Test of Lorentz invariance through observation of the longitudinal development of ultrahigh-energy extensive air showers

An idea that Lorentz invariance can be violated was proposed by Coleman and Glashow to overcome the astrophysical problems of air showers of ultrahigh energies $E>10^{20}$ eV. This statement can be tested by analyzing experimental data on these showers. The longitudinal development of showers and the distribution of the depths of shower maxima were calculated in the model of quark-gluon strings with allowance made for the Landau–Pomeranchuk–Migdal effect and the interactions of ultrahigh-energy neutral pions. Comparison of the calculations with available experimental data provides a new bound $|c_{\gamma}-c_{{\pi}^o}|< {10}^{-20}$ for the possible difference between the speeds of photons and neutral pions. This bound becomes $|c_{\gamma}-c_{{\pi}^o}|< {10}^{-22}$ when one takes the upper limiting value for the observed depth of maximum.