Gravitational Waves from First-Order Phase Transition in an Electroweakly Interacting Vector Dark Matter Model

We discuss gravitational waves (GWs) in an electroweakly interacting vector dark matter (DM) model. In the model, the electroweak gauge symmetry is extended to SU(2)$_0 \times$SU(2)$_1 \times$SU(2)$_2 \times$U(1)$_Y$ and spontaneously broken into SU(2)$_L \times$U(1)$_Y$ at TeV scale. The model has an exchange symmetry between SU(2)$_0$ and SU(2)$_2$. This symmetry stabilizes some massive vector bosons associated with the spontaneous symmetry breaking described above, and an electrically neutral one is a DM candidate. In a previous study, it was found that the gauge couplings of SU(2)$_0$ and SU(2)$_1$ are relatively large to explain the measured value of the DM energy density via the freeze-out mechanism. With the large gauge couplings, the gauge bosons potentially have a sizable effect on the scalar potential. In this paper, we focus on the phase transition of SU(2)$_0 \times$SU(2)$_1 \times$SU(2)$_2 \rightarrow$ SU(2)$_L$. We calculate the effective potential at finite temperature and find that the phase transition is first-order and strong in a wide range of the parameter space. The strong first-order phase transition generates GWs. We calculate the GW spectrum and find that it will be possible to detect the GWs predicted in the model by future space-based GW interferometers. We explore the regions of the parameter space probed by the GW detection. We find that the GW detection can probe the region where the mass of $h<^>{\prime }$, a CP-even scalar in the model, is a few TeV.