Probing a possible vacuum refractive index with γ-ray telescopes

We have used a stringy model of quantum space-time foam to suggest that the vacuum may exhibit a non-trivial refractive index depending linearly on γ-ray energy: η−1∼Eγ/MQG1, where MQG is some mass scale typical of quantum gravity that may be ∼1018GeV: see [J. Ellis, N.E. Mavromatos, D.V. Nanopoulos, Phys. Lett. B 665 (2008) 412] and references therein. The MAGIC, HESS and Fermi γ-ray telescopes have recently probed the possible existence of such an energy-dependent vacuum refractive index. All find indications of time-lags for higher-energy photons, but cannot exclude the possibility that they are due to intrinsic delays at the sources. However, the MAGIC and HESS observations of time-lags in emissions from AGNs Mkn 501 and PKS 2155-304 are compatible with each other and a refractive index depending linearly on the γ-ray energy, with MQG1∼1018 GeV. We combine their results to estimate the time-lag Δt to be expected for the highest-energy photon from GRB 080916c measured by the Fermi telescope, which has an energy ∼13.2GeV, assuming the redshift z=4.35±0.15 measured by GROND. In the case of a refractive index depending linearly on the γ-ray energy we predict Δt=26±11 s. This is compatible with the time-lag Δt⩽16.5 s reported by the Fermi Collaboration, whereas the time-lag would be negligible in the case of a refractive index depending quadratically on the γ-ray energy. We suggest a strategy for future observations that could distinguish between a quantum-gravitational effect and other interpretations of the time-lags observed by the MAGIC, HESS and Fermi γ-ray telescopes.