Planck-scale corrections to axion models

It has been argued that quantum gravitational effects will violate all nonlocal symmetries. Peccei-Quinn symmetries must therefore be an "accidental" or automatic consequence of local gauge symmetry. Moreover, higher-dimensional operators suppressed by powers of $M_{\mathrm{Pl}}$ are expected to explicitly violate the Peccei-Quinn symmetry. Unless these operators are of dimension $d\ge 10$, axion models do not solve the strong $\mathrm{CP}$ problem in a natural fashion. A small gravitationally induced contribution to the axion mass has little if any effect on the density of relic axions. If $d=10,11,or 12$ these operators can solve the axion domain-wall problem, and we describe a simple class of Kim-Shifman-Vainshtein-Zakharov axion models where this occurs. We also study the astrophysics and cosmology of "heavy axions" in models where $5\le d\le 10$.