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6.2 Time of flight

The simplest astrophysical observations that provide interesting constraints on Planck scale Lorentz violation are time of flight measurements of photons from distant sources [20106Jump To The Next Citation Point109Jump To The Next Citation Point]. This is also one of two processes (the other being birefringence) that can be directly applied to kinematic models. With a modified dispersion relation of the form (15View Equation) and the assumption that the velocity is given by v = @E/@p 15, the velocity of a photon is given by
(n) n-2 v = 1 + (n---1)fg--E----. (74) g 2EnP-l2
If n /= 2, the velocity is a function of energy, and the time of arrival difference DT between two photons at different energies travelling over a time T is
(n) n-2 n- 2 DT = DvT = (n---1)fg--(E-1-----E2---)T, (75) 2EnP-l2
where E1,2 are the photon energies. The large time T plays the role of an amplifier in this process, compensating for the small ratio E/EPl.16 For n = 1 there are much better low energy constraints, while for n = 4 the constraints are far too weak to be useful. Hence we shall concentrate on n = 3 type dispersion, where this constraint has been most often applied. The best limits [53] are provided by observations of rapid flares from Markarian 421, a blazar at a redshift of approximately z = 0.03, although a number of other objects give comparable results [25062]. The most rapid flare from Markarian 421 showed a strong correlation of flux at 1TeV and 2 TeV on a timescale of 280 s. If we assume that the flare was emitted from the same event at the source, the time of arrival delay between 1TeV and 2TeV photons must be less than 280 s. Combining all these factors yields the limit |f (3)|< 128.

A possible problem with the above bound is that in a single emission event it is not known if the photons of different energies are produced simultaneously. If different energies are emitted at different times, that might mask a LV signal. One way around this is to look for correlations between time delay and redshift, which has been done for a set of gamma ray bursts (GRBs) in [109Jump To The Next Citation Point]. Since time of flight delay is a propagation effect that increases over time, a survey of GRBs at different redshifts can separate this from intrinsic source effects. This enables constraints to be imposed (or LV to be observed) despite uncertainty regarding source effects. The current data from GRBs limit f(3) to be less than O(103) [109]. Therefore significant observational progress must be made in order to reach O(1) bounds on (3) f. Improvements on this limit might come from observations of GRBs with new instruments such as GLAST, however concerns have been raised that source effects may severely impair this approach as well [242106]. Higher order dispersion corrections seem unlikely to ever be probed with time of flight measurements.

The limit (3) |f |< 128 can be easily applied to the EFT operators in Equation (40View Equation). From Equation (41View Equation) we trivially see that the constraint on q is |q| < 64, again comparing the 2TeV peak to the 1TeV. It might seem that we can get a better constraint by demanding the time delay between 2TeV right handed and left handed photons is less than 280s. However, the polarization of the flare is unknown, so it is possible (although perhaps unlikely) that only one polarization is being produced. If one can show that both polarizations are present, then one can further improve this constraint. However, the time of flight constraints are much weaker than other constraints that can be derived on the operators in Equation (40View Equation) from birefringence, so this line of research would not be fruitful.

DSR theories may also predict a time of flight signal, where the speed of light is effectively given by the group velocity of an n = 3 type dispersion relation.17 If there is such a frequency dependence, it is not expected that DSR also yields birefringence as in the EFT case. An n = 3 type dispersion for photons without birefringence would hence be a strong signal for DSR or something similar. Coupled with the fact that DSR does not affect threshold reactions or exhibit sidereal effects, time of flight analyses provide the only currently realistic probe of DSR theories. Unfortunately, since the invariant energy scale is usually taken to be the Planck energy, time of flight constraints are still one to two orders of magnitude below what is needed to constrain/probe DSR. As an aside, note that the actual measurement of the dependence of the speed of light with frequency in a telescope such as GLAST [260Jump To The Next Citation Point] has a few subtleties in a DSR framework. Let us make the (unrealistic) assumption that the situation is as good as it could possibly be experimentally: there is a short, high energy GRBs from some astrophysical source where all the photons are emitted from the same point at the same time. The expected observational signal is then a correlation between the photon time of arrival and energy. The time of arrival is fairly straightforward to measure, but the reconstruction of the initial photon energy is not so easy. GLAST measures the initial photon energy by calorimetry - the photon goes through a conversion foil and converts to an electron-positron pair. The pair then enters a calorimeter, which measures the energy by scintillation. The initial particle energy is then only known by reconstruction from many events. Energy reconstruction requires addition of the multitude of low energy signals back into the single high energy incoming photon. Usually this addition in energy is linear (with corrections due to systematics/experimental error). However, if we take the DSR energy summation rules as currently postulated the energies of the low energy events add non-linearly, leading to a modified high energy signal. One might guess that since the initial particle energy is well below the Planck scale, the non-linear corrections make little difference to the energy reconstruction. However, to concretely answer such a question, the multi-particle sector of DSR must be properly understood (for a discussion of the problems with multi-particle states in DSR see [186]).

Finally, while photons are the most commonly used particle in time of flight tests, other particles may also be employed. For example, it has been proposed in [81] that neutrino emission from GRBs may also be used to set limits on n = 3 dispersion. Observed neutrino energies can be much higher than the TeV scale used for photon measurements, hence one expects that any time delay is greatly magnified. Neutrino time delay might therefore be a very precise probe of even n > 3 dispersion corrections. Of course, first an identifiable GRB neutrino flux must be detected, which has not happened yet [5]. Assuming that a flux is seen and able to be correlated on the sky with a GRB, one must still disentangle the signal. In a DSR scenario, where time delay scales uniformly with energy this is not problematic, at least theoretically. However, in an EFT scenario there can be independent coefficients for each helicity, thereby possibly masking an energy dependent signal. For n = 3 this complication is irrelevant if one assumes that all the neutrinos are left-handed (as would be expected if produced from a standard model interaction) as only f(3) nL would then apply. For n > 3 the possible operators are not yet known, so it is not clear what bounds would be set by limits on neutrino time of flight delays.

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