High energy cosmic neutrino observations provide a sensitive test of Lorentz invariance violation, which may be a consequence of quantum gravity theories. We consider a class of non-renormalizable, Lorentz invariance violating operators that arise in an effective field theory description of Lorentz invariance violation in the neutrino sector inspired by Planck-scale physics and quantum gravity models. We assume a conservative generic scenario for the redshift distribution of extragalactic neutrino sources and employ Monte Carlo techniques to describe superluminal neutrino propagation, treating kinematically allowed energy losses of superluminal neutrinos caused by both vacuum pair emission and neutrino splitting. We consider EFTs with both non-renormalizable CPT-odd and non-renormalizable CPT-even operator dominance. We then compare the spectra derived using our Monte Carlo calculations in both cases with the spectrum observed by IceCube in order to determine the implications of our results regarding Planck-scale physics. We find that if the drop off in the neutrino flux above ~2 PeV is caused by Planck scale physics, rather than by a limiting energy in the source emission, a potentially significant pileup effect would be produced just below the drop off energy in the case of CPT-even operator dominance. However, such a clear drop off effect would not be observed if the CPT-odd, CPT-violating term dominates.
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