  We argue that extensions of the Standard Model (SM) with a strongly
first-order electroweak phase transition generically predict significant
deviations of the Higgs couplings to gluons, photons, and Z bosons from their
SM values. Precise experimental measurements of the Higgs couplings at the LHC
and at the proposed next-generation facilities will allow for a robust test of
the phase transition dynamics. To illustrate this point, in this paper we focus
on the scenario in which loops of a new scalar field are responsible for the
first-order phase transition, and study a selection of benchmark models with
various SM gauge quantum numbers of the new scalar. We find that the current
LHC measurement of the Higgs coupling to gluons already excludes the
possibility of a first-order phase transition induced by a scalar in a sextet,
or larger, representation of the SU(3)_c. Future LHC experiments (including
HL-LHC) will be able to definitively probe the case when the new scalar is a
color triplet. If the new scalar is not colored, an electron-positron Higgs
factory, such as the proposed ILC or TLEP, would be required to test the nature
of the phase transition. The extremely precise measurement of the
Higgsstrahlung cross section possible at such machines will allow for a
comprehensive and definitive probe of the possibility of a first-order
electroweak phase transition in all models we considered, including the case
when the new scalar is a pure gauge singlet.
