Towards a precision calculation of $N_{\mathrm{e}\mathrm{f}\mathrm{f}}$ in the Standard Model II: Neutrino decoupling in the presence of flavour oscillations and finite-temperature QED

We present in this work a new calculation of the standard-model benchmark value for the effective number of neutrinos, $N_{\mathrm{e}\mathrm{f}\mathrm{f}}^{\mathrm{S}\mathrm{M}}$, that quantifies the cosmological neutrino-to-photon energy densities. The calculation takes into account neutrino flavour oscillations, finite-temperature effects in the quantum electrodynamics plasma to ${\cal O}(e^3)$,where$e$ is the elementary electric charge, and a full evaluation of the neutrino–neutrino collision integral. We provide furthermore a detailed assessment of the uncertainties in the benchmark $N_{\mathrm{e}\mathrm{f}\mathrm{f}}^{\mathrm{S}\mathrm{M}}$ value, through testing the value’s dependence on (i)~optional approximate modelling of the weak collision integrals, (ii)~measurement errors in the physical parameters of the weak sector, and (iii)~numerical convergence, particularly in relation to momentum discretisation. Our new, recommended standard-model benchmark is $N_{\mathrm{e}\mathrm{f}\mathrm{f}}^{\mathrm{S}\mathrm{M}}=3.0440\pm 0.0002$, where the nominal uncertainty is attributed predominantly to errors incurred in the numerical solution procedure ($\mathrm{\mid }\delta N_{\mathrm{e}\mathrm{f}\mathrm{f}}\mathrm{\mid }\sim {10}^{-4}$), augmented by measurement errors in the solar mixing angle ${\sin }^2{\theta }_{12}$ ($|\delta N_{\rm eff}| \sim10^{-4}$).