Probing a Possible Vacuum Refractive Index with Gamma-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
gamma-ray energy: eta -1 ~ E_gamma/M_QG1, where M_QG1 is some mass scale
typical of quantum gravity that may be ~ 10^18 GeV: see Phys. Lett. B 665, 412
(2008) and references therein. The MAGIC, HESS and Fermi gamma-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 gamma-ray energy, with M_QG1 ~ 10^18
GeV. We combine their results to estimate the time-lag Delta t to be expected
for the highest-energy photon from GRB 080916c measured by the Fermi telescope,
which has an energy ~ 13.2 GeV, assuming the redshift z = 4.2 \pm 0.3 measured
by GROND. In the case of a refractive index depending linearly on the gamma-ray
energy we predict Delta t = 25 \pm 11 s. This is compatible with the time-lag
Delta t <= 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 gamma-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
gamma-ray telescopes.
