In cosmology, the roles that axions play are pretty complicated, but I’m going to try to do my best to explain them.
80% of the matter in the universe is made up of “dark matter”, and we don’t really have any idea what it is. A lot of cosmologists think that, along with dark energy, dark matter may be left over from something that happened during the big bang. Cosmologists have proposed that dark matter could be composed of the LSP (the lightest supersymmetric particle), gravitinos (the supersymmetric partner of the graviton, which is the “gravity” particle that we haven’t found yet), or, interestingly enough, axions!
The reason why axions make a great dark matter candidate is because, as I briefly mentioned earlier, they basically don’t interact with any of the other particles in the standard model with the exception of photons. If axions were the constituents of dark matter, the question “why doesn’t dark matter interact with the rest of the matter we know of?” would be solved.
There are two types of dark matter – “hot” dark matter, and “cold” dark matter, and axions could possibly constitute both. To understand why, let’s take a look at a special theory in cosmology about axions (there are, of course, other theories about axions in cosmology, but this one is my favorite).
In cosmology, there are these hypothesized things called “cosmic strings”. Contrary to their name, these aren’t the same sorts of strings that are studied in string theory – rather, they are a certain kind of “topological defect” that physicist Tom Kibble hypothesized played a role in the evolution of the universe, and they form when a symmetry is broken.
Let’s suppose that these cosmic strings exist, as do axions, and look at the early universe. While the universe is evolving and the temperature of the universe is changing, different symmetries are being broken. At a certain temperature, we have the global U(1) symmetry that Peccei and Quinn proposed, along with its (at first) massless axions. Not long after, this symmetry is broken, and axions acquire mass while these massive cosmic strings are formed.
As the universe is expanding, these cosmic strings begin to radiate axions. The axions emitted by these strings could be either relativistic or non-relativistic. If they are relativistic, then they make up what is called “hot” dark matter, and would be very fast and energetic. If, on the other hand, the axions are non-relativistic, they could make up “cold” dark matter, because they would be very slow.
There is a problem with the axions being relativistic, however, because axions that were relativistic would be waaay too energetic to make dark energy: due to the way that they decay to photons, they would all decay quickly and there would be no dark matter in the universe today!
If the axions were non-relativistic, then they would be radiated by these big cosmic strings as the strings begin to loop around one another and form “domain walls”, which are two-dimensional topological defects (strings are one-dimensional topological defects). At this point, all of the axions would fall to the minimum of their potential where they remain now, and form clusters throughout the universe, becoming the dark matter we are trying to understand today.