see also The Strong CP Problem
An obvious question about the axion hypothesis is how natural it really is. Why introduce a global PQ “symmetry” if it is not actually a symmetry? What is the sense in constraining a theory so that the classical Lagrangian possesses a certain symmetry if the symmetry is actually anomalous? It could be argued that the best evidence that PQ “symmetries” are natural comes from string theory, which produces them without any contrivance.
Axions In String Theory by Peter Svrcek, Edward Witten
The axion can play a cosmological role23 for larger values of fa. At high temperatures, the instanton effects which give rise to the axion mass go away. In the very early universe, then, the value of θ is undetermined, and presumably varies slowly from place to place. Inflation would pick out a particular value in our horizon volume. If subsequent reheating is to a temperature below fa, this value is frozen in place until much later, when the universe cools and the axion mass appears. The axion field then begins coherent oscillations around its minimum. If the initial value of θ is O(1), then the energy stored in these oscillations would overclose the universe if fa is larger than about 1012 GeV. If fa is near this value, this energy (which is in the form of axions with near zero momentum) could be the cold dark matter. Things are more complicated if there is no inflation, or reheating after inflation to a temperature above fa. In this case a network of cosmic-string defects forms in the axion field24, and the evolution of this network is a complicated numerical problem. The axions from string decay contribute at least as much dark matter as the axions from the initial misalignment, and possibly much more; see Ref.  for more details.
Axions: Past, Present, and Future by Mark Srednicki
For a great discussion of the history of axion models, see Axions: Past, Present, and Future by Mark Srednicki