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Extensions of the Higgs sector of the Standard Model allow for a richcosmological history around the electroweak scale. We show that besides thepossibility of strong first-order phase transitions, which have been thoroughlystudied in the literature, also other important phenomena can occur, like thenon-restoration of the electroweak symmetry or the existence of vacua in whichthe Universe becomes trapped, preventing a transition to the electroweakminimum. Focusing on the next-to-minimal two-Higgs-doublet model (N2HDM) oftype II and taking into account the existing theoretical and experimentalconstraints, we identify the scenarios of electroweak symmetry non-restoration,vacuum trapping and first-order phase transition in the thermal history of theUniverse. We analyze these phenomena and in particular their relation to eachother, and discuss their connection to the predicted phenomenology of the N2HDMat the LHC. Our analysis demonstrates that the presence of a global electroweakminimum of the scalar potential at zero temperature does not guarantee that thecorresponding N2HDM parameter space will be physically viable: the existence ofa critical temperature at which the electroweak phase becomes the deepestminimum is not sufficient for a transition to take place, necessitating ananalysis of the tunnelling probability to the electroweak minimum for areliable prediction of the thermal history of the Universe.

Fate of electroweak symmetry in the early Universe: Non-restoration and trapped vacua in the N2HDM