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

TL;DR

This paper refines the theoretical calculation of the effective number of neutrinos, $N_{ m eff}$, in the early universe by including higher-order finite-temperature QED corrections to the equation of state and weak interaction rates, achieving a more precise value.

Contribution

It provides the first detailed calculation of ${ m O}(e^3)$ corrections to $N_{ m eff}$ and assesses finite-temperature effects on weak interaction rates, improving the precision of cosmological neutrino predictions.

Findings

Reconfirmed the ${ m O}(e^2)$ correction of about 0.010 to $N_{ m eff}$.

Discovered a new ${ m O}(e^3)$ correction of approximately -0.001 to $N_{ m eff}$.

Estimated that these effects slightly lower $N_{ m eff}$ from 3.044 to about 3.043.

Abstract

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 $N_{\rm eff} = 3.043$.