Modern ideas in quantum gravity predict the possibility of Lorenz invariance violation (LIV) manifested, for example, by energy-dependent modification of the standard relativistic dispersion relation. In a recent paper, Jacob and Piran proposed that time of flight delays in high-energy neutrinos emitted by gamma-ray bursts (GRBs) located at cosmological distances can become a valuable tool for setting limits on LIV theories. However, current advances in observational cosmology suggest that our Universe is dominated by dark energy with relatively little guidance on its nature, thus leading to several cosmological scenarios that are compatible with observations.

In this paper we raise the issue of how important, in the context of testing LIV theories, is our knowledge of the background cosmological model. Specifically we calculate expected time lags for high-energy (100 TeV) neutrinos in different cosmological models. Out of many particular models of dark energy we focus on five as representative for various competing approaches: ΛCDM, quintessence, quintessence with time-varying equation of state, brane-world and generalized Chaplygin gas.

The conclusion is that better understanding of a dark energy dominated Universe is crucial for testing LIV theories with cosmological sources like GRBs. Theoretically one may also invert this argument by saying that if an LIV dispersion relation was proven experimentally and its parameters were constrained then time delays from GRBs could become a new kind of cosmological test.