Abstract
Once there is a global $U(1)$ symmetry broken at a high-energy scale $f_a$, we will have a light axion-like particle (ALP) from the spontaneous symmetry breaking. In a large parameter space,
the global $U(1)$ symmetry breaking induces a strongly first order phase transition, thereby producing stochastic gravitational waves (GWs). A broad range of the $U(1)$ breaking scale $f_a$
can be probed in current and future GW experiments, i.e. from $10^3$ GeV to $10^8$ GeV. All the ALP couplings to the Standard Model particles are proportional to inverse powers of the energy
scale $f_a$, and the GW prospects are largely complementary to the current low-energy laboratory, astrophysical and cosmological probes of ALPs.
