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We present a dedicated complementarity study of gravitational wave andcollider measurements of the simplest extension of the Higgs sector: thesinglet scalar augmented Standard Model. We study the following issues: (i) theelectroweak phase transition patterns admitted by the model, and the proportionof parameter space for each pattern; (ii) the regions of parameter space thatgive detectable gravitational waves at future space-based detectors; and (iii)the current and future collider measurements of di-Higgs production, as well assearches for a heavy weak diboson resonance, and how these searches interplaywith regions of parameter space that exhibit strong gravitational wave signals.We carefully investigate the behavior of the normalized energy released duringthe phase transition as a function of the model parameters, address subtleissues pertaining to the bubble wall velocity, and provide a description ofdifferent fluid velocity profiles. On the collider side, we identify the subsetof points that are most promising in terms of di-Higgs and weak dibosonproduction studies while also giving detectable signals at LISA, setting thestage for future benchmark points that can be used by both communities.

Collider and Gravitational Wave Complementarity in Exploring the Singlet Extension of the Standard Model