Influence of cosmological expansion on the threshold effects in the annihilation reaction of hard photons with CMB photons

The redshift (z) dependence of the dispersion relations for free particles is analyzed by taking into account the Lorentz invariance violation. A nonlinear algebraic equation is derived for the momenta of the particles involved in the annihilation reaction of a hard photon from a γ-ray source with a soft cosmic microwave background (CMB) photon near the threshold of this reaction. The solutions of this threshold equation are constructed and analyzed as a function of the redshift. We show that the threshold of the reaction under consideration tends to decrease with increasing z; the energy spectra of γ-ray sources at energies of ～10 TeV must be cut off in accordance with the calculated z dependence. We also calculate the time delay of the light signals from γ-ray sources that corresponds to the Lorentz invariance violation for photons. We discuss the possibility of improving the standard constraints on the Lorentz invariance violation parameters for fields of various physical natures.