  Annihilation of extremely energetic cosmic neutrinos on the relic-neutrino
background can give rise to absorption lines at energies corresponding to
formation of the electroweak gauge boson $Z^{0}$. The positions of the
absorption dips are set by the masses of the relic neutrinos. Suitably intense
sources of extremely energetic ($10^{21}$ -- $10^{25}$-eV) cosmic neutrinos
might therefore enable the determination of the absolute neutrino masses and
the flavor composition of the mass eigenstates. Several factors--other than
neutrino mass and composition--distort the absorption lines, however. We
analyze the influence of the time-evolution of the relic-neutrino density and
the consequences of neutrino decay. We consider the sensitivity of the
lineshape to the age and character of extremely energetic neutrino sources, and
to the thermal history of the Universe, reflected in the expansion rate. We
take into account Fermi motion arising from the thermal distribution of the
relic-neutrino gas. We also note the implications of Dirac vs. Majorana relics,
and briefly consider unconventional neutrino histories. We ask what kinds of
external information would enhance the potential of cosmic-neutrino absorption
spectroscopy, and estimate the sensitivity required to make the technique a
reality.
