Relic neutrino decoupling with flavour oscillations revisited

TL;DR

This work revisits relic neutrino decoupling in the early Universe by solving momentum-dependent Boltzmann equations for the three-flavour neutrino density matrix with full collision integrals for both diagonal and off-diagonal components. The analysis shows that three-flavour oscillations modestly modify spectral distortions but do not alter the total radiation energy density, yielding $N_{ m eff}=3.045$ independent of mass ordering. Non-standard neutrino-electron interactions can shift $N_{ m eff}$ to $3.040$ or $3.059$, depending on the NSI parameters. The results reinforce the robustness of the standard $N_{ m eff}$ prediction while highlighting small spectral effects and the potential for NSI to leave subtle imprints on future cosmological measurements.

Abstract

We study the decoupling process of neutrinos in the early universe in the presence of three-flavour oscillations. The evolution of the neutrino spectra is found by solving the corresponding momentum-dependent kinetic equations for the neutrino density matrix, including for the first time the proper collision integrals for both diagonal and off-diagonal elements. This improved calculation modifies the evolution of the off-diagonal elements of the neutrino density matrix and changes the deviation from equilibrium of the frozen neutrino spectra. However, it does not vary the contribution of neutrinos to the cosmological energy density in the form of radiation, usually expressed in terms of the effective number of neutrinos, N_eff. We find a value of N_eff=3.045, in agreement with previous theoretical calculations and consistent with the latest analysis of Planck data. This result does not depend on the ordering of neutrino masses. We also consider the effect of non-standard neutrino-electron interactions (NSI), predicted in many theoretical models where neutrinos acquire mass. For two sets of NSI parameters allowed by present data, we find that N_eff can be reduced down to 3.040 or enhanced up to 3.059.

Figures (6)

• Figure 1: Evolution of the distortions of the neutrino spectra for the dimensionless momentum ${y=5}$ with standard neutrino interactions, as a function of $x$ or the photon temperature. Outer (inner) lines correspond to the case with no neutrino mixing (with oscillations and masses in the NH case). The upper two lines correspond to electron neutrinos and the lower lines to muon and tau neutrinos (slightly different in the case with oscillations).
• Figure 2: Collision terms for the real parts of the off-diagonal elements of the neutrino density matrix in eq. \ref{['densityMatrix']}: $a_1$ (full lines), $b_1$ (dashed lines), and $c_1$ (dotted lines). They are shown for a dimensionless momentum ${y=5}$ and were calculated with the full expression (black lines) or with damping functions (grey lines), accounting only for the processes involving the interactions of neutrinos and $e^\pm$.
• Figure 3: Final distortions of the flavour neutrino spectra as a function of the comoving momentum. Outer (inner) lines correspond to the case with no neutrino mixing (with oscillations and masses in the NH case). The IH case overlaps the NH one at this scale.
• Figure 4: Same as Figure \ref{['fig:delta_alpha']} for the mass neutrino states and NH. The spectral distortions of the mass states for the IH case overlap with those for NH at this scale.
• Figure 5: Same as Figure \ref{['fig:evol_i']} for the two cases with non-standard neutrino interactions, including flavour oscillations and masses in the NH case.