  The only invariant speed in special relativity is c; therefore, if some
neutrinos travel at even tiny speeds above c, normal special relativity is
incomplete and any superluminal speed may be possible. I derive a limit on
superluminal neutrino speeds v >> c at high energies by noting that such speeds
would increase the size of the neutrino horizon. The increased volume of the
Universe visible leads to a brighter astrophysical neutrino background. The
nondetection of "guaranteed" neutrino backgrounds from star-forming galaxies
and ultrahigh energy cosmic rays (UHECRs) constrains v/c at TeV--ZeV energies.
I find that v/c <= 820 at 60 TeV from the nondetection of neutrinos from
star-forming galaxies. The nondetection of neutrinos from UHECRs constrains v/c
to be less than 2500 at 0.1 EeV in a pessimistic model and less than 4.6 at 4
EeV in an optimistic model. The UHECR neutrino background nondetection is
strongly inconsistent with a naive quadratic extrapolation of the OPERA results
to EeV energies. The limits apply subject to some caveats, particularly that
the expected pionic neutrino backgrounds exist and that neutrinos travel faster
than c when they pass the detector. They could be improved substantially as the
expected neutrino backgrounds are better understood and with new experimental
neutrino background limits. I also point out that extremely subluminal speeds
would result in a much smaller neutrino background intensity than expected.
