Abstract. We point out that previous studies of possible Lorentz-violating effects in astronomical time-of-flight data did not take into account the entire implications of the universe's cosmological expansion. We present the derivation of the accurate formulation of the problem and show that the resulting correction of the limits on Lorentz violation is significant.

Lorentz-invariance violation (LIV) arises in various quantumgravity1,2 theories, but typically at Planck energies that are not accessible on Earth. To test LIV, we must turn to astronomical observations 2-11 . Time-of-flight measurements from astronomical sources have set the present limits on the LIV energy scale. According to existing models, gamma-ray bursts (GRBs) are accompanied by very high-energy neutrinos 12,13 . At these energies, the background level in neutrino detectors such as IceCube (currently under construction in Antarctica) is extremely low. We show that the detection of even a single neutrino from the same direction as a GRB, months after the burst, would be statistically significant and imply that the neutrino was associated with the burst. The detection of several delayed neutrinos from different bursts with compatible relations between their delay times, energies and distances would enable us to generically determine (or set limits on) LIV at levels that cannot be reached by any other method.History tells us that symmetries, observed over a large range of parameters and believed to be fundamental properties of our physical world, may lose their significance later when observations are made over a larger range of parameters and when a new physical understanding arises. Such apparent symmetries often emerge as leading-order approximations of more complex symmetries, found to describe more accurately the larger range of observations. The Lorentz symmetry might be such an apparent symmetry, as Lorentz-invariance might be violated or deformed at very high energies. This possibility, which was initially motivated by attempts [14][15][16] to resolve the GZK paradox 17,18 , arises naturally in various theories of quantum gravity (see, for example, refs 1,2 for a review).Here, we consider a simple phenomenological approach 16 for LIV with a symmetry-breaking energy scale of ξE pl (E pl being the Planck energy ∼1.2 × 10 28 eV). We consider a single LIV scale (ξ) for all particles. This arises naturally in any theory in which the modification originates from a small-scale structure of spacetime. Certain theories, for example the one proposed by Coleman and Glashow 15 , allow different LIV scales for different particles. In such a case, the time delay between low-energy photons and high-energy neutrinos, which we consider here, essentially limits the neutrinos LIV scale.Taking into account only the leading-order correction, we expect, for particles with E ξE pl , a generic approximate dispersion relation:

We investigate the implications of energy-dependence of the speed of photons, one of the candidate effects of quantum-gravity theories that has been most studied recently, from the perspective of observations in different reference frames. We examine how a simultaneous burst of photons would be measured by two observers with a relative velocity, establishing some associated conditions for the consistency of theories. For scenarios where the Lorentz transformations remain valid these consistency conditions allow us to characterize the violations of Lorentz symmetry through an explicit description of the modification of the quantum-gravity scale in boosted frames with respect to its definition in a preferred frame. When applied to relativistic scenarios with a deformation of Lorentz invariance that preserves the equivalence of inertial observers, we find an insightful characterization of the necessity to adopt in such frameworks non-classical features of spacetime geometry, e.g. events that are at the same spacetime point for one observer cannot be considered at the same spacetime point for other observers. Our findings also suggest that, at least in principle (and perhaps one day even in practice), measurements of the dispersion of photons in relatively boosted frames can be particularly valuable for the purpose of testing these scenarios.

We examine what the absorbed spectra of extra-galactic TeV gamma-ray sources, such as blazars, would look like in the presence of Lorentz invariance violation (LIV).Pair-production with the extra-galactic background light modifies the observed spectra of such sources, and we show that a violation of Lorentz invariance would generically have a dramatic effect on this absorption feature. Inspecting this effect, an experimental task likely practical in the near future, can provide unique insight on the possibility of LIV.

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