It has been predicted by several background-independent approaches to Quantum Gravity (like LQG or spinfoams) that the physical dispersion relations in vacuum could take the following form:
$$ p^2 = E^2 \left( 1 + \xi \frac{E}{E_P} + {\cal O}\left( \frac{E^2}{E_P^2} \right) \right), $$
where $E$ is the observed energy of the massless particle, $p$ is the observed absolute value of the 3-momentum and $E_P = \text{const}$ is Planck's energy. The dimensionless coefficient $\xi$ is to be calculated by theory.
This anzatz is in apparent contradiction with Special Relativity, but actually this dispersion relations are a consequence of spacetime being discrete at the Planck scale. The idea is that it may be correct far beyond the domain of validity of SR. For observable energies $E \ll E_P$ we have $E = p$.
My question is: what are the most restrictive experimental bounds on the value of $\xi$? I've heard claims that tight bounds have been obtained from investigating the spectrum of distant astrophysical gamma-ray sources. Ideally, I want a reference to a peer-reviewed paper explicitly stating the experimental bound on $\xi$.
Answer
The astrophysical gamma ray sources to which you are referring are gamma ray bursts, particularly the short bursts. These systems are ideal for testing Lorentz invariance because they can lead to the emission of high energy (tens of GeV) photons in a short interval of time, allowing for measurements of the potential delay in arrival of photons of different energies.
The strictest limit set on Lorentz invariance so far in the literature comes from Abdo et al 2009, in which they use the detection of a photon of approximately 30 GeV from the short GRB 090510. In your notation, they find
$$|\xi| < 0.82 \qquad $$
at about the 99% confidence limit. With less conservative assumptions, the limit drops even lower. To gain a more complete understanding of their analysis, you need to read the supplemental information, where they enumerate the various limits as they make their assumptions less and less conservative.
(Note that what they call $\xi_1$ in their paper is actually $1/\xi$ in your notation, and what they report is $\xi_1 > 1.22$).
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