Gravitational waves and primordial black holes from axion domain walls in level crossing

In this paper, we investigate the nano-Hertz gravitational waves (GWs) emission and the massive primordial black holes (PBHs) formation from the light QCD axion scenario. We consider the axion domain walls formation from the level crossing induced by the mass mixing between the light $Z_{\mathcal N}$ QCD axion and axion-like particle. A general mixing case is considered that the heavy and light mass eigenvalues do not necessarily have to coincide with the axion masses. In order to form the domain walls, the axions should start to oscillate slightly before the level crossing. The domain walls must annihilate before dominating the Universe to avoid the cosmological catastrophe. Then we focus our attention on the GWs emission from the domain walls annihilation and the PBHs formation from the domain walls collapse. We show the predicted GWs spectra with the peak frequency $\sim 0.2\, \rm nHz$ and the peak amplitude $\sim 5\times 10^{-9}$, which can be tested by the future pulsar timing array projects. In addition, during the domain walls annihilation, the closed walls could shrink to the Schwarzschild radius and collapse into the PBHs. We find that the PBHs in the mass range of $\mathcal{O}(10^5-10^8) M_\odot$ could potentially form in this scenario and account for a small fraction $\sim 10^{-5}$ of the cold dark matter.