First Combined Study on LIV from Observations of Energy-dependent Time Delays from Multiple-type Gamma-ray Sources -- Part I. Motivation, Method Description and Validation through Simulations of H.E.S.S., MAGIC and VERITAS Datasets

Gamma-ray astronomy has become one of the main experimental ways to test the modified dispersion relations (MDRs) of photons in vacuum, obtained in some attempts to formulate a theory of Quantum Gravity. The MDRs in use imply time delays which depend on the energy, and which increase with distance following some function of redshift. The use of transient, or variable, distant and highly energetic sources, already allows us to set stringent limits on the energy scale related to this phenomenon, usually thought to be of the order of the Planck energy, but robust conclusions on the existence of MDR-related propagation effects still require the analysis of a large population of sources. In order to gather the biggest sample of sources possible for MDR searches at teraelectronvolt energies, the H.E.S.S., MAGIC and VERITAS collaborations enacted a joint task force to combine all their relevant data to constrain the Quantum Gravity energy scale. In the present article, the likelihood method used, to combine the data and provide a common limit, is described in detail and tested through simulations of recorded data sets for a gamma-ray burst, three flaring active galactic nuclei and two pulsars. Statistical and systematic errors are assessed and included in the likelihood as nuisance parameters. In addition, a comparison of two different formalisms for distance dependence of the time lags is performed for the first time. In a second article, to appear later, the method will be applied on all relevant data from the three experiments.