The cold axion populations

We give a systematic discussion of the contributions to the cosmological energy density in axions from vacuum realignment, string decay and wall decay. We call these the cold axion populations because their kinetic energy per particle is at all times much less than the ambient temperature. In case there is no inflation after the Peccei-Quinn phase transition, the value of the axion mass for which axions contribute the critical energy density for closure is estimated to be of order $6 \cdot 10^{-6}$ eV, with large uncertainties. It is emphasized that there are two groups of cold axions differing in velocity dispersion by a factor of order $10^3$.