  We have previously shown that a very small amount of Lorentz invariance
violation (LIV), which suppresses photomeson interactions of ultrahigh energy
cosmic rays (UHECRs) with cosmic background radiation (CBR) photons, can
produce a spectrum of cosmic rays that is consistent with that currently
observed by the Pierre Auger Observatory (PAO) and HiRes experiments. Here, we
calculate the corresponding flux of high energy neutrinos generated by the
propagation of UHECR protons through the CBR in the presence of LIV. We find
that LIV produces a reduction in the flux of the highest energy neutrinos and a
reduction in the energy of the peak of the neutrino energy flux spectrum, both
depending on the strength of the LIV. Thus, observations of the UHE neutrino
spectrum provide a clear test for the existence and amount of LIV at the
highest energies. We further discuss the ability of current and future proposed
detectors make such observations.
