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 currently 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 $\xi$.

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 $\xi$.