$N_{\text{eff}}$ in low-scale seesaw models versus the lightest neutrino mass

We evaluate the contribution to $N_{\text{eff}}$ of the extra sterile states in low-scale type I seesaw models (with three extra sterile states). We explore the full parameter space and find that at least two of the heavy states always reach thermalization in the early Universe, while the third one might not thermalize provided the lightest neutrino mass is below $\mathcal{O}({10}^{-3}\mathrm{eV})$. Constraints from cosmology therefore severely restrict the spectra of heavy states in the range 1 eV–100 MeV. The implications for neutrinoless double beta decay are also discussed.

Figure 1

Leptonic contribution to $C_{\alpha }(T)/\sqrt{g_{*}(T)}$ taken from Refs. [56, 57] for $\alpha =e$ (top/blue), $\mu$ (middle/magenta), $\tau$ (bottom/yellow).

Figure 2

Contours of $\text{Min}[f_{s_1}(T_{\max })]=0.1$, 1, 10 on the plane $(M_1,m_1)$.

Figure 3

Minimum of $h_2$ in bins of $h_1$ in the full allowed parameter space with fixed $m_1={10}^{-[5-2]}\mathrm{eV}$. The dashed line corresponds to the analytical bound $m_1^{\text{th}}=3.25\times {10}^{-3}\mathrm{eV}$.

Figure 4

Allowed spectra of the heavy states $M_i$ for $m_1\ge m_1^{\text{th}}$.

Figure 5

$\sum _{j=2,3}\mathrm{\Delta }{N}_{\text{eff}}^{(j)\mathrm{BBN}}$ for $m_1\le m_1^{\text{th}}$, as a function of $M_2$ and $M_3$. The thick lines correspond to present BBN bounds.

Figure 6

Allowed spectra of the heavy states $M_i$ for $m_1\le m_1^{\text{th}}$. The unconstrained mass could be any $i=1,2,3$.

Figure 7

Contour plots for $\mathrm{\Delta }{N}_{\text{eff}}^{(1)\mathrm{BBN}}={10}^{-1},{10}^{-2},{10}^{-3}$ defined by the ratio of the energy density of the $j=1$ sterile state and one standard neutrino as a function of $m_1$ and $M_1$. The solid (dashed) lines correspond to the contours of the ratio of sterile to active number (energy) densities. The shaded region corresponds to ${\mathrm{\Omega }}_{s_1}{h}^2\ge 0.1199$ and the dashed straight line is roughly the one corresponding to decay at recombination. The heavier neutrino masses have been fixed to $M_{2,3}=1\mathrm{GeV}$, 10 GeV and the unconstrained parameters have been chosen to minimize $f_1(T_{\max })$ and $f_2(T_{\max })$. The light neutrino spectrum has been assumed to be normal (NH).

Figure 8

$m_{\beta \beta }$ as a function of the lightest neutrino mass: contribution from the active neutrinos (red and blue regions) and the maximum contribution of the lightest sterile neutrino, for $M_1=1\mathrm{eV}$ (solid), 100 eV (dashed), 1 keV (dotted), for NH (blue) and IH (red) restricting ${\mathrm{\Omega }}_{s_1}{h}^2\le 0.12$ and $f_{s_1}(T_{\max })\le 1$, for $M_{2,3}\gg 100\mathrm{MeV}$, as a function of the lightest neutrino mass. The shaded region is ruled out for $M_1\in [1\mathrm{eV}–100\mathrm{MeV}]$ by the thermalization bound on the lightest neutrino mass, $m_1\le {10}^{-3}\mathrm{eV}$.