Massive sterile neutrinos in the early Universe: From thermal decoupling to cosmological constraints

We consider relatively heavy neutrinos ${\nu }_H$, mostly contributing to a sterile state ${\nu }_s$, with mass in the range $10\mathrm{MeV}\lesssim m_s\lesssim m_{\pi }\sim 135\mathrm{MeV}$, which are thermally produced in the early Universe in collisional processes involving active neutrinos and freezing out after the QCD phase transition. If these neutrinos decay after the active neutrino decoupling, they generate extra neutrino radiation but also contribute to entropy production. Thus, they alter the value of the effective number of neutrino species $N_{\mathrm{eff}}$ as, for instance, measured by the cosmic microwave background (CMB), as well as affect primordial nucleosynthesis (BBN), notably ${}^4\mathrm{He}$ production. We provide a detailed account of the solution of the relevant Boltzmann equations. We also identify the parameter space allowed by current Planck satellite data and forecast the parameter space probed by future stage-4 ground-based CMB observations, expected to match or surpass BBN sensitivity.

Figure 1

Comparison between the comoving extra active neutrino energy using $x_d=0.1$ (red curve) and $x_d=0.2$ (black curve), for parameters $m_s=100\mathrm{MeV}$ and ${\tau }_s=0.045\mathrm{s}$.

Figure 2

$N_{\mathrm{eff}}$ evolution in $x$ for $m_s=30\mathrm{MeV}$, ${\tau }_{\tau s}=0.15\mathrm{s}$ (red, solid line) and $m_s=100\mathrm{MeV}$, ${\tau }_{\tau s}=0.055\mathrm{s}$ (black, dashed line).

Figure 3

Comoving energy density evolution for the sterile neutrino vs $x$ for $m_s=30\mathrm{MeV}$, ${\tau }_{\tau s}=0.15\mathrm{s}$ (red, solid line) and $m_s=100\mathrm{MeV}$, ${\tau }_{\tau s}=0.055\mathrm{s}$ (black, dashed line).

Figure 4

Evolution of the dimensionless temperature $z$ vs $x$ for $m_s=30\mathrm{MeV}$, ${\tau }_{es}=0.15\mathrm{s}$ (red, dash-dotted line) ${\tau }_{\tau s}=0.15\mathrm{s}$ (purple, solid line) and $m_s=100\mathrm{MeV}$, ${\tau }_{es}=0.055\mathrm{s}$ (blue, dashed line), and ${\tau }_{\tau s}=0.055\mathrm{s}$ (black, dotted line).

Figure 5

Comoving energy density evolution for the three active neutrinos vs $x$ for $m_s=30\mathrm{MeV}$, ${\tau }_{\tau s}=0.15\mathrm{s}$ (dashed lines) and $m_s=100\mathrm{MeV}$, ${\tau }_{\tau s}=0.055\mathrm{s}$ (solid lines). Each sets of curves, from top to bottom, represents the ${\nu }_{\tau }$, ${\nu }_{\mu }$, and ${\nu }_e$ energy density evolution, respectively, reflecting the assumed sterile mixing with ${\nu }_{\tau }$ and the active neutrino mixing matrix. Here, $x_d=0.2$ is assumed.

Figure 6

${\nu }_{\tau }$ and ${\nu }_e$ distribution functions vs $y$ at $x=1$ for the same cases reported in Fig. 5.

Figure 7

Bounds in the plane $(m_s,{\tau }_s)$ obtained from CMB (red curve) and BBN-$Y_p$ (blue curve), as well as forecast sensitivity of CMB-S4 (black curve), for a sterile neutrino mixed with ${\nu }_{\tau }$ (or ${\nu }_{\mu }$). The $2\sigma$ excluded region is the one above the curves.

Figure 8

Bounds in the plane $(m_s,{\theta }_{\tau s})$ obtained from CMB (red curve) and BBN-$Y_p$ (blue curve), as well as forecast sensitivity of CMB-S4 (black curve), for a sterile neutrino mixed with ${\nu }_{\tau }$ (or ${\nu }_{\mu }$). The $2\sigma$ excluded region is the one under the curves.

Figure 9

Bounds in the plane $(m_s,{\tau }_s)$ obtained from CMB (red curve) and BBN-$Y_p$ (blue curve), as well as forecast sensitivity of CMB-S4 (black curve), for a sterile neutrino mixed with ${\nu }_e$. The $2\sigma$ excluded region is the one above the curves.

Figure 10

Bounds in the plane $(m_s,{\theta }_{es})$ obtained from CMB (red curve) and BBN-$Y_p$ (blue curve), as well as forecast sensitivity of CMB-S4 (black curve), for a sterile neutrino mixed with ${\nu }_e$. The $2\sigma$ excluded region is the one under the curves.

Figure 11

Comparison between our results (red, solid line) and Ref. [11] (black, dashed line) on the comoving density of the sterile species for $m_s=100\mathrm{MeV}$ and ${\tau }_s=0.055\mathrm{s}$.

Figure 12

Comparison of results from the BBN constraints for the decay time of a sterile neutrino mixed only with active tauonic (or muonic) neutrino in Refs. [11, 18] and the one from our code.

Figure 13

Comparison of results from the BBN constraints for the decay time of a sterile neutrino mixed only with active electron neutrino in Refs. [11, 18] and the one from our code.

Figure 14

Comparison of results from the CMB constraints for the decay time of a sterile neutrino mixed only with active tauonic (or muonic) neutrino in Ref. [18] and the one from our code obtained using their same value of ${\mathrm{\Theta }}_{\mathrm{Obs}}$ as a benchmark.

Figure 15

Comparison of results from the CMB constraints for the decay time of a sterile neutrino mixed only with an active electron neutrino in Ref. [18] and the one from our code obtained using their same value of ${\mathrm{\Theta }}_{\mathrm{Obs}}$ as a benchmark.

Figure 16

$\mathrm{\Delta }{N}_{\mathrm{eff}}$ vs ${\tau }_s$ for $m_s=50\mathrm{MeV}$ in the limit of zero b.r. into active neutrino states (solid red line) and zero b.r. into electromagnetic sector (black dashed line).