Neutrino Decays over Cosmological Distances and the Implications for
  Neutrino Telescopes
  We discuss decays of ultra-relativistic neutrinos over cosmological distances
by solving the decay equation in terms of its redshift dependence. We
demonstrate that there are significant conceptual differences compared to more
simplified treatments of neutrino decay. For instance, the maximum distance the
neutrinos have traveled is limited by the Hubble length, which means that the
common belief that longer neutrino lifetimes can be probed by longer distances
does not apply. As a consequence, the neutrino lifetime limit from supernova
1987A cannot be exceeded by high-energy astrophysical neutrinos. We discuss the
implications for neutrino spectra and flavor ratios from gamma-ray bursts as
one example of extragalactic sources, using up-to-date neutrino flux
predictions. If the observation of SN 1987A implies that \nu_1 is stable and
the other mass eigenstates decay with rates much smaller than their current
bounds, the muon track rate can be substantially suppressed compared to the
cascade rate in the region IceCube is most sensitive to. In this scenario, no
gamma-ray burst neutrinos may be found using muon tracks even with the full
scale experiment, whereas reliable information on high-energy astrophysical
sources can only be obtained from cascade measurements. As another consequence,
the recently observed two cascade event candidates at PeV energies will not be
accompanied by corresponding muon tracks.
