Towards a precision calculation of the effective number of neutrinos
  $N_{\rm eff}$ in the Standard Model I: The QED equation of state
  We revisit several aspects of Standard Model physics at finite temperature
that drive the theoretical value of the cosmological parameter $N_{\rm eff}$,
the effective number of neutrinos in the early universe, away from 3. 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 ${\cal O}(e^2)$, where $e$ is the elementary electric charge, the
well-established correction of $\delta N_{\rm eff}^{(2)} \simeq 0.010$ 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 ${\cal O}(e^3)$ we find a new and significant correction of $\delta N_{\rm
eff}^{(3)} \simeq -0.001$ that has so far not been accounted for in any
precision calculation of $N_{\rm eff}$, significant because this correction is
potentially larger than the change in $N_{\rm eff}$ 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 that the ${\cal O}(e^2)$
thermal electron mass correction induces a change of $\delta N_{\rm
eff}^{m_{\rm th}} \lesssim 10^{-4}$. A complete assessment of the various
effects considered in this work on the final value of $N_{\rm eff}$ will
necessitate an account of neutrino energy transport beyond the instantaneous
decoupling approximation. However, relative to $N_{\rm eff} = 3.044$ obtained
in the most recent such calculation, we expect the new effects found in this
work to lower the number to $N_{\rm eff} = 3.043$.
