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Possible cosmogenic neutrino constraints on Planck-scale Lorentz violation

We study, within an effective field theory framework, $O(E^{2}/\Mpl^{2})$ Planck-scale suppressed Lorentz invariance violation (LV) effects in the neutrino sector, whose size we parameterize by a dimensionless parameter $η_ν$. We find deviations from predictions of Lorentz invariant physics in the cosmogenic neutrino spectrum. For positive O(1) coefficients no neutrino will survive above $10^{19} \eV$. The existence of this cutoff generates a bump in the neutrino spectrum at energies of $10^{17} \eV$. Although at present no constraint can be cast, as current experiments do not have enough sensitivity to detect ultra-high-energy neutrinos, we show that experiments in construction or being planned have the potential to cast limits as strong as $η_ν \lesssim 10^{-4}$ on the neutrino LV parameter, depending on how LV is distributed among neutrino mass states. Constraints on $η_ν < 0$ can in principle be obtained with this strategy, but they require a more detailed modeling of how LV affects the neutrino sector.