Lepton asymmetry, neutrino spectral distortions, and big bang nucleosynthesis

We calculate Boltzmann neutrino energy transport with self-consistently coupled nuclear reactions through the weak-decoupling-nucleosynthesis epoch in an early universe with significant lepton numbers. We find that the presence of lepton asymmetry enhances processes which give rise to nonthermal neutrino spectral distortions. Our results reveal how asymmetries in energy and entropy density uniquely evolve for different transport processes and neutrino flavors. The enhanced distortions in the neutrino spectra alter the expected big bang nucleosynthesis light element abundance yields relative to those in the standard Fermi-Dirac neutrino distribution cases. These yields, sensitive to the shapes of the neutrino energy spectra, are also sensitive to the phasing of the growth of distortions and entropy flow with time/scale factor. We analyze these issues and speculate on new sensitivity limits of deuterium and helium to lepton number.

Figure 1

Normalized number density plotted against $\epsilon$ for three choices of degeneracy parameter: nondegenerate ($\xi =0$, solid blue), degenerate with an excess ($\xi =3.0$, dashed red), and degenerate with a deficit ($\xi =-3.0$, dash-dotted green).

Figure 2

${\epsilon }_{\max }$ versus $N_{\mathrm{bins}}$ for contours of constant error in $L_{\nu }^{\star }$. The exact value of $L_{\nu }^{\star }$ is 0.1. The black line on the contour space gives the value of ${\epsilon }_{\max }$ with the smallest error as a function of $N_{\mathrm{bins}}$.

Figure 3

Relative differences in neutrino/antineutrino occupation numbers plotted against $\epsilon$ at $T_{\mathrm{cm}}=1\mathrm{keV}$. The relative differences are with respect to FD with zero degeneracy parameter. The solid lines show the evolution for a scenario where $L_{\nu }^{\star }=0.1$. ${\nu }_e$ and ${\overline{\nu }}_e$ curves are colored red, and ${\nu }_{\mu }$ and ${\overline{\nu }}_{\mu }$ curves are colored green. Neutrinos have thick line widths and antineutrinos have thin line widths. Plotted for comparison are black dashed curves representing the equilibrium relative differences. The top dashed curve corresponds to a $\nu$ spectrum with ${\xi }_{\nu }=0.1458$, the middle horizontal curve corresponds to a $\nu$ spectrum with ${\xi }_{\nu }=0$, and the bottom dashed curve corresponds to a $\overline{\nu }$ spectrum with ${\xi }_{\overline{\nu }}=-0.1458$.

Figure 4

Relative differences in electron neutrino/antineutrino occupation numbers plotted against $T_{\mathrm{cm}}$. The relative differences are with respect to FD with the same degeneracy parameters as Fig. 3. The solid lines show the evolution for a scenario where $L_{\nu }^{\star }=0.1$. The ${\overline{\nu }}_e$ (thin red curves) has a larger relative change than the ${\nu }_e$ (thick red curves). Plotted for comparison is the relative difference for ${\nu }_e$ in a $L_{\nu }^{\star }=0$ scenario (blue dash-dot curves). The relative differences are plotted for three values of $\epsilon$, from bottom to top: $\epsilon =3$, 5, 7.

Figure 5

Relative differences in neutrino/antineutrino occupation numbers plotted against $\epsilon$ at $T_{\mathrm{cm}}=1\mathrm{keV}$. The relative differences are with respect to FD with the same corresponding degeneracy parameters as Fig. 3. The solid lines are for a scenario where $L_{\nu }^{\star }=0.1$. Plotted for comparison is the relative difference for ${\nu }_e$ (blue dash-dot curve) and for ${\nu }_{\mu }$ (blue dotted curve) in a $L_{\nu }^{\star }=0$ scenario.

Figure 6

Absolute change in the neutrino/antineutrino energy distributions plotted against $\epsilon$ at $T_{\mathrm{cm}}=1\mathrm{keV}$. The changes are with respect to the same degeneracy parameters as those in Fig. 5. Furthermore, the line colors and styles correspond to the same species and scenarios as Fig. 5.

Figure 7

Absolute change in the neutrino/antineutrino number distributions plotted against $\epsilon$ at $T_{\mathrm{cm}}=1.0\mathrm{keV}$. The changes are with respect to the same degeneracy parameters, and the nomenclature of line colors and line styles is the same as those in Fig. 5. The numbers given on the plot ($\delta L_i^{\star }$ for $i=e$, $\mu$) show the relative error accumulated over the course of a simulation.

Figure 8

Relative differences in neutrino occupation numbers plotted against $\epsilon$ at $T_{\mathrm{cm}}=1\mathrm{keV}$ for various values of $L_{\nu }^{\star }$. The relative differences are with respect to FD with the corresponding degeneracy parameter, namely, $\xi =0.1458$ ($L_{\nu }^{\star }={10}^{-1}$), $\xi =0.0730$ ($L_{\nu }^{\star }=5\times {10}^{-2}$), $\xi =0.0146$ ($L_{\nu }^{\star }={10}^{-2}$), and $\xi =0$ ($L_{\nu }^{\star }=0$). Shown are two sets of curves: the set with the larger relative differences correspond to the ${\nu }_e$ spectral distortions, and the set with the smaller differences are ${\nu }_{\mu }$. Within each set, the increase in $L_{\nu }^{\star }$ leads to a decrease in $\delta f_{\xi }$. For the antineutrinos, the relative differences behave in the opposite manner: increase in $L_{\nu }^{\star }$ leads to an increase in $\delta f_{\xi }$.

Figure 9

Absolute change in the neutrino/antineutrino number distributions plotted against $\epsilon$ at $T_{\mathrm{cm}}=1.0\mathrm{keV}$. The numbers given on the plot ($\delta L_i^{\star }$ for $i=e$, $\mu$) show the relative error accumulated over the course of a simulation.

Figure 10

Absolute change in the neutrino/antineutrino number distributions plotted against $\epsilon$ at $T_{\mathrm{cm}}=1.0\mathrm{keV}$. The numbers given on the plot ($\delta L_i^{\star }$ for $i=e$, $\mu$) show the relative error accumulated over the course of a simulation.

Figure 11

The relative changes in $R$ plotted against $T_{\mathrm{cm}}$ for ${\nu }_e$ and ${\nu }_{\mu }$ with different processes included in the transport calculation. Red lines correspond to ${\nu }_e$ and green lines correspond to ${\nu }_{\mu }$. The process scheme is all processes (solid curves), annihilation only (dash-dot curves), or elastic scattering only (dotted curves).

Figure 12

The relative changes in $\mathrm{\Sigma }$ plotted against $T_{\mathrm{cm}}$. Line colors and styles correspond to the same transport processes and neutrino flavors in Fig. 11.

Figure 13

Same as Fig. 12 except zoomed in on the solid curves.

Figure 14

Relative changes in the primordial abundances plotted against the absolute value of the comoving lepton number. Positive changes in abundances correspond to negative comoving lepton numbers, and negative changes correspond to positive comoving lepton numbers. The solid lines use the dark-radiation model described in the text. Individual points using the full Boltzmann-transport calculation are plotted for three decades of $L_{\nu }^{\star }$. Squares correspond to $L_{\nu }^{\star }>0$, and circles for $L_{\nu }^{\star }<0$. The baryon density is fixed to be ${\omega }_b=0.022068$ (equivalent to a baryon-to-photon ratio $n_b/n_{\gamma }=6.0747\times {10}^{-10}$) for all calculations in both scenarios [47]. The mean neutron lifetime is taken to be 885.7 s.