A Limit on the Total Lepton Number in the Universe from BBN and the CMB

Abstract

At temperatures below the QCD phase transition, any substantial lepton number in the Universe can only be present within the neutrino sector. In this work, we systematically explore the impact of a non-vanishing lepton number on Big Bang Nucleosynthesis (BBN) and the Cosmic Microwave Background (CMB). Relying on our recently developed framework based on momentum averaged quantum kinetic equations for the neutrino density matrix, we solve the full BBN reaction network to obtain the abundances of primordial elements. We find that the maximal primordial total lepton number $L$ allowed by BBN and the CMB is $-0.12 \,(-0.10) \leq L \leq 0.13\,(0.12) $ for NH (IH), while specific flavor directions can be even more constrained. This bound is complementary to the limits obtained from avoiding baryon overproduction through sphaleron processes at the electroweak phase transition since, although numerically weaker, it applies at lower temperatures and is obtained completely independently. We publicly release the C++ code COFLASY-C on GitHub which solves for the evolution of the neutrino quantum kinetic equations numerically.

Figures (7)

• Figure 1: Typical example of the time evolution for large amplitude of initial flavor asymmetries ($\xi_e= -0.5, \xi_\mu= 0.75, \xi_\tau=0.25$) in NH (solid) and IH (dashed). The full solution depicted here essentially overlaps with the adiabatic approximation $(V_s =0)$Domcke:2025lzg.
• Figure 2: Impact of neutrino asymmetries on the evolution of the neutron fraction in the plasma for the same initial conditions as in Fig. \ref{['fig:time-evolution']}.
• Figure 3: Prediction of the primordial element abundances within the SM (red dashed) compared to a scenario with non-vanishing neutrino chemical potential (black solid). Specifically we choose NH and the same initial conditions are in Fig. \ref{['fig:time-evolution']} and Fig. \ref{['fig:Xn-example']}. The IH result is very similar. We use $\eta_B = 6.109\times 10^{-10}$ and $\tau_n = 878.4\,{\rm s}$ for the BBN network. Note that the helium mass fraction is $4\times$ the number fraction shown here.
• Figure 4: Constraints on primordial lepton flavor asymmetries for increasing total lepton number (from left to right) for normal (upper) and inverted (lower) hierarchy. The black region is the currently allowed region at 95% CL, $\Delta \chi^2 < 5.99$. The dashed red contours show the improvement expected for the upcoming sensitivity of $\Delta N_{\mathrm{eff}} = 0.1$ centered around the SM value $N_\text{eff} = 3.044$, with the corresponding total lepton number excluded when the black region no longer intersects with the dashed red contour. In the case of $L=0$ we further highlight with a yellow star the scenario of successful generation of the observed baryon asymmetry via tauphobic leptoflavorgenesis, which currently cannot be excluded, see also our previous analysis Domcke:2025lzg.
• Figure 5: Sensitivity to the primordial lepton flavor asymmetries for increasing total lepton number (from left to right) for normal (upper) and inverted (lower) hierarchy. The light black region is the $\Delta \chi^2 < 5.99$ would be the allowed region assuming a future sensitivity on $Y_P$ of 0.5% and on $N_{\mathrm{eff}}$ of 0.05 as described in Sec. \ref{['sec:res']}. The red lines are contours of $\Delta N_{\mathrm{eff}}$ with solid, dashed and dash-dotted corresponding respectively to $\Delta N_{\mathrm{eff}} = 0.17, 0.1, 0.05$, while the solid (dashed) cyan line represents $\Delta Y_P/Y_{\rm P} = \pm 1.2 \,(\pm 0.5)\%$ centered around the Standard Model value $Y_{\rm P} = 0.247$. See main text for details.