Consistent treatment of flavour-specific neutrino self-interactions in cosmology

We study the impact of flavour-specific neutrino self-interactions on the CMB anisotropy spectrum and consistently take into account neutrino mixing. Starting from the neutrino quantum kinetic equations, we derive the neutrino Boltzmann hierarchies for the neutrino mass states. Our work shows that most scenarios of flavour-specific neutrino self-interactions are well approximated by a combined fluid approach. In practice, this implies that existing CMB bounds on the effective neutrino coupling are weakened by a factor $\sqrt{2}$ for scenarios in which only two neutrino flavours interact (irrespective of which flavours). If only one neutrino flavour interacts the bounds are weaker by a factor $\sqrt{6}$, with the only caveat that the combined fluid approach is less reliable for only ${\nu }_e$ interacting.