  Theories with extended Higgs sectors constructed in view of cosmological
ramifications (gravitational wave signal, baryogenesis, dark matter) are often
faced with conflicting requirements for their couplings; in particular those
influencing the strength of a phase transition may be large. Large couplings
compromise perturbative studies, as well as the high-temperature expansion that
is invoked in dimensionally reduced lattice investigations. With the example of
the inert doublet extension of the Standard Model (IDM), we show how a resummed
2-loop effective potential can be computed without a high-T expansion, and use
the result to scrutinize its accuracy. With the exception of Tc, which is
sensitive to contributions from heavy modes, the high-T expansion is found to
perform well. 2-loop corrections weaken the transition in IDM, but they are
moderate, whereby a strong transition remains an option.
