Imperfect Axion Precludes the Domain Wall Problem

The QCD axion needs not be an exact pseudoscalar for solving the strong $CP$ problem. Its imperfectness can play a profound role cosmologically. We propose effective operators, where the Peccei-Quinn field linearly couples to standard model particles, provide a dynamical solution to the domain wall problem that prevails in postinflationary axion models with discrete symmetry. Such interactions generate a thermal potential that drives the axion field to a universal value throughout the Universe at high temperatures thus preventing the birth of domain walls when the QCD potential switches on. We discuss generic conditions for this mechanism to work and several concrete examples. Combining with existing electric dipole moment and fifth force constraints, a lower bound on the axion mass is obtained around ${10}^{-5}\mathrm{eV}$. Our findings make a strong case for complementary axion searches with both quality preserving and violating interactions.