Big bang nucleosynthesis

The abundances of primordial $^4He$ and $D$ can be used to determine the equivalent number $N_\nu^{\rm eff}$ of light neutrinos in equilibrium at the time of neutrino decoupling ($t \sim$ 1 s, $T \sim 1$ MeV) and the baryon density $h^2 \Omega_B$, where $h \sim 0.65 \pm 0.05$:

$$1.7 < N_\nu^{\rm eff} < 3.3; \ \ \ \ h^2 \Omega_B \sim 0.017(3).$$ (19)

$N_\nu^{\rm eff}$ is actually an effective parameter, incorporating any contribution to the $n/p$ ratio at the time of decoupling, i.e.,

$$N_\nu^{\rm eff} = n_{\rm active}^{\rm eff} + n_{\rm sterile}^{\rm eff} + n_{\rm asym}^{\rm eff},$$ (20)

where $n_{\rm active}^{\rm eff} \sim 3$ is from the active neutrinos, and $n_{\rm sterile}^{\rm eff}$ represents the number of light sterile neutrino species present at decoupling. It has long been known that sterile neutrinos could be generated in equilibrium numbers by mixing with active neutrinos prior to nucleosynthesis for a wide range of $\Delta m^2$ and $\sin^2 2 \theta$. In particular, it was believed that $n_{\rm sterile}^{\rm eff} \sim 1$ for mixing parameters corresponding to atmospheric $\nu_\mu \mbox{$\rightarrow$}\nu_s$ oscillations, in conflict with (). (The recent Super K data also directly exclude pure $\nu_\mu \mbox{$\rightarrow$}\nu_s$). For the solar neutrinos, the rates have always allowed a small mixing angle (SMA) $\nu_e \mbox{$\rightarrow$}\mbox{$\nu_s$}$ solution analogous to the SMA $\nu_e \mbox{$\rightarrow$}\mbox{$\nu_\mu$}$, but in this case the mixing would not have been sufficient to generate $\nu_s$ cosmologically⁶.

There has been considerable interest in ways to modify (or lower) the prediction for $N_\nu^{\rm eff}$, motivated by: (a) the observed value, which depends on the somewhat controversial determinations of the relic $^4He$ abundance, may eventually settle at a value lower than 3. (b) If there really is a light sterile neutrino, as suggested by the LANL data, then the possible contributions of $n_{\rm sterile}^{\rm eff}$ become crucial. There are several canonical ways to change the prediction:

• If $\nu_\tau$ is unstable, then the active contribution to $N_\nu^{\rm eff}$ may fall below 3, depending on the $\nu_\tau$ lifetime and the energy density in its decay products.
• Large lepton asymmetries $\Delta L_a \simeq (n_{\nu_a} - n_{\bar{\nu}_a})/n_\gamma$ lead to increased energy densities $n_{\rm asym}^{\rm eff} > 0$, increasing $N_\nu^{\rm eff}$. For $a=\mu, \tau$ this is the only effect. However, a positive $\Delta L_e$ can actually decrease $N_\nu^{\rm eff}$ because it preferentially drives the rate for $\mbox{$\nu_e$}n \mbox{$\rightarrow$}e^- p$ compared to its inverse, and thus decreases the $n$ and therefore the $^4He$ abundance. Neither of these effects are important unless the asymmetry is very much larger than the baryon asymmetry $\sim 10^{-9}$.
• A massive $\tau$ neutrino in the range between $\sim$ 0.5 MeV and the laboratory limit $\sim$ 18 MeV would increase $n_{\rm active}^{\rm eff}$ above 3, because the large rest energy would be more important than the reduced number density. This range is therefore most likely excluded.

Foot and Volkas and others [49] have recently reexamined the effects of sterile neutrinos on nucleosynthesis, and argued that they could actually decrease $N_\nu^{\rm eff}$. The current status was discussed by Kirilova [50] and Wong [51]. There may be a strong interplay between active-sterile mixing and a nonzero asymmetry $\Delta L_a$ (e.g., a small preexisting $\Delta L_a \sim 10^{-9}$ or one generated by the oscillations). In particular, such effects can suppress the production of $\nu_s$, deplete the number of $\nu_e$, distort the active neutrino spectrum and therefore the reaction rates (a very important effect), and amplify (or generate) a small initial $\Delta L_e$ asymmetry. The net result is that the bounds on active-sterile mixing may be weakened, especially for small mixing. However, there is still considerable debate as to the details and the size of these effects.