After the discovery of the Higgs boson, understanding the nature of electroweak symmetry breaking and the associated electroweak phase transition has become the most pressing question in particle physics. Answering this question is a priority for experimental studies. Data from the LHC and future lepton collider-based Higgs factories may uncover new physics coupled to the Higgs boson, which can induce the electroweak phase transition to become first order. Such a phase transition generates a stochastic background of gravitational waves, which could potentially be detected by a space-based gravitational wave interferometer. In this paper, we survey a few classes of models in which the electroweak phase transition is strongly first order. We identify the observables that would provide evidence of these models at the LHC and next-generation lepton colliders, and we assess whether the corresponding gravitational wave signal could be detected by eLISA. We find that most of the models with first-order electroweak phase transition can be covered by the precise measurements of Higgs couplings at the proposed Higgs factories. We also map out the model space that can be probed with gravitational wave detection by eLISA.
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