Nonequilibrium neutrino statistical mechanics in the expanding Universe

We study neutrino decoupling in the early Universe ($t\sim sec,T\sim \mathrm{MeV}$) by integrating the Boltzmann equations that govern the neutrino phase-space distribution functions. In particular, we compute the distortions in the ${\nu }_e$ and $\frac{{\nu }_{\mu }}{{\nu }_{\tau }}$ phase-space distributions that arise in the standard cosmology due to $e^{\pm }$ annihilations. These distortions are nonthermal, with the effective neutrino temperature increasing with neutrino momentum, approaching a 0.7% increase for electron neutrinos and a 0.3% increase for $\mu$ and $\tau$ neutrinos at the highest neutrino momenta, and correspond to an increase in the energy density of ${\nu }_e\text{'}\mathrm{s}$ of about 1.2% and in the energy density of $\frac{{\nu }_{\mu }}{{\nu }_{\tau }\text{'}\mathrm{s}}$ of about 0.5% (roughly one additional relic neutrino per ${\mathrm{cm}}^{-3\mathrm{}}$ per species). The distortion for electron neutrinos is larger than that for $\mu$ and $\tau$ neutrinos because electron neutrinos couple to $e^{\pm }\text{'}\mathrm{s}$ through both charged- and neutral-current interactions. Our results graphically illustrate that neutrino decoupling is a continuous process which is momentum dependent. Because of subtle cancellations, these distortions lead to only a tiny change in the predicted primordial ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ abundance, $\Delta Y\simeq 1-2\times {10}^{-4\mathrm{}}$.