Figure 1

Different constraints on $N_1$ mass and mixing in scenario I. The blue region is excluded by x-ray observations, the dark gray region $M_1<1\mathrm{keV}$ by the Tremaine-Gunn bound [110, 111, 112]. For points on the upper solid black line, the observed ${\Omega }_{\mathrm{DM}}$ is produced for ${\mu }_{\alpha }=0$ [4]; points on the lower solid black line give the correct ${\Omega }_{\mathrm{DM}}$ for $|{\mu }_{\alpha }|=1.24\times {10}^{-4}$, the maximal asymmetry we found at $T=100\mathrm{MeV}$. The region between these lines is accessible for $0\le |{\mu }_{\alpha }|\le 1.24\times {10}^{-4}$. Observations of the matter distribution in the Universe constrain the DM free streaming length. Without resonant production (${\mu }_{\alpha }=0$), this implies that $M_1>8\mathrm{keV}$ [59], which excludes the upper black curve and makes resonant production necessary. Combining both production mechanisms ($|{\mu }_{\alpha }|\gtrsim {10}^{-5}$), this bound relaxes to $M_1>2\mathrm{keV}$ [59]. However, we do not display it here because it depends on ${\mu }_{\alpha }$ in a complicated way and the calculation currently includes considerable uncertainties [59]; cf. also Refs. 113, 114.Reuse & Permissions

Figure 2

Constraints on $N_{2,3}$ masses $M_{2,3}\simeq M$ and mixing $U^2\equiv \mathrm{tr}({\theta }^{†}\theta )$ in scenarios I (red line) and II (blue line) for normal (upper panel) and inverted (lower panel) hierarchies of neutrino masses. In the regions within the blue and red lines, no physics beyond the $\nu \mathrm{MSM}$ is needed to explain the observed ${\Omega }_B$ and ${\Omega }_{\mathrm{DM}}$, respectively.Reuse & Permissions