  Resonant annihilation of extremely high-energy cosmic neutrinos on big-bang
relic anti-neutrinos (and vice versa) into Z-bosons leads to sizable absorption
dips in the neutrino flux to be observed at Earth. The high-energy edges of
these dips are fixed, via the resonance energies, by the neutrino masses alone.
Their depths are determined by the cosmic neutrino background density, by the
cosmological parameters determining the expansion rate of the universe, and by
the large redshift history of the cosmic neutrino sources. We investigate the
possibility of determining the existence of the cosmic neutrino background
within the next decade from a measurement of these absorption dips in the
neutrino flux. As a by-product, we study the prospects to infer the absolute
neutrino mass scale. We find that, with the presently planned neutrino
detectors (ANITA, Auger, EUSO, OWL, RICE, and SalSA) operating in the relevant
energy regime above 10^{21} eV, relic neutrino absorption spectroscopy becomes
a realistic possibility. It requires, however, the existence of extremely
powerful neutrino sources, which should be opaque to nucleons and high-energy
photons to evade present constraints. Furthermore, the neutrino mass spectrum
must be quasi-degenerate to optimize the dip, which implies m_{nu} >~ 0.1 eV
for the lightest neutrino. With a second generation of neutrino detectors,
these demanding requirements can be relaxed considerably.
