# 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?

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For all those that come across this in the future, I recently found this article: arxiv.org/pdf/gr-qc/0512056v1, which gives a nice description. – Benjamin Horowitz Oct 21 '11 at 4:37

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.)