Gravitational waves from domain wall collapses and dark matter in the SM with a complex scalar

We study domain wall induced by spontaneously broken $\mathbb{Z}_2$ symmetry and its gravitational wave signature in the standard model with a complex scalar in connection with dark matter physics. In a minimal setup, a linear term of the singlet field is added to the scalar potential as an explicit $\mathbb{Z}_2$ breaking term to make the domain wall unstable. We obtain its minimal size from cosmological constraints and show that the parameter space that can be probed by current and future pulsar time array experiments requires the vacuum expectation value of the singlet field to be greater than $\mathcal{O}(10-100)$ TeV, along with a singlet-like Higgs mass of $\mathcal{O}(1-100)$ TeV. However, such a region is severely restricted by the dark matter relic density, which places an upper bound on the singlet vacuum expectation value at approximately 200 TeV, and limits the dark matter mass to about half of the singlet-like Higgs boson mass.