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Theories with more than one scalar field often exhibit phase transitionsproducing potentially detectable gravitational wave (GW) signal. In this workwe study the semi-annihilating $\mathbb{Z}_3$ dark matter model, whose darksector comprises an inert doublet and a complex singlet, and assess itsprospects in future GW detectors. Without imposing limits from requirement ofproviding a viable dark matter candidate, i.e. taking into account only otherexperimental and theoretical constraints, we find that the first order phasetransition in this model can be strong enough to lead to a detectable signal.However, direct detection and the dark matter thermal relic density constraintcalculated with the state-of-the-art method including the impact of earlykinetic decoupling, very strongly limit the parameter space of the modelexplaining all of dark matter and providing observable GW peak amplitude.Extending the analysis to underabundant dark matter thus reveals region withdetectable GWs from a single-step or multi-step phase transition.

Phase Transitions and Gravitational Waves in a Model of $\mathbb{Z}_{3}$ Scalar Dark Matter