Abstract
Lorentz Invariance (LI) requires the space-time structure to be the same for all observers, but various quantum gravity theories suggest that it may be violated when approaching the Planck scale. Even a small violation of LI could easily affect the Ultra-High-Energy Cosmic Ray (UHECRs) propagation on a cosmological scale. Moreover, at the extreme energies, like those available in the collisions of UHECRs in the atmosphere, one should also expect a change in the development of Extensive Air-Showers (EAS). In this contribution, Lorentz Invariance Violation (LIV) has been introduced as a perturbation term in the single particle dispersion relation considering a phenomenological approach. As a result, the kinematics of the interactions in both the extragalactic propagation and in the shower development in the atmosphere is affected. The comparison between the model predictions and the data measured by the Pierre Auger Observatory are considered to constrain the LIV scenarios. The impact of LIV has been tested considering the measured energy spectrum and the composition and the resulting upper limits on the photon flux. Finally, the effects on the shower cascade in the atmosphere are studied considering the muon content distribution.