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We discuss decays of ultra-relativistic neutrinos over cosmological distancesby solving the decay equation in terms of its redshift dependence. Wedemonstrate that there are significant conceptual differences compared to moresimplified treatments of neutrino decay. For instance, the maximum distance theneutrinos have traveled is limited by the Hubble length, which means that thecommon belief that longer neutrino lifetimes can be probed by longer distancesdoes not apply. As a consequence, the neutrino lifetime limit from supernova1987A cannot be exceeded by high-energy astrophysical neutrinos. We discuss theimplications for neutrino spectra and flavor ratios from gamma-ray bursts asone example of extragalactic sources, using up-to-date neutrino fluxpredictions. If the observation of SN 1987A implies that \nu_1 is stable andthe other mass eigenstates decay with rates much smaller than their currentbounds, the muon track rate can be substantially suppressed compared to thecascade rate in the region IceCube is most sensitive to. In this scenario, nogamma-ray burst neutrinos may be found using muon tracks even with the fullscale experiment, whereas reliable information on high-energy astrophysicalsources can only be obtained from cascade measurements. As another consequence,the recently observed two cascade event candidates at PeV energies will not beaccompanied by corresponding muon tracks.

Neutrino Decays over Cosmological Distances and the Implications for Neutrino Telescopes