Dynamics of the Peccei Quinn Scale

Invoking the Peccei-Quinn (PQ) solution to the strong CP problem substitutes the puzzle of why ${\theta }_{qcd}$ is so small with the puzzle of why the PQ symmetry is of such high quality. Cosmological and astrophysical considerations raise further puzzles. This paper explores this issues in several contexts: string theory and field theory, and theories without and with low energy supersymmetry. Among the questions studied are whether requiring axion dark matter can account for the quality of the PQ symmetry, to which the answer is sometimes yes. In non-supersymmetric theories, we find $f_a={10}^{12}$ GeV is quite plausible. In gauge mediation, cosmological constraints on pseudomoduli place $f_a$ in this range, and require that the gravitino mass be of order an MeV.