# How does Inflation solve the Magnetic Monopole problem?

Cosmological Inflation was proposed by Alan Guth to explain the flatness problem, the horizon problem and the magnetic monopole problem. I think I pretty much understand the first two, however I don't quite understand how a period of exponential expansion fully explains monopole problem.

From Weinburg's Cosmology, the issue is essentially that various grand unified theories predict that the standard models $SU(3)\times SU(2)\times U(1)$ arose from the breaking of an original simple symmetry group. For many of these theories, a crazy particle known as a "magnetic monopole" is created at a certain energy (sometimes quoted at around $M = 10^{16} GeV$). So my question is why does a period of rapid expansion somehow or other result in a low density of magnetic monopoles (assuming they exist/existed at all)?

I would think, like in nucleosynthesis, that the primary factor in monopole creation is energy density, and since inflation is still a "smooth" process, at some point the universe would hit the proper energy density to create magnetic monopoles. How does the rate of expansion at the time they were created effect overall present density?

At the time when the monopoles are created, they're created at a density of order 1 per Hubble volume -- that is, there's one in each "observable Universe" at that time. In general, when a symmetry breaks, topological defects form that are separated on a length scale of order (speed of propagation of the field)(time scale over which the symmetry breaks). The first is of order $c$, and the second is of order the Hubble time, so monopoles are separated by a distance of order the Hubble length.
You should take "of order" here very liberally -- I don't actually care if I'm off by factors of $10^5$ or $10^{10}$ or anything measly like that! After all, inflation blows up lengths by something like $10^{20}$ or more. So one monopole per horizon volume becomes one per $10^{60}$ horizon volumes. (Also, the horizon volume continues to change after inflation is over, but not by anything like this sort of factor.)