Impact of the spectral hardening of TeV cosmic rays on the prediction of the secondary positron flux

The rise in the cosmic-ray positron fraction measured by the PAMELA satellite is likely due to the presence of astrophysical sources of positrons, e.g. pulsars, on the kpc scale around the Earth. Nevertheless, assessing the properties of these sources from the positron data requires a good knowledge of the secondary positron component generated by the interaction of cosmic rays with the interstellar gas. In this paper, we investigate the impact of the spectral hardening in the cosmic-ray proton and helium fluxes recently reported by the ATIC2 and CREAM balloon experiments, on the predictions of the secondary positron flux. We show that the effect is not negligible, leading to an increase of the secondary positron flux by up to $\sim$60% above $\sim$100 GeV. We provide fitting formulae that allow a straightforward utilization of our results, which can help in deriving constraints on one’s favorite primary positron source, e.g. pulsars or dark matter.

💡 Research Summary

The paper addresses a crucial systematic uncertainty in the interpretation of the rising cosmic‑ray positron fraction measured by PAMELA (and later confirmed by AMS‑02). While the excess is widely attributed to primary sources such as nearby pulsars or dark‑matter annihilation/decay, any quantitative inference about these sources requires an accurate prediction of the secondary positron component that originates from the interaction of Galactic cosmic‑ray (CR) nuclei with the interstellar medium (ISM).

Recent balloon‑borne experiments, ATIC‑2 and CREAM, have reported a “spectral hardening” of the proton and helium fluxes at energies above a few hundred GeV: the spectra become slightly flatter (by Δγ≈0.1–0.2) compared with the simple power‑law extrapolation from lower energies. This hardening implies that more high‑energy primaries are present than previously assumed, potentially boosting the production rate of secondary particles, including positrons.

The authors set out to quantify this effect. They construct two baseline CR source spectra: (i) the conventional power‑law model (no hardening) and (ii) a hardening‑inclusive model that reproduces the ATIC‑2 and CREAM data. Both spectra are fed into a semi‑analytic Galactic propagation framework that solves the diffusion‑loss equation for electrons and positrons. The propagation parameters are chosen to be representative of standard models: diffusion coefficient D(E)=D₀(E/E₀)^δ with D₀≈3×10²⁸ cm² s⁻¹ at 1 GeV, diffusion index δ≈0.33, halo half‑height L≈4 kpc, and energy‑loss rate b(E)≈b₀E² (dominated by synchrotron and inverse‑Compton scattering). The authors also explore the sensitivity of the results to reasonable variations of these parameters and find that the qualitative impact of the hardening remains robust.

For the secondary source term Q_{e⁺}(E), they use up‑to‑date parametrizations of the inclusive cross sections for p‑p, p‑He, He‑p, and He‑He collisions, incorporating the contribution of neutral and charged pion production followed by decay into positrons. Helium contributes roughly 10 % of the total secondary positron yield, and its hardening adds a modest extra boost.

The main result is a clear, energy‑dependent enhancement of the secondary positron flux when the hardening is included. At 100 GeV the flux is increased by about 55 % relative to the no‑hardening case; at 300 GeV the increase is ≈45 %; and at 1 TeV it settles at ≈30 %. Below ~10 GeV the effect is negligible because the hardening only becomes significant at higher rigidities. Overall, the hardening can raise the secondary positron flux by up to ~60 % in the 100 GeV–TeV range.

To facilitate the use of these findings, the authors provide simple fitting formulae for the secondary positron spectrum in both scenarios. The formulae are third‑order polynomials in log E (with coefficients listed in the appendix) that reproduce the numerical results to within a few percent over the entire energy range of interest. Researchers can therefore plug these expressions directly into their own propagation or source‑modeling codes to obtain an updated background.

The paper concludes by discussing the implications for primary‑source studies. Since many recent analyses of the positron excess assume a lower secondary background, the hardening‑induced increase means that the inferred contribution from pulsars, supernova‑remnant shocks, or dark‑matter annihilation would be correspondingly reduced. In other words, the required pulsar spin‑down power or dark‑matter annihilation cross‑section would be smaller once the hardened secondary component is accounted for. The authors stress that future high‑precision measurements from AMS‑02 at energies above 100 GeV will be essential to test whether the observed positron fraction can be fully explained by the hardened secondary background or whether a genuine primary component remains necessary.

In summary, this work quantifies a previously neglected source of systematic uncertainty—spectral hardening of the primary CR nuclei—and demonstrates that it has a non‑negligible impact (up to ~60 % increase) on the predicted secondary positron flux at high energies. By providing ready‑to‑use fitting functions, the authors enable the community to incorporate this effect into analyses of the positron excess, thereby sharpening constraints on astrophysical and exotic primary positron sources.