Constraining tau-sterile neutrino mixing with high-energy astrophysical neutrinos

Whether a sterile neutrino mixes with the three active ones remains an open question. Its mixing with the tau neutrino, $|U_{τ4}|^2$, is the least constrained, since terrestrial experiments probe it only indirectly and over limited ranges of sterile-neutrino mass. We show that TeV-PeV astrophysical neutrinos probe it differently. Their sources make almost no $ν_τ$, but oscillations over cosmological distances turn about one third of them into $ν_τ$ before they reach Earth. Mixing with a sterile neutrino would make these $ν_τ$ scarcer, lowering the $ν_τ$ fraction that neutrino telescopes measure, whatever the sterile-neutrino mass. In contrast, sterile mixing with $ν_e$ or $ν_μ$ is nearly undetectable. Fitting the flavor composition measured by IceCube, with the composition at the sources left free, we find $|U_{τ4}|^2 < 0.88$ at 90% C.L. Combining IceCube with present and planned neutrino telescopes, including IceCube-Gen2 and HUNT, could lower this limit to 0.24-0.29 by 2040 and to 0.12-0.14 by 2050. By 2050, these limits would match some present terrestrial limits, though not the strongest.