Pseudo-Goldstone Dark Matter from Primordial Black Holes: Gravitational Wave Signatures and Implications for KM3-230213A Event at KM3NeT

In many well-motivated new physics models, the pseudo-Nambu-Goldstone boson (pNGB) from U(1) symmetry breaking emerges as a promising dark matter candidate. Its coupling, suppressed by the symmetry breaking scale, prevents thermal equilibrium in the early Universe for high scale symmetry breaking. Thus, pNGB dark matter is predominantly produced via non-thermal mechanisms, such as the freeze-in process through a new portal coupling. In this work, we explore a novel mechanism for the production of pNGB dark matter even with feeble Higgs portal coupling-arising from Hawking radiation or superradiance of primordial black holes. We systematically investigate the production of light and heavy pNGB dark matter, both for Schwarzschild and Kerr black holes. We also discuss its potential gravitational wave signatures from domain wall collapse, density perturbations, and Hawking radiation. If the ultraviolet (UV) model is considered, the recent $\mathcal{O}$(100) PeV neutrino event KM3-230213A at KM3NeT can be naturally explained.