Limits to Quantum Gravity Effects on Energy Dependence of the Speed of Light from Observations of TeV Flares in Active Galaxies

We have used data from a TeV $\gamma$-ray flare associated with the active galaxy Markarian 421 to place bounds on the possible energy dependence of the speed of light in the context of an effective quantum gravitational energy scale. Recent theoretical work suggests that such an energy scale could be less than the Planck mass and perhaps as low as ${10}^{16}$ GeV. The limits derived here indicate this energy scale to be in excess of $6\times {10}^{16}$ GeV for at least one approach to quantum gravity in the context of D-brane string theory. To the best of our knowledge, this constitutes the first convincing limit on such phenomena in this energy regime.