Possible cosmogenic neutrino constraints on Planck-scale Lorentz violation

We study, within an effective field theory framework, $O(E^2\mathrm{/}{\text{Mpl}}^2)$ Planck-scale suppressed Lorentz invariance violation (LV) effects in the neutrino sector, whose size we parameterize by a dimensionless parameter ${\eta }_{\nu }$. We find deviations from predictions of Lorentz invariant physics in the cosmogenic neutrino spectrum. For positive O(1) coefficients no neutrino will survive above ${10}^{19}\text{eV}$. The existence of this cutoff generates a bump in the neutrino spectrum at energies of ${10}^{17}\text{eV}$. 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 $\eta_{\nu} \lesssim 10^{-4}$ on the neutrino LV parameter, depending on how LV is distributed among neutrino mass states. Constraints on ${\eta }_{\nu }<0$ can in principle be obtained with this strategy, but they require a more detailed modeling of how LV affects the neutrino sector.