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We revisit several aspects of Standard Model physics at finite temperaturethat drive the theoretical value of the cosmological parameter $N_{\rm eff}$,the effective number of neutrinos in the early universe, away from 3. Our chieffocus is finite-temperature corrections to the equation of state of the QEDplasma in the vicinity of neutrino decoupling at $T \sim 1$ MeV, where $T$ isthe photon temperature. Working in the instantaneous decoupling approximation,we recover at ${\cal O}(e^2)$, where $e$ is the elementary electric charge, thewell-established correction of $\delta N_{\rm eff}^{(2)} \simeq 0.010$ across arange of plausible neutrino decoupling temperatures, in contrast to anerroneous 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 $\delta N_{\rmeff}^{(3)} \simeq -0.001$ that has so far not been accounted for in anyprecision calculation of $N_{\rm eff}$, significant because this correction ispotentially larger than the change in $N_{\rm eff}$ induced between includingand excluding neutrino oscillations in the transport modelling. In addition tothe QED equation of state, we make a first pass at quantifyingfinite-temperature QED corrections to the weak interaction rates that directlyaffect the neutrino decoupling process, and find that the ${\cal O}(e^2)$thermal electron mass correction induces a change of $\delta N_{\rmeff}^{m_{\rm th}} \lesssim 10^{-4}$. A complete assessment of the variouseffects considered in this work on the final value of $N_{\rm eff}$ willnecessitate an account of neutrino energy transport beyond the instantaneousdecoupling approximation. However, relative to $N_{\rm eff} = 3.044$ obtainedin the most recent such calculation, we expect the new effects found in thiswork to lower the number to $N_{\rm eff} = 3.043$.

Towards a precision calculation of the effective number of neutrinos $N_{\rm eff}$ in the Standard Model I: The QED equation of state