Gravitational wave signature of generic disappearance of $Z_2$-symmetry breaking domain walls

Breaking of discrete parity at high scale gives rise to $Z_2$-domain walls. The metastability of such walls can make them relatively long lived and contradict standard cosmology. We consider two classes of theories with similar underlying feature, the left right symmetric theories and two Higgs doublet models. These theories effectively possess two steps of breaking of $Z_2$ discrete symmetries. The domains formed at a high energy scale in the first step further decompose into subdomains near the electroweak scale in the second step. Then a QCD instanton induced energy difference can remove the subdomain walls as well as the domain walls successfully. This result holds regardless of whether the domain wall removal is guided by small symmetry breaking terms or purely statistical outcome of a homogeneous vacuum. We then investigate the gravitational waves arising from the collapse of such domain walls and show that the peak frequency of these waves lies in the $10^{-9}$ - $10^{-7}$ Hz band, depending on annihilation temperatures of $10^{-2}$ - 1 GeV, which is sensitive to pulsar timing based experiments such as SKA and NANOGrav.