Probing High-Quality Axions with Gravitational Waves

We present a systematic study of gravitational wave (GW) signals from phase transitions and topological defects in a unified high-quality axion framework. The gauged $U(1)_g$ symmetry forbids any bias term that could lift the vacuum degeneracy, restricting the theory to the phenomenologically viable case $N_{\rm DW}=1$. Requiring the axion to account for the observed dark matter (DM) abundance and satisfy the high-quality condition constrains the gauge symmetry-breaking scale to $f_g \in [1.6\times10^{11},\,10^{16}]\,\mathrm{GeV}$ for the QCD axion, leading to a well-defined band of GW signals, part of which is consistent with current pulsar timing array observations. Two-step first-order phase transitions are common in this framework, with the lower-scale transition generating GWs with $f^{\rm peak} \gtrsim \mathcal{O}(10^7)\,\mathrm{Hz}$. For axion-like realizations, generic post-inflation models predict GW spectra that are nearly degenerate with the QCD axion case. We conclude that GWs alone cannot distinguish between these scenarios, highlighting the need for complementary probes.