Discriminating between different modified dispersion relations from gamma-ray observations

The fact that the standard dispersion relation for photons in vacuum could be modified because of their interaction with the quantum nature of spacetime has been proposed more than two decades ago. A quantitative model by Jacob and Piran has been tested extensively using distant highly energetic astrophysical sources, searching for energy-dependent time delays in photon arrival times. Since no delay was firmly measured, lower limits were set on the energy scale $\mathrm{\ensuremath{\Lambda}}$ related to these effects. In recent years, however, different but equally well-grounded expressions beyond the Jacob and Piran model were obtained for the photon dispersion relation, leading to different expressions for the dependence of lag versus redshift. This article introduces a general parametrization of modified dispersion relations in homogeneous and isotropic, i.e. cosmological, symmetry which directly leads to a general parametrized lag versus redshift dependence encompassing both existing and new models. This parametrization could be used in the future to compare the predicted time lags of the different models and test them against observations. To investigate this possibility, realistic datasets are simulated, mimicking different types of extragalactic sources as detected by current and future instruments. When no lag is injected in the simulated data, each lag-redshift model leads, as expected, to a different value for the limit on $\mathrm{\ensuremath{\Lambda}}$, and the Jacob and Piran model gives the most stringent bound. When a lag at $\mathrm{\ensuremath{\Lambda}}\ensuremath{\sim}{E}_{P}$ in the Jacob and Piran model is injected, it is detected for all the other lag-redshift relations considered, although leading to different values. Finally, the possibility to discriminate between several lag-redshift models is investigated, emphasizing the importance of an evenly distributed sample of sources across a wide range of redshifts.