Gravitational Wave Imprints of a High-Quality Axion and the Origin of Flavor Hierarchies

Abstract: Axions, arising from an anomalous global Peccei-Quinn symmetry $U(1){\text{PQ}}$, offer a compelling solution to the strong CP problem but are vulnerable to Planck-suppressed operators. Gauged abelian flavor symmetries $U(1)_F$, invoked to explain the flavor hierarchies via the Froggatt-Nielsen mechanism, can naturally shield the axion from such effects, yielding an accidental high-quality flavored axion with unit domain wall number. Such constructions predict two complementary signatures: (i) flavor-changing neutral currents from $K\toπa$ decays, typically associated with high flavor scales $Λ{\text{FN}}\gtrsim f_a$, and (ii) stochastic Gravitational Waves (GWs) sourced by the evolution and decay of gauged flavonic and axionic cosmic-string networks. In addition, global axionic strings can efficiently radiate axions, potentially accounting for the observed dark matter relic abundance. We show that the resulting characteristic plateau--valley structure in the GW spectrum provides a distinctive and powerful probe of high-quality flavored axion dark matter models, complementary to low-energy flavor experiments.