Sterile neutrinos are one of the leading dark matter candidates. Their masses may originate from a vacuum expectation value of a scalar field. If the sterile neutrino couplings are very small and their direct coupling to the inflaton is forbidden by the lepton number symmetry, the leading dark matter production mechanism is the freeze-in scenario. We study this possibility in the neutrino mass range up to 1 GeV, taking into account relativistic production rates based on the Bose-Einstein statistics, thermal masses and phase transition effects. The specifics of the production mechanism and the dominant mode depend on the relation between the scalar and sterile neutrino masses as well as on whether or not the scalar is thermalized. We find that the observed dark matter abundance can be produced in all of the cases considered. We also revisit the freeze-in production of a Higgs portal scalar, pointing out the importance of a fusion mode, as well as the thermalization constraints.
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