Towards a precision calculation of the effective number of neutrinos N_eff in the Standard Model: the QED equation of state

We revisit and quantify in this work several aspects of Standard Model physics at finite temperature that drive the theoretical value of the cosmological parameter, the effective number of neutrinos $N_{eff}$, away from 3 in the early universe. Our chief focus is finite-temperature corrections to the equation of state of the QED plasma in the vicinity of neutrino decoupling at $T\sim 1$ MeV, where $T$ is the photon temperature. Working in the instantaneous decoupling approximation, we recover at $O\left(e^2\right)$, where $e$ is the elementary electric charge, the well-established correction of $\delta N_{eff}^{\left(2\right)}\simeq 0.010$ across a range of plausible neutrino decoupling temperatures, in contrast to an erroneous claim in the recent literature which found twice as large an effect. At $O\left(e^3\right)$ we find a new and significant correction of $\delta N_{eff}^{\left(3\right)}\simeq -0.001$ that has so far not been accounted for in any precision neutrino decoupling calculation of $N_{eff}$, significant because this correction is in fact larger than—or at least comparable to—the change in $N_{eff}$ induced between including and excluding neutrino oscillations in the transport modelling. In addition to the QED equation of state, we make a first pass at quantifying finite-temperature QED corrections to the weak interaction rates that directly affect the neutrino decoupling process, and find in this connection that the $O\left(e^2\right)$ thermal electron mass correction induces a change of $\delta N_{eff}^{m_{th}}\lesssim {10}^{-4}$. A complete assessment of the various effects considered in this work on the final value of $N_{eff}$ will necessitate an account of neutrino energy transport beyond the instantaneous decoupling approximation. However, relative to $N_{eff}=3.044$ obtained in the most recent such calculation, we expect the new effects found in this work to lower the number to $N_{eff}=3.043$.