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

TL;DR

This work delivers a state-of-the-art calculation of the SM benchmark value for the effective number of neutrinos, N_{\rm eff}^{\rm SM}, by integrating neutrino flavour oscillations, finite-temperature QED corrections to ${\cal O}(e^3)$, and a full neutrino–neutrino collision integral into a precision decoupling framework. The authors quantify uncertainties from modelling choices, oscillation parameters, and numerical convergence, and demonstrate that including the full collision integral and the ${\cal O}(e^3)$ QED corrections shifts N_{\rm eff}^{\rm SM} to 3.0440 with a tight ±0.0002 error. They also show that the dominant uncertainty arises from numerical solution, with a subdominant contribution from the solar mixing angle sin^2 θ_{12}. The results align with recent independent calculations and refine the SM baseline used to constrain beyond-Standard-Model light relics via cosmological measurements.

Abstract

We present in this work a new calculation of the standard-model benchmark value for the effective number of neutrinos, $N_{\rm eff}^{\rm SM}$, 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 $N_{\rm eff}^{\rm SM}$ 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 $N_{\rm eff}^{\rm SM} = 3.0440 \pm 0.0002$, where the nominal uncertainty is attributed predominantly to errors incurred in the numerical solution procedure ($|δN_{\rm eff}| \sim10^{-4}$), augmented by measurement errors in the solar mixing angle $\sin^2θ_{12}$ ($|δN_{\rm eff}| \sim10^{-4}$).

Figures (7)

• Figure 1: Diagrammatic expression for the ${\cal O}(e^2)$ and ${\cal O}(e^3)$ corrections to the finite-temperature QED partition function. The filled black circles represent one-particle irreducible photon self-energies at finite temperature.
• Figure 2:
• Figure 3: Change in $N_{\rm eff}^{\rm SM}$ in a full calculation including the complete neutrino--neutrino collision integral, with respect to variations in the number of $y$-nodes and their range. Top: Variations in $N_y$ and $y_{\rm min}$ using the Newton--Cotes quadrature method. Bottom left: Variations in $N_y$ and $y_{\rm max}$ using the Newton--Cotes quadrature method. Bottom right: Variations in $N_y$ and $y_{\rm max}$ using the Gauss--Laguerre quadrature method. The setting used in our benchmark full calculation including the complete neutrino--neutrino collision integral ($y_{\rm min}=0.01$ where applicable, $y_{\rm max}=30$, and $N_y = 80$) is marked by a red dot on each plot.
• Figure 4: Change in $N_{\rm eff}^{\rm SM}$ with respect to variations in the initialisation time $x_{\rm in}$. The default choice is $x_{\rm in} = 0.001$. The solid red and blue lines denote the full calculation including the full neutrino--neutrino collision term using respectively the GL and the NC methods, while the green dotted line represents the minimum set-up computed with NC.
• Figure 5: Left: Fractional change in percent in the comoving energy density relative to the initial state at $x_{\rm in}=0.001$, under variations of the quadrature method (Gauss--Laguerre vs Newton--Cotes), the number of $y$-nodes $N_y$, and modelling of the collision integral ${\cal I}[\varrho]$. Right: Fractional change in percent in the comoving number density.