Stable laws and cosmic ray physics
Résumé
In the new precision era for cosmic ray astrophysics, theoretical predictionscannot content themselves with average trends, but need to correctly take intoaccount intrinsic uncertainties. The space-time discreteness of the cosmic raysources, joined with a substantial ignorance of their precise epochs andlocations (with the possible exception of the most recent and close ones) playsan important role in this sense. We elaborate a statistical theory to deal withthis problem, relating the composite probability P({\Psi}) to obtain a flux{\Psi} at the Earth to the single-source probability p({\psi}) to contributewith a flux {\psi}. The main difficulty arises since p({\psi}) is a fat taildistribution, characterized by power-law or broken power-law behaviour up tovery large fluxes for which central limit theorem does not hold, and leading towell-known stable laws as opposed to Gaussian distributions. We find thatrelatively simple recipes provide a satisfactory description of the probabilityP({\Psi}). We also find that a naive Gaussian fit to simulation results wouldunderestimate the probability of very large fluxes, i.e. several times abovethe average, while overestimating the probability of relatively milderexcursions. At large energies, large flux fluctuations are prevented by causalconsiderations, while at low energies a partial knowledge on the recent andnearby population of sources plays an important role. A few proposal have beendiscussed in the literature to account for spectral breaks recently reported incosmic ray data in terms of local contributions. We apply our newly developedtheory to assess their probabilities, finding that they are relatively small.