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We perform a comprehensive study of a model in which the Higgs sector isextended to contain two Higgs doublet fields, with the four types ofpossibilities to couple to standard fermions, as well as an additional lightpseudoscalar Higgs boson which mixes with the one of the two doublets. This2HD+a model includes also a stable isosinglet massive fermion that has thecorrect thermal relic abundance to account for the dark matter in the Universe.We summarize the theoretical constraints to which the model is subject and thenperform a detailed study of the phenomenological constraints. In particular, wediscuss the bounds from the LHC in the search for light and heavy scalarresonances and invisible states and those from high precision measurements inthe Higgs, electroweak and flavor sectors, addressing the possibility ofexplaining the deviation from the standard expectation of the anomalousmagnetic moment of the muon and the $W$-boson mass recently observed atFermilab. We also summarize the astrophysical constraints from direct andindirect detection dark matter experiments. We finally conduct a thoroughanalysis of the cosmic phase transitions and the gravitational wave spectrumthat are implied by the model and identify the parameter space in which theelectroweak vacuum is reached after single and multiple phase transitions. Wethen discuss the prospects for observing the signal of such gravitational wavesin near future experiments such as LISA, BBO or DECIGO.

The 2HD+a model: collider, dark matter and gravitational wave signals