A precision calculation of relic neutrino decoupling

TL;DR

This work presents a precision calculation of relic neutrino decoupling by solving the full density-matrix Boltzmann equations for three-flavor neutrinos, including oscillations and finite-temperature QED corrections up to next-to-leading order. It analyzes both flavor- and mass-basis formulations, demonstrating consistent results with N_eff = 3.044 and quantifying tiny shifts due to neutrino mixing and QED corrections. The study provides momentum-dependent spectral distortions and a simple transformation between flavor and mass distributions, informing future cosmological constraints and relic-neutrino detection efforts. Overall, it establishes a robust framework for precision neutrino cosmology relevant to upcoming CMB-LSS measurements.

Abstract

We study the distortions of equilibrium spectra of relic neutrinos due to the interactions with electrons, positrons, and neutrinos in the early Universe. We solve the integro-differential kinetic equations for the neutrino density matrix, including three-flavor oscillations and finite temperature corrections from QED up to the next-to-leading order $\mathcal{O}(e^3)$ for the first time. In addition, the equivalent kinetic equations in the mass basis of neutrinos are directly solved, and we numerically evaluate the distortions of the neutrino spectra in the mass basis as well, which can be easily extrapolated into those for non-relativistic neutrinos in the current Universe. In both bases, we find the same value of the effective number of neutrinos, $N_{\rm eff} = 3.044$, which parameterizes the total neutrino energy density. The estimated error for the value of $N_{\rm eff}$ due to the numerical calculations and the choice of neutrino mixing parameters would be at most 0.0005.

Figures (5)

• Figure 1: The time evolution of the comoving photon temperature $z(x)$ as a function of the normalized scale factor $x=m_ea$ in the case both with neutrino mixing and QED finite temperature corrections up to $\mathcal{O}(e^3)$ .
• Figure 2: The time evolution of the distortions of flavor neutrinos for a fixed momentum ($y=5$) as a function of the normalized scale factor $x=m_ea$ in the case with QED finite temperature corrections up to $\mathcal{O}(e^3)$. Upper (lower) dotted line is for $\nu_e\ (\nu_{\mu,\tau})$ without neutrino oscillations. Inner three lines are for flavor neutrinos with neutrino oscillations.
• Figure 3: The final distortions of flavor neutrino spectra as a function of the comoving momentum $y$ in the case with QED finite temperature corrections up to $\mathcal{O}(e^3)$. Dotted lines represent those for $\nu_e\ (\nu_{\mu,\tau})$ without neutrino oscillations, while solid and dashed lines represent those for flavor neutrinos with neutrino oscillations.
• Figure 4: The time evolution of the distortions of massive neutrinos for a fixed momentum ($y=5$) in the case with QED finite temperature corrections up to $\mathcal{O}(e^3)$.
• Figure 5: The final distortions of massive neutrino spectra as a function of the comoving momentum $y$ in the case with QED finite temperature corrections up to $\mathcal{O}(e^3)$.