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Sound Speed and Viscosity of Semi-Relativistic Relic Neutrinos

Generalized fluid equations, using sound speed $c_{\rm eff}^2$ and viscosity $c_{\rm vis}^2$ as effective parameters, provide a convenient phenomenological formalism for testing the relic neutrino "null hypothesis," i.e. that that neutrinos are relativistic and free-streaming prior to recombination. In this work, we relax the relativistic assumption and ask "to what extent can the generalized fluid equations accommodate finite neutrino mass?" We consider both the mass of active neutrinos, which are largely still relativistic at recombination $m^2 / T^2 \sim 0.2$, and the effect of a semi-relativistic sterile component. While there is no one-to-one mapping between mass/mixing parameters and $c_{\rm eff}^2$ and $c_{\rm vis}^2$, we demonstrate that the existence of a neutrino mass could induce a bias to measurements of $c_{\rm eff}^2$ and $c_{\rm vis}^2$ at the level of $0.01 m^2 / T^2 \sim 10^{-3}$.