Unveiling secret interactions among sterile neutrinos with big-bang nucleosynthesis

Short-baseline neutrino anomalies suggest the existence of low-mass [$m\sim \mathcal{O}(1)\mathrm{eV}$] sterile neutrinos ${\nu }_s$. These would be efficiently produced in the early universe by oscillations with active neutrino species, leading to a thermal population of the sterile states seemingly incompatible with cosmological observations. In order to relieve this tension it has been recently speculated that new “secret” interactions among sterile neutrinos, mediated by a massive gauge boson $X$ (with $M_X\ll M_W$), can inhibit or suppress the sterile neutrino thermalization, due to the production of a large matter potential term. We note however, that they also generate strong collisional terms in the sterile neutrino sector that induce an efficient sterile neutrino production after a resonance in matter is encountered, increasing their contribution to the number of relativistic particle species $N_{\text{eff}}$. Moreover, for values of the parameters of the ${\nu }_s\text{−}{\nu }_s$ interaction for which the resonance takes place at temperature $T\lesssim \text{few}\mathrm{MeV}$, significant distortions are produced in the electron (anti)neutrino spectra, altering the abundance of light element in big bang nucleosynthesis (BBN). Using the present determination of ${}^4\mathrm{He}$ and deuterium primordial abundances we determine the BBN constraints on the model parameters. We find that ${}^2\mathrm{H}/\mathrm{H}$ density ratio exclude much of the parameter space if one assumes a baryon density at the best fit value of Planck experiment, ${\mathrm{\Omega }}_B{h}^2=0.02207$, while bounds become weaker for a higher ${\mathrm{\Omega }}_B{h}^2=0.02261$, the 95% C.L. upper bound of Planck. Due to the large error on its experimental determination, the helium mass fraction $Y_p$ gives no significant bounds.

Figure 1

Neutrino refractive and collisional rates (normalized in terms of the Hubble rate) versus temperature $T$ for $G_X={10}^3$ $G_F$. Left panel corresponds to $g_X={10}^{-1}$, while right panel to $g_X={10}^{-2}$. The curves correspond to the active-sterile vacuum term (solid curve), the secret matter potential for ${\rho }_{ss}={10}^{-2}$ (dashed curve), the standard collisional term (dotted curve) and the collisional term associated with $G_X^2$ (dot-dashed curve).

Figure 2

Resonance temperature $T_{\text{res}}$ in the plane $(G_X,g_X)$. Dashed curves represent constant $T_{\text{res}}$ contours, while on solid curves $M_X$ is constant. The values shown for both parameters are expressed in MeV. The case of $T_{\text{res}}=1\mathrm{MeV}$ is highlighted with a tick dot-dashed style, and correspond to the order of magnitude of BBN onset in the standard case, when neutron to proton density ratio freezes out.

Figure 3

Flavor evolution as functions of temperature $T$ for different cases for $G_X={10}^3$ $G_F$. Upper panel is the standard case without secret interactions. Middle and lower plots are for $g_X={10}^{-1}$ and $g_X={10}^{-2}$, respectively. In the left panels we report ${\rho }_{ee}$ (continuous curve), ${\rho }_{\mu \mu }$ (dotted curve) and ${\rho }_{ss}$ (dashed curve). Right panels show the corresponding $\mathrm{\Delta }{N}_{\text{eff}}$.

Figure 4

The asymptotic values of $\mathrm{\Delta }{N}_{\text{eff}}$ versus $G_X$ and $g_X$. Colors from blue (lower right corner) to red (upper left corner) correspond to increasing values. Dashed curves show some reference values.

Figure 5

${}^4\mathrm{He}$ results. The dark region is the $1.5\sigma$ allowed parameter space for the helium mass fraction $Y_p$ using the experimental result of Eq. (13), varying the baryon density parameter in the range $0.02153\le {\mathrm{\Omega }}_B{h}^2\le 0.02261$, corresponding to the 95% C.L. Planck range. The solid line bounds the permitted region if we fix ${\mathrm{\Omega }}_B{h}^2=0.02207$, the best fit quoted by Planck collaboration. At $2\sigma$ the whole region shown for $G_X$ and $g_X$ would be allowed, while at $1\sigma$ it is all excluded.

Figure 6

${}^2\mathrm{H}/\mathrm{H}$ results. Left panel: the region below each curve is the allowed one at the number of $\sigma$ shown for each case, using the experimental determination for ${}^2\mathrm{H}/\mathrm{H}$ of Eq. (14) and for ${\mathrm{\Omega }}_B{h}^2=0.02207$, the best fit quoted by Planck collaboration. Right panel: the same as in the left panel but for a higher baryon density ${\mathrm{\Omega }}_B{h}^2=0.02261$, i.e. the Planck upper limit at 95% C.L.