Addressing the Impact of Solar Modulation Systematic Uncertainties on Cosmic-Ray Propagation Models

We perform a comprehensive analysis of cosmic-ray propagation using the time-dependent AMS-02 flux measurements covering a full solar cycle, with particular emphasis on the role of solar modulation. We fit two representative Galactic propagation scenarios, convection- and re-acceleration-dominated models, in combination with three solar modulation prescriptions: the standard force-field approximation, an extended force-field model with a rigidity break, and the heliospheric propagation code $\texttt{HelMod}$. The inclusion of time-resolved antiproton data provides a unique probe of charge-sign-dependent modulation effects and low-energy systematics. We find that the force-field approximation can describe positively charged nuclei reasonably well outside the solar maximum in convection-dominated models, but fails during periods of high solar activity and for antiprotons at all times. In re-acceleration scenarios, strong degeneracies between solar modulation and low-energy propagation lead to unphysical results when simple modulation models are employed. Across all models, we identify systematic uncertainties of order 10-15% in the reconstructed local interstellar spectra and propagation parameters, driven by limitations in current solar modulation modelling. Compared to the percent level error of current measurements, these uncertainties significantly limit the precision of cosmic-ray studies. Future time-dependent measurements spanning a full 22-year solar cycle will be crucial to reduce these uncertainties.

💡 Research Summary

This paper presents a comprehensive study of how uncertainties in solar modulation affect the determination of Galactic cosmic‑ray (CR) propagation parameters. Using the time‑dependent flux measurements from AMS‑02 that span nearly a full solar cycle (May 2011 – November 2022), the authors simultaneously fit Galactic propagation models and three distinct solar‑modulation prescriptions.

The Galactic propagation is modeled with the numerical code GALPROP and two benchmark scenarios are considered: (i) a convection‑dominated model (no re‑acceleration, a single power‑law injection spectrum, bulk wind speed v₀,c as a free parameter) and (ii) a re‑acceleration‑dominated model (Alfvén speed v_A free, allowing for diffusive re‑acceleration of low‑energy particles). Both models include a break in the diffusion coefficient at a few GV and a second high‑rigidity break around 200 TV, as is standard in recent CR analyses.

For solar modulation three approaches are employed:

1. The classic force‑field approximation, which compresses the entire heliospheric transport into a single modulation potential φ.
2. An extended force‑field model that introduces a rigidity‑dependent break, i.e. two potentials φ₁ and φ₂ for low and high rigidity, to capture the observed charge‑sign and rigidity dependence.
3. The full three‑dimensional heliospheric propagation code HelMod, which solves the Parker transport equation including convection, diffusion, gradient and curvature drifts, and adiabatic energy losses, thus accounting for charge‑sign effects and the polarity reversal of the solar magnetic field.

The AMS‑02 data are divided into three time‑integrated periods based on sunspot activity: pre‑MAX (May 2011 – Nov 2012), MAX (Dec 2012 – Nov 2015) and post‑MAX (Dec 2015 – Apr 2021). In addition, a 7‑year integrated dataset (May 2011 – May 2018) and high‑energy measurements from CALET and DAMPE are used to constrain the high‑rigidity region where solar modulation is negligible.

A Bayesian inference framework (MultiNest) is used to explore the high‑dimensional parameter space, fitting propagation parameters, nuclear production cross‑section nuisance parameters, and the solar‑modulation parameters simultaneously. This joint fitting allows the authors to quantify degeneracies between low‑energy propagation (e.g. diffusion coefficient, Alfvén speed) and the solar‑modulation description.

Key findings:

• In the convection‑dominated scenario the simple force‑field model reproduces the fluxes of positively charged nuclei (protons, He, C, O, etc.) within ~10 % during periods of low solar activity (pre‑MAX and post‑MAX). However, during the solar maximum the force‑field potential varies rapidly and fails to describe the data, especially for antiprotons, which are sensitive to charge‑sign dependent drift effects.
• In the re‑acceleration scenario, low‑energy propagation parameters are strongly correlated with the modulation potential. When only the simple force‑field is used, the fit drives the Alfvén speed or diffusion parameters to unphysical values (e.g., negative v_A), indicating that the model cannot disentangle propagation from modulation with such a simplistic description.
• The extended force‑field model improves the description of antiprotons by allowing a rigidity break, but still leaves residual discrepancies of order 10‑15 % across all periods, reflecting the limited physical realism of the phenomenological approach.
• HelMod, which incorporates drift effects and the polarity reversal, provides the best overall fit, especially for antiprotons, yet the large number of free heliospheric parameters and the current uncertainties in the solar wind and magnetic field models translate into a systematic uncertainty of roughly 10‑15 % on the reconstructed local interstellar spectra (LIS) and on the Galactic propagation parameters.
• A clear time‑lag of 2–14 months between sunspot number and CR flux variations is identified, with the lag decreasing with increasing rigidity. Ignoring this lag introduces additional bias in the inferred modulation potentials.

Overall, the authors demonstrate that the dominant source of systematic error in modern CR analyses is not the experimental precision (which is at the percent level) but the incomplete knowledge of solar modulation. The 10‑15 % systematic uncertainty they quantify limits the ability to extract precise propagation parameters, to test secondary production models, and to search for exotic contributions such as dark‑matter annihilation signals.

The paper concludes that future measurements covering a full 22‑year Hale cycle, combined with refined heliospheric modeling (e.g., improved HelMod inputs, better treatment of drift and turbulence), are essential to reduce the modulation‑induced uncertainties. Moreover, the joint fitting framework presented here sets a benchmark for forthcoming analyses that aim to exploit the ever‑increasing precision of CR data while properly accounting for solar‑modulation systematics.