  We perform a complementarity study of gravitational waves and colliders in
the context of electroweak phase transitions choosing as our template the xSM
model, which consists of the Standard Model augmented by a real scalar. We
carefully analyze the gravitational wave signal at benchmark points compatible
with a first order phase transition, taking into account subtle issues
pertaining to the bubble wall velocity and the hydrodynamics of the plasma. In
particular, we comment on the tension between requiring bubble wall velocities
small enough to produce a net baryon number through the sphaleron process, and
large enough to obtain appreciable gravitational wave production. For the most
promising benchmark models, we study resonant di-Higgs production at the
high-luminosity LHC using machine learning tools: a Gaussian process algorithm
to jointly search for optimum cut thresholds and tuning hyperparameters, and a
boosted decision trees algorithm to discriminate signal and background. The
multivariate analysis on the collider side is able either to discover or
provide strong statistical evidence of the benchmark points, opening the
possibility for complementary searches for electroweak phase transitions in
collider and gravitational wave experiments.
