On extrapolations below the Planck scale in models with Lorentz symmetry violation (I)

12 pages, LaTex. This paper contains un updated version of part of the material presented at the 35th COSPAR Scientific Assembly, Paris, France, 18 - 25 July 2004Most current models of Lorentz symmetry violation (LSV) at the Planck scale involve power-like extrapolations of the Lorentz-beaking term down to accelerator and even much lower energies. It is therefore assumed that no intermediate energy scale alters this behaviour. But this is not the only possible scenario: a more sophisticate energy-dependence is possible, and would even be natural, involving significant effective thresholds at intermediate energies. Such thresholds may exist between the Planck scale and the highest cosmic-ray energies, or between ultra-high cosmic-ray energies and the TeV scale, leading to interesting scenarios. In many cases, experimental predictions of LSV patterns can be dramatically modified and space experiments become necessary irrespective of AUGER results. By combining both kinds of experiments, future results of cosmic-ray observations will hopefully be able to test, for a large family of models involving various patterns of Planck-scale physics, the possible existence of an absolute local rest frame in the real world