Active-sterile neutrino oscillations in the early Universe with full collision terms

TL;DR

This work advances the cosmological treatment of active–sterile neutrino oscillations by implementing the full collision term in momentum-dependent quantum kinetic equations, including nonzero electron mass and Pauli blocking. It demonstrates that common approximations—namely the equilibrium and Chu–Cirelli schemes—can introduce biases in the predicted sterile-neutrino production and in the resulting $\Delta N_{ m eff}$, especially at low conversion temperatures. The authors introduce the A/S approximation, which separates annihilation and scattering contributions to repopulation and incorporates Pauli blocking into damping, achieving near-full-term accuracy ($\sim10^{-3}$) across most of the parameter space, while remaining computationally tractable. These results sharpen cosmological constraints on eV-mass sterile neutrinos and provide a robust framework for interpreting upcoming precision measurements of $N_{ m eff}$ and related observables in the early Universe.

Abstract

Sterile neutrinos are thermalised in the early Universe via oscillations with the active neutrinos for certain mixing parameters. The most detailed calculation of this thermalisation process involves the solution of the momentum-dependent quantum kinetic equations, which track the evolution of the neutrino phase space distributions. Until now the collision terms in the quantum kinetic equations have always been approximated using equilibrium distributions, but this approximation has never been checked numerically. In this work we revisit the sterile neutrino thermalisation calculation using the full collision term, and compare the results with various existing approximations in the literature. We find a better agreement than would naively be expected, but also identify some issues with these approximations that have not been appreciated previously. These include an unphysical production of neutrinos via scattering and the importance of redistributing momentum through scattering, as well as details of Pauli blocking. Finally, we devise a new approximation scheme, which improves upon some of the shortcomings of previous schemes.

Figures (7)

• Figure 1: Convergence test using the full collision term. We compare the difference in the neutrino number and energy densities between using 200 momentum bins and 40 (blue long dashed), 80 (red dot-dashed), 100 (green dashed) and 150 bins (cyan dotted).
• Figure 2: Different contributions to the collision terms and their effects on the neutrino number and energy densities relative to the no-oscillations case. The red solid and dot-dash lines denote a scenario with only scattering and no annihilation. The blue dashed and dotted lines represent scattering only amongst the oscillating active neutrinos $\nu_\alpha$; the thick black dashed and dotted lines denote the same scenario, but now computed using 150 momentum bins instead of the canonical 100.
• Figure 3: Comparison of the full treatment (black solid lines) with the equilibrium approximation (red dot-short dash), the CC approximation (blue long dashed), and the A/S approximation (purple dot-long dash) introduced in this work. Setting $m_e=0$ (cyan dotted) has almost no effect, while ignoring Pauli blocking in the full expression (green dashed) gives almost the same result as the equilibrium approximation.
• Figure 4: The active and sterile neutrino momentum distributions computed from the full treatment (black solid lines), the equilibrium approximation (red fit-short dash), the CC approximation (blue dashed), and the A/S approximation (purple dot-long dash) for the benchmark mixing parameters $\delta m^2 = 0.1\electronvolt^2$ and $\sin^2(2\theta) = 0.025$ at three different temperatures. The distributions have been normalised to the relativistic Fermi--Dirac distribution $f_0$.
• Figure 5: The active momentum distributions at $T=10$ MeV calculated using a collision term incorporating only scattering processes ( top) and only annihilations ( bottom). The difference between the full treatment (black solid) and the A/S approximation (purple long-dot dash) is much larger for the scatterings than for the annihilations. We have used the benchmark mixing parameters $\delta m^2=0.1\electronvolt^2$ and $\sin^2(2\theta)=0.025$, and the distributions have been normalised to the relativistic Fermi--Dirac distribution $f_0$.