On the detectability of Lorentz invariance violation through anomalous multi-TeV $\gamma$-ray spectra of blazars

Cosmic opacity for very high-energy gamma rays ($E>10$ TeV) due to the interaction with the extragalactic background light can be strongly reduced because of possible Lorentz-violating terms in the particle dispersion relations expected, e.g., in several versions of quantum gravity theories. We discuss the possibility to use very high energy observations of blazars to detect anomalies of the cosmic opacity induced by LIV, considering in particular the possibility to use -- besides the bright and close-by BL Lac Mkn 501 -- extreme BL Lac objects. We derive the modified expression for the optical depth of $\gamma$ rays considering also the redshift dependence and we apply it to derive the expected high-energy spectrum above 10 TeV of Mkn 501 in high and low state and the extreme BL Lac 1ES 0229+200. We find that, besides the nearby and well studied BL Lac Mkn 501 -- especially in high state --, suitable targets are extreme BL Lac objects, characterized by quite hard TeV intrinsic spectra likely extending at the energies relevant to detect LIV features.