Wednesday, March 30, 2005

Topological Defects

Today's cosmology/relativity/particle physics seminar at Syracuse was given by Brandon Carter from the Observatoire de Paris-Meudon. Brandon spoke about the stability properties of the "solid dark matter" model of the accelerating universe, proposed by Bucher and Spergel. The main idea of Bucher and Spergel is that, under certain circumstances, topological defects may lead to an accelerated expansion. I'd like to post about this, and may do so tomorrow. However, as a prelude, I thought I'd better provide a version of what topological defects are.

Topological defects are extended solutions to field theories that can arise when the vacuum structure of the field theory is topologically nontrivial. As a somewhat simple example (and, I admit, a clumsy one, but its the best I can do right now), let us model a field theory by standing many pencils on their ends on a table top, and connecting the pencils to their nearest neighbor pencils by springs. What is the vacuum configuration of these pencils? Obviously, because of gravity, each individual pencil would like to lie down on the table, but doesn't care which direction it is pointing as long as it is lying down. So the vacuum configuration of the theory is all the pencils lying down, facing in the same direction, because if any pencil faces in a direction different from that of its neighbor, then there will be energy bound up in the spring which is stretched between them, which can be reduced by the two pencils aligning. Obviously, there are an infinite number of equivalent vacua, corresponding to all the pencils aligning in any of the possible directions in the plane of the table.

Now suppose that the table is very big (maybe even infinite), and pencils that are very far apart from each other fall down into different vacuum states, because causality doesn't permit information to travel between them so that they can align. You could imagine, in fact, that pencils that trace out a very large circle all fall down pointing outwards in different directions along that circle. All other pencils inside that circle will try to align with the pencil closest to them, in order to reduce the energy in the springs. However, if you think through this for a moment, you'll see that there will always be one pencil, the one at the center, which is equally pulled in all directions and so will remain standing up - now stably. In fact, a few pencils on each side of this one will be partially standing up, because of the spring tension.

These few pencils, and particularly the one at the center, represent what is meant by a topological defect. It is a small region of space, in which the field configuration is out of the vacuum manifold, but which remains metastably in that configuration because the topological properties of the vacua chosen by the field at infinity are nontrivial. I won't harp on about how these topological properties are defined, because it's more technical than the sketch above and won't buy us much clarity here.

In the model system above, the topological defect is point-like - it is just a single point in space. In three spatial dimensions one can have either point-like defects (monopoles), line-like defects (strings) or membrane-like defects (domain walls). Which, if any, of these exist depends on the particular particle physics model one considers. However, there are some generic situations that are important. The most well-known occurs if one has a grand unified theory, which unifies the strong, weak and electromagnetic forces so that they are described by one simple gauge group. In this case, when the group breaks to the standard model of particle physics, as it must, monopoles are inevitably formed. These can be problematic in the context of cosmology, and were one of the original motivations behind the development of the inflationary universe.
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