Clockwork axion footprint on nano-hertz stochastic gravitational wave background

The recent Pulsar Timing Arrays (PTAs) nano-Hz gravitational wave (GW) background signal can be naturally induced by the annihilation of domain walls (DWs) formed at a symmetry-breaking scale $f\simeq 200$~TeV in the clockwork axion framework. Based on our first successful and precise prediction, we for the first time suggest that the recent PTA observations strongly support the novel mechanism of the QCD instanton-induced DW annihilation in the clockwork axion framework. We also for the first time discover a novel correlation between dark matter (DM) relic abundance and nano-Hz GW background, which in turn indicates a natural connection between the axion decay constant and the symmetry-breaking scale in the clockwork framework. We find that the GW signal has a peak $h^2{\mathrm{\Omega }}_{\mathrm{G}\mathrm{W}}\simeq {10}^{-6.6}-{10}^{-6.1}$ at about 50~nHz, which is definite and testable for future PTA data at frequencies $\gtrsim 25$~nHz and CMB-S4 experiment. We also propose various phenomena that may appear in PTAs and future GW interferometers.