Extending the Standard Model with three right-handed neutrinos and a simple QCD axion sector can account for neutrino oscillations, dark matter and baryon asymmetry; at the same time, it solves the strong CP problem, stabilizes the electroweak vacuum and can implement critical Higgs inflation (satisfying all current observational bounds). We perform here a general analysis of dark matter (DM) in such a model, which we call the $a u$MSM. Although critical Higgs inflation features a (quasi) inflection point of the inflaton potential we show that DM cannot receive a contribution from primordial black holes in the $a u$MSM. This leads to a multicomponent axion-sterile-neutrino DM and allows us to relate the axion parameters, such as the axion decay constant, to the neutrino parameters. We include several DM production mechanisms: the axion production via misalignment and decay of topological defects as well as the sterile-neutrino production through the resonant and non-resonant mechanisms and in the recently proposed CPT-symmetric universe.
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