Other effects which could also produce an energy dependent delay in photon arrival times include energy dependent dispersion due to the strong gravitational field near the neutron star, purely electromagnetic dispersion, an energy dependence in the emission location, or an intrinsic energy dependence in the emission time. The effect of any energy dependent dispersion due to the strong gravitational field near the neutron star is likely to be small because, even if emitted from the neutron star surface, photons traverse the region of high gravitational fields within about 0.1 ms. Allowing a fractional change in the speed of light equal to the dimensionless field strength at the neutron star surface, , where is the neutron star mass and is the neutron star radius, the difference in arrival times would be only 0.02 ms. The actual energy dependent change in the speed of light is likely to be much smaller than 0.2. Any significant purely electromagnetic dispersion at MeV energies and above can be excluded based on the dispersions measured at lower energies.
An energy dependence in the photon emission location or intrinsic emission time could produce a significant energy dependent time delay. While the possibility that precise tuning of the emission locations or times for various energy photons could cancel an energy dependent dispersion arising from quantum gravity effects, we consider such a coincidence unlikely, although not excluded, and interpret our lack of detection of any energy dependence in arrival times as constraining both the energy dependent dispersion and the emission location and time. In this case, the average emission location, projected along our line of sight, for photons at energies in the 70-100 MeV band must lie within 110 km of that for photons above 2 GeV, within 50 km of that for 0.5-1.0 GeV photons, and within 150 km of that for radio photons.
It is encouraging that the analysis shows that it is possible to time the Crab pulsar at gamma-ray energies to an accuracy of 0.07 ms (95% confidence) given adequate statistics. Detection of pulsations from the Crab at 50-100 GeV could improve the limit on by two orders of magnitude. The key question is whether the pulsations of the Crab and other gamma-ray pulsars continue to such high energies. Observations of the Crab near 1 TeV show only unpulsed emission (Vacanti et al. 1991) and the cutoff energy of the pulsed emission is unknown. If the Crab does pulse at 50-100 GeV, detection of the pulses may be possible in the near term with low energy threshold atmospheric Cherenkov telescopes (ACTs), such as STACEE (Bhattacharya et al. 1997) and CELESTE (Giebels et al. 1998), or in the longer term with a space-borne gamma-ray detector such as GLAST (Gehrels et al. 1998). The Crab pulsed signal may extend only to the lowest energies accessible with the ACTs. Thus, measurement of a timing difference between two energy bands might require contemporaneous measurements at other wavelengths. Both optical (Smith et al. 1978) and x-ray timing (Rots et al. 1998) can exceed the accuracy of gamma-ray timing. However, the emission location for x-ray and optical photons may differ from that of gamma-ray photons. If quantum gravity does produce a first order correction to the dispersion relation for electromagnetic waves, then measurement of the pulse arrival time of the Crab at 50 GeV with an accuracy of 0.1 ms could be used to place a lower bound on . This is within the range, , for the energy scale for quantum gravity effects preferred in string theory (Witten 1996).
If future measurements do reveal an energy dependence in pulsar photon arrival times, then it will be difficult to distinguish an energy dependent dispersion from an intrinsic energy dependence in the emission location or emission time. This problem is common to all of the suggested astronomical tests of quantum gravity effects. Convincing proof for quantum gravity effects will likely require detection of energy dependent time delays in at least two different classes of objects, preferably at vastly difference distances, i.e. pulsars versus AGN or gamma-ray bursts, with all of the detections compatible with the same value of .
© European Southern Observatory (ESO) 1999
Online publication: April 19, 1999