, and our only observational understanding comes from studying patterns in the CMB. The trick with cosmic strings is that they actually serve to connect the local gravitational field (the geometry) to the global structure of the universe (the topology).
The game is this – take a spacetime with cosmic strings running around everywhere, and take a flat surface which intersects some of them. This surface can always be taken as flat, so the intersections are conical points. If you measure an angular coordinate around each point, you won’t get , you’ll get something a bit smaller or a bit larger since the surfaces are twisted up around the points. It turns out that if you add up all the twists, you had better get an integer – the genus of the surface. The genus is essentially the number of holes in the surface. A sphere has , a torus , two tori attached to each other have , and so on.
Now we consider what kind of observational evidence there is for cosmic strings. The short answer, none! People have been looking for them in the CMB, but so far they’ve only been able to say “if cosmic strings exist, they must be in such-and-such numbers and have energies of such-and-such.” If we use these limits, we find that to a very good approximation, if cosmic strings exist, a surface passing through them must have genus 1, and therefore be a torus (the surface of a doughnut)!
Ok big deal – but here’s where the foliations come in. For example, if we parametrize our spatial (3-dimensional) manifold with tori, the result is a 3-torus. So this actually implies that space is not a sphere, but is a solid torus (like a doughnut). The mathematics behind this statement are actually quite profound, and were worked out in the early days of foliation theory by the likes of Reeb, Thurston, and Novikov. But the idea is that such foliations of 3-manifolds are very stable, and a single closed surface greatly restricts the kinds of foliations allowed for the manifold as a whole.
The archive paper where I discuss this in more detail can be found here. This idea that space is not a sphere is not new, and there is actually some evidence for it in the CMB, in the form of a repeating pattern (or a preferred direction) in space. But my primary interest is pointing out that this is an independent way of measuring the topology of the universe, since it’s based on local observations of strings in the CMB rather than overall patterns. If strings don’t actually exist, it can still be used to study the presence of the conical singularities, but I expect the restrictions on the topology are much less strict. Perhaps I’ll look into that further into that, but for the moment I’m happy with this. It’s a new way to determine information about the global topology of the universe, and it’s a great combination of pure mathematics, theoretical physics, and observational cosmology.