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keV scale $ν_R$ dark matter and its detection in $β$ decay experiment

We study dark matter(DM) in the model with one keV scale right-handed neutrino $ν_{R1}$ and two GeV scale right-handed neutrinos $ν_{R2,3}$, the $ν$SM. We find that one of the GeV scale right-handed neutrinos can have much longer lifetime than the other when two GeV scale right-handed neutrinos are degenerate. We show that mass and mixing of light neutrinos can be explained in this case. Significant entropy release can be generated in a reheating produced by the decay of one of the GeV scale $ν_{R}$. The density of $ν_{R1}$ DM can be diluted by two orders of magnitude and the mixing of $ν_{R1}$ with active neutrinos is allowed to be much larger, reaching the bound from X-ray observation. This mixing can lead to sizeable rate of $ν_{R1}$ capture by radioactive nuclei. The $ν_{R1}$ capture events are mono-energetic electrons with keV scale energy away from the beta decay spectrum. This is a new way to detect DM in the universe.