Thermalisation of sterile neutrinos in the early Universe in the 3+1 scheme with full mixing matrix

Abstract: In the framework of a 3+1 scheme with an additional inert state, we consider the thermalisation of sterile neutrinos in the early Universe taking into account the full $4\times4$ mixing matrix. The evolution of the neutrino energy distributions is found solving the momentum-dependent kinetic equations with full diagonal collision terms, as in previous analyses of flavour neutrino decoupling in the standard case. The degree of thermalisation of the sterile state is shown in terms of the effective number of neutrinos, $N_{\rm eff}$, and its dependence on the three additional mixing angles ($\theta_{14}$, $\theta_{24}$, $\theta_{34}$) and on the squared mass difference $\Delta m^2_{41}$ is discussed. Our results are relevant for fixing the contribution of a fourth light neutrino species to the cosmological energy density, whose value is very well constrained by the final Planck analysis. For the preferred region of active-sterile mixing parameters from short-baseline neutrino experiments, we find that the fourth state is fully thermalised ($N_{\rm eff}\simeq 4$).

Thermalisation of Sterile Neutrinos in the 3+1 Scheme

This paper provides an extensive analysis of sterile neutrino thermalisation in the early Universe within the 3+1 framework, considering the full mixing matrix to incorporate the interactions between active and sterile neutrino states. The authors have employed a numerical solution approach to solve the momentum-dependent kinetic equations, emphasizing full diagonal collision terms akin to standard flavor neutrino decoupling models. The study introduces a novel computational tool, FortEPiaNO, to facilitate these calculations, further extending it to accommodate up to six neutrinos, although primarily focusing on a four-neutrino system for this analysis.

The primary contribution of the paper lies in the detailed examination of the active-sterile mixing parameters' effects, particularly considering the mixing angles $\theta_{14}$, $\theta_{24}$, and $\theta_{34}$ and their influence on the degree of thermalisation expressed in terms of the effective number of neutrinos, $N_{\rm eff}$. The authors report that for the active-sterile mixing parameters favored by short-baseline experiments, the fourth neutrino state achieves full thermalisation at an $N_{\rm eff} \simeq 4$.

Key findings from the paper hinge on the comparison of different mixing angles' impacts on the thermalisation process. The study finds that $\theta_{14}$ generally results in less effective sterile neutrino production compared to $\theta_{24}$ and $\theta_{34}$. This suggests that a larger value of $\theta_{14}$ is required to achieve similar $N_{\rm eff}$ outcomes as with $\theta_{24}$ or $\theta_{34}$, due to the differing active-sterile interactions and, importantly, the presence of the surrounding cosmic plasma's matter potential, which can delay the onset of $\nu_e \rightarrow \nu_s$ oscillations.

The research underscores a potential tension between cosmological observations and laboratory experiments concerning sterile neutrinos. While short-baseline neutrino experiments suggest significant active-sterile mixing, cosmological data, particularly from the Planck satellite, limits the contribution to $N_{\rm eff}$ to a maximum of around 3.3, conflicting with the fully thermalised state anticipated from mixing parameter values preferred by terrestrial experiments.

The implications of these results suggest that if the existence of a light sterile neutrino mixed with active neutrinos is confirmed by both experiments and observations, an additional mechanism must be introduced to suppress the sterile state’s thermalisation during the early Universe. Such mechanisms could include the influence of a substantial neutrino-antineutrino asymmetry or potentially undiscovered secret interactions involving neutrinos.

In summary, this paper expands our understanding of neutrino mixing in cosmological contexts and highlights the complexity and challenges of reconciling experimental and observational data on neutrinos. Future developments, potentially encompassing new experimental insights or more complex cosmological models, could offer resolutions to the current discrepancies in $N_{\rm eff}$ resulting from sterile neutrinos in the 3+1 scheme.