The PAMELA anomaly indicates a nearby cosmic ray accelerator

We discuss the recently observed excesses' in cosmic ray electron and positron fluxes which have been widely interpreted as signals of dark matter. By considering the production and acceleration of secondary electrons and positrons in nearby supernova remnants, we predict an additional, harder component that becomes dominant at high energies. The unknown spatial distribution of the supernova remnants introduces a stochastic uncertainty which we estimate analytically. Fitting the prediction for different source distributions to the total electron + positron flux measured by Fermi--LAT fixes all free parameters and allows us to postdict’ the rise in the positron fraction seen by PAMELA. A similar rise in the B/C ratio is predicted at high energies.

💡 Research Summary

The paper addresses the long‑standing “PAMELA anomaly” – the unexpected rise in the cosmic‑ray positron fraction reported by PAMELA and the excess in the total electron‑plus‑positron (e⁺+e⁻) flux measured by Fermi‑LAT. While many interpretations invoke exotic physics such as dark‑matter annihilation or decay, the authors propose a purely astrophysical explanation based on secondary electron and positron production and subsequent diffusive shock acceleration (DSA) inside nearby supernova remnants (SNRs).

Key Elements of the Model

1. Discrete Source Distribution
   Supernova remnants are not a smooth, continuous source population; they are discrete in both space and time. The authors treat each SNR as an individual burst that injects electrons and positrons at a distance L and a time t ago. The contribution of a single source to the observed flux is given by the Green’s function of the diffusion–energy‑loss equation, G_disk(E, L, t). The total flux is the sum over all N sources. Because the source locations and ages are random variables, the resulting flux is also a random variable.

2. Statistical Treatment
   The standard central limit theorem cannot be applied directly because the Green’s function has a heavy‑tailed probability distribution, leading to a divergent variance. Instead, the authors invoke a generalized central limit theorem and describe the flux distribution with a stable (Lévy) distribution. This yields analytic expressions for the mean flux ⟨J⟩ and for quantiles (68 % and 90 % confidence intervals). The quantiles have a harder energy dependence than the mean, implying that stochastic fluctuations grow with energy – a crucial point for interpreting high‑energy data.

3. Primary Electron Injection
   Primary electrons are injected with a power‑law spectrum R_{e⁻}=R₀ E^{−γ_e} exp(−E/E_cut). The spectral index γ_e≈2.4 and cutoff energy E_cut≈20 TeV are derived from γ‑ray observations of SNRs by Imaging Atmospheric Cherenkov Telescopes (HESS, MAGIC, VERITAS). The normalization R₀ is fixed by matching the low‑energy (≈10 GeV) electron flux measured by PAMELA, corresponding to a total injected energy of ~7×10⁴⁷ erg per SNR.

4. Secondary Production and Acceleration
   Traditionally, secondary e⁻ and e⁺ are produced only during the propagation of cosmic‑ray nuclei in the interstellar medium (ISM), yielding a soft spectrum that falls off faster than the primary component. The authors argue that a fraction of these secondaries can be re‑accelerated inside the SNR shock. The acceleration efficiency grows linearly with particle momentum because the diffusion coefficient in the shock region scales as D∝p (Bohm‑like diffusion). The resulting source spectrum for secondaries is a sum of two power laws: \