Figure 1

The final distribution functions of neutrinos. (a) and (c) are cases for no oscillations (${\nu }_e$ is displayed by solid lines and ${\nu }_{\mu }$ by dashed lines) and (b) and (d) incorporate the oscillations (${\nu }_e$ is displayed by solid lines, ${\nu }_{\mu }^{\prime }$ by dashed lines and ${\nu }_{\tau }^{\prime }$ by dot-dashed lines). The equilibrium distributions are drawn by dotted lines in order to show how much they are thermalized. For $T_R=15\mathrm{MeV}$, in (a) and (b), whether the oscillations are present or not, all the lines overlap and this means every neutrino species is fully thermalized for high reheating temperature. For $T_R=2.5\mathrm{MeV}$, in (c) and (d), distributions are away from equilibrium form. When the oscillations are taken into account, distributions of ${\nu }_e$ and ${\nu }_{\mu }^{\prime }$ get close as seen in (d).Reuse & Permissions

Figure 2

We draw the sums of the distribution functions, $f_{{\nu }_e}+f_{{\nu }_{\mu }}$ (no oscillation) and $f_{{\nu }_e}+f_{{\nu }_{\mu }^{\prime }}$ (including oscillation) with the dashed line and the solid line, respectively. The latter is larger showing that the oscillations make the thermalization more efficient in total.Reuse & Permissions

Figure 3

The effective neutrino number $N_{\nu }$ as a function of the reheating temperature $T_R$ (shown on the bottom abscissa) or the decay width $\Gamma$ (shown on the top abscissa). The cases with and without the oscillations are drawn, respectively, by the solid and dashed lines. The horizontal line denotes $N_{\nu }=3.04$ with which $N_{\nu }$ for high $T_R$ should coincide (see the text).Reuse & Permissions

Figure 4

The ${}^4\mathrm{He}$ abundance (mass fraction) $Y_p$ as a function of the reheating temperature $T_R$ (shown on the bottom abscissa) or the decay width $\Gamma$ (shown on the top abscissa). The cases with and without the oscillations are drawn, respectively, by the solid and dashed curves. Thinner curves are calculated with Fermi distributed neutrinos with $N_{\nu }$ of Fig. 3 (namely, only the change in the expansion rate due to the incomplete thermalization is taken into account). The horizontal line represents “standard” $Y_p$ calculated by BBN with neutrinos obeying the Fermi distribution and $N_{\nu }=3.04$. The baryon-to-photon ratio is fixed at $\eta =5\times {10}^{-10}$.Reuse & Permissions

Figure 5

The weak interaction rate ${\Gamma }_{n\to p}$ and the expansion rate $H$ as functions of temperature, where ${\Gamma }_{n\to p}$ and $H$ first become equal. We plot for $T_R=2.5\mathrm{MeV}$ with and without the oscillations. For $T_R=15\mathrm{MeV}$, the oscillations do not make any difference.Reuse & Permissions

Figure 6

(a) $T_{\mathrm{np}}$, freeze-out temperature of the neutron-to-proton ratio, and (b) $2/[1+\exp (\Delta m/T_{\mathrm{np}})]$, as functions of $T_R$.Reuse & Permissions

Figure 7

Contour plots for the light element abundances. ${}^4\mathrm{He}$ mass fraction is plotted in (a) with the oscillations and in (b) without. D and ${}^7\mathrm{Li}$ are plotted, respectively, in (c) and (d) where dotted lines express the case with the oscillations and dashed lines express the case without. Shaded areas represent uncertainties in the observed abundances expressed in Eqs. (28, 29, 30) (for D and ${}^7\mathrm{Li}$, they are drawn against the contours considering the oscillations). Darker areas are for $1\sigma$ and lighter for $2\sigma$.Reuse & Permissions

Figure 8

${\chi }^2$ contour plot using data of D and ${}^4\mathrm{He}$. For no oscillation case, the allowed regions at 68% and 95% confidence levels are drawn with solid and dashed lines. For the case with oscillations, the 68% allowed region does not appear and only the 95% region is indicated by the shaded area.Reuse & Permissions