Invoking the Peccei-Quinn (PQ) solution to the strong $CP$ problem substitutes the puzzle of why ${\theta }_{\mathrm{qcd}}$ 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 these 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 nonsupersymmetric theories, we find $f_a={10}^{12}\mathrm{GeV}$ 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.

• Received 17 August 2009

DOI:https://doi.org/10.1103/PhysRevD.80.125017