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 [109]. 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
to be less than
[109]. Therefore
significant observational progress must be made in order to reach
bounds on
. 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 [242, 106]. Higher order dispersion corrections seem unlikely to ever be probed with time of flight
measurements.
The limit can be easily applied to the EFT operators in Equation (40
). From
Equation (41
) we trivially see that the constraint on
is
, again comparing the
peak to
the
. It might seem that we can get a better constraint by demanding the time delay between
right handed and left handed photons is less than
. 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 (40
) 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 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
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 [260
] 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 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
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
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
would then apply. For
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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