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We consider models with the $U(1)_X$ gauge symmetry, which is spontaneouslybroken by dark Higgs mechanism. We discuss patterns of the electroweak phasetransition and detectability of gravitational waves (GWs) when strongly firstorder phase transition (1stOPT) occurs. It is pointed out that the colliderbounds on the properties of the discovered Higgs boson exclude a part ofparameter space that could otherwise generate detectable GWs. We find that GWsproduced from multi-step PT can be detected at future observations such as LISAand DECIGO if the dark photon mass is $m_X^{} \gtrsim 25$ GeV with the$U(1)_X^{}$ gauge coupling being $g_X^{} \gtrsim 0.5$. In addition, we showthat most of the parameter regions can be covered by precision measurements ofvarious Higgs boson couplings and direct searches for the singlet scalar bosonat future collider experiments. Furthermore, we expect the complementarity ofthe detection of GW observations from the strongly 1stOPT, collider bounds anddark photon searches in the models of the dark gauge symmetry.

Exploring first order phase transition in $U(1)$ extended models by complementarity between collider measurements and cosmological observations