The trap in the early Universe: impact on the interplay between gravitational waves and LHC physics in the 2HDM

We analyze the thermal history of 2HDM and determine parameter regions featuring a first-order electroweak phase transition (FOEWPT) also much less studied phenomena like high-temperature (EW) symmetry non-restoration possibility vacuum trapping (i.e. Universe remains trapped in an EW-symmetric throughout cosmological evolution, despite at $T=0$ EW breaking is deeper). show that presence impedes parameter-space previously considered suitable for realization baryogenesis. Focusing then on do feature such transition, we space would yield stochastic gravitational wave signal potentially detectable by future LISA observatory very contrived, will be well probed direct searches Higgs bosons HL-LHC, (possibly) via measurements self-coupling boson 125 GeV. This has important impact interplay between LHC regarding exploration scenarios 2HDM: absence new physics indications HL-LHC severely limit prospects detection LISA. Finally, demonstrate as consequence predicted enhancement GeV ILC able to probe majority yielding FOEWPT through self-coupling, with large improvement precision respect HL-LHC.