Stable laws and cosmic ray physics
LAPTh, Université Savoie Mont Blanc & CNRS, 9 chemin de Bellevue, BP 110, 74941 Annecy-le-Vieux, France
Received: 14 October 2016
Accepted: 19 December 2016
Context. In the new “precision era” for cosmic ray astrophysics, scientists making theoretical predictions cannot content themselves with average trends, but need to correctly take into account intrinsic uncertainties. The space-time discreteness of the cosmic ray sources, together with a substantial ignorance of their precise epochs and locations (with the possible exception of the most recent and close ones) play an important role in this sense.
Aims. We elaborate a statistical theory to deal with this problem, relating the composite probability P(Ψ) to obtain a flux Ψ at the Earth and the single-source probability p(ψ) to contribute with a flux ψ. The main difficulty arises from the fact that p(ψ) is a “heavy tail” distribution, characterized by power-law or broken power-law behavior up to very large fluxes, for which the central limit theorem does not hold, and leading to distributions different from Gaussian. The functional form of the distribution for the aggregated flux is nonetheless unchanged by its own convolution, that is, it belongs to the so-called stable laws class.
Methods. We analytically discuss the regime of validity of the stable laws associated with the distributions arising in cosmic ray astrophysics, as well as the limitations to the treatment imposed by causal considerations and partial source catalog knowledge. We validate our results with extensive Monte Carlo simulations, for different regimes of propagation parameters and energies.
Results. We find that relatively simple recipes provide a satisfactory description of the probability P(Ψ). We also find that a naive Gaussian fit to simulation results would underestimate the probability of very large fluxes, that is, several times above the average, while overestimating the probability of relatively milder excursions. At large energies, large flux fluctuations are prevented by causal considerations, while at low energies, a partial knowledge of the recent and nearby population of sources plays an important role. A few proposals have been recently discussed in the literature to account for spectral breaks reported in cosmic ray data in terms of local contributions. We apply our newly developed theory to assess their probabilities, finding that they are relatively small, typically at the 0.1% level or smaller, never exceeding 1%.
Conclusions. The use of heavy tail distributions is relevant in assessing how likely a measured cosmic ray flux is to depart from the average expectation in a given model. The existing mathematical theory leading to stable laws can be adapted to the case of interest via some recipes that closely reproduce numerical simulations and are relatively easy to implement.
Key words: astroparticle physics / cosmic rays / diffusion / supernovae: general / methods: statistical
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