On the origin of dark matter axions
  We discuss the possible sources of dark matter axions in the early universe. In the standard thermal scenario, an axion string network forms at the Peccei-Quinn phase transition $T\sim \fa$ and then radiatively decays into a cosmological background of axions; to be the dark matter, these axions must have a mass $\ma \sim 100 \mu eV$ with specified large uncertainties. An inflationary phase with a reheat temperature below the PQ-scale $T_{reh} \lapp \fa$ can also produce axion strings through quantum fluctuations, provided that the Hubble parameter during inflation is large $H_1 \gapp \fa$; this case again implies a dark matter axion mass $\ma \sim 100 \mu eV$. For a smaller Hubble parameter during inflation $H_1 \lapp \fa$, `anthropic tuning' allows dark matter axions to have any mass in a huge range below $\ma\lapp 1 meV$. 
E.P.S. Shellard and R.A. Battye