Fast and Flexible Neutrino Decoupling Part I: The Standard Model

Abstract

Cosmological determinations of the number of relativistic neutrino species, $N^{ }_{\rm eff}$, are becoming increasingly accurate, and further improvements are expected both from CMB and BBN data. Given this context, we update the evaluation of $N^{ }_{\rm eff}$ and the current entropy density via the momentum-averaged approach. This allows for a numerically fast description of neutrino decoupling, easily portable to an array of new physics scenarios. We revisit all aspects of this approach, including collision terms with full electron mass dependence, finite temperature QED corrections to the equation of state, neutrino oscillations, and the modelling of neutrino ensembles with effective chemical potentials. For integrated observables, our results differ by less than $0.04\%$ from the solution of the momentum-dependent evolution equation. We outline how to extend the approach to BSM settings, and will highlight its power in Part II. To facilitate the practical implementation, we release a Mathematica and Python code within nudec_BSM_v2, easily linkable to BBN codes.

Figures (4)

• Figure 1: The MB correction factors for the energy density and number density transfer rates, determined by matching to the linear response regime. These correct eqs. (\ref{['Q^MB']}) and (\ref{['J^MB']}) as laid out by the replacements in eqs. (\ref{['newMB1']})--(\ref{['newMB3']}).
• Figure 2: Evolution of the neutrino temperature and (effective) chemical potential in the Standard Model, as a function of $T_\gamma^{ }$. In this plot, the "fast oscillation" case from eq. (\ref{['varrho']}) is assumed, see table \ref{['tab:SM_summary']} for the corresponding cosmological parameters. The increase in the ratio $T_\gamma^{ }/T_\nu^{ }$ with time is chiefly due to heating from $e^+_{ } e^-_{ } \to 2\gamma\,$, while the deviation in the parameter $\mu_\nu^{ }$ from zero tracks the spectral distortions in the neutrino momentum distribution.
• Figure 3: The difference of the (weighted) neutrino distribution function between our results and Fortepiano Bennett:2020zkv. The panels display the same information in different ways, moving from left to right: the differential energy density (${\rm d}N^{ }_{\hbox{\scriptsize eff}} \equiv \frac{120}{7\pi^4} (\frac{11}{4})^{4/3} \frac{ y^3\, {\rm d}y}{z^4} f^{ }_{\nu_\alpha}$, cf. eqs. (\ref{['e_fit']}) and (\ref{['def_Neff_alt']})); the differential number density (${\rm d}\Omega^{ }_\nu / \Omega_\nu^{(0)} \equiv \frac{2}{3\zeta(3)} \frac{11}{4} \frac{ y^2 \, {\rm d}y }{z^3} f^{ }_{\nu_\alpha}\,$, cf. eqs. (\ref{['Omega_nu']}), (\ref{['n_fit']}), and table \ref{['thermo_massless']}); and the momentum distribution itself. On the $x$-axis, $a$ is normalized to $1/T_\nu$ at the initial temperature, i.e. $a \equiv a/(a T^{ }_\nu)^{ }_{\hbox{\tiny\rm{ini}}}$. In the nomenclature of table \ref{['tab:SM_summary']}, we show the comparison with oscillations, having included the neutrino chemical potentials. In the left and middle panels, the spectral distortions are $\lesssim 0.03\%$ across comoving neutrino momenta.
• Figure 4: Comparison of the neutrino distribution function between our results and those obtained in Fortepiano Bennett:2020zkv. The four upper panels show the relative difference in terms of the neutrino energy density and the four lower ones for the number density. We can see that the spectral distortions are quite small, $< 8\times 10^{-4}$. The left panels correspond to solutions with $\mu_\nu = 0$, and the right ones with $\mu_\nu \neq 0$. Including the chemical potentials brings the differences to $< 4\times 10^{-4}$.