Lepton Number and Expansion of the Universe

We show that the non-integer effective number of neutrinos $N^{\mathrm{eff}}_\nu$ can be understood as an effect of lepton $L$ asymmetry in the early Universe carried by the Dirac neutrino cosmic background. We show that $N_\nu^{\mathrm{eff}}=3.36\pm0.34$ (CMB only) and $N_\nu^{\mathrm{eff}}= 3.62\pm0.25$ (CMB and $H_0$) require a ratio between baryon number $B$ and lepton number to be $1.16 \times 10^{-9}\leqslant B/|L|\leqslant 1.51 \times 10^{-9}$. These values are close to the baryon-to-photon ratio $0.57\times 10^{-9}\leqslant B/N_\gamma \leqslant 0.67\times10^{-9}$. Thus instead of the usual $|L|\ll N_\gamma$ and $B\simeq |L|$, we propose to use $0.4 \leqslant |L|/N_\gamma\leqslant 0.52$ and $B\ll|L|$ as another natural choice, which resolves the tension between Planck-CMB and $H_0$ and leads to a non-integer value of $N_\nu^{\mathrm{eff}}>3$.