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

This study demonstrates that current solar modulation models introduce significant systematic uncertainties (10–15%) in cosmic-ray propagation parameters and local interstellar spectra, particularly for antiprotons and during solar maximum, thereby limiting the precision of current cosmic-ray studies despite high-accuracy AMS-02 measurements.

Original authors: Isabelle John, Alessandro Cuoco, Mattia Di Mauro

Published 2026-02-23

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CC BY 4.0

Original authors: Isabelle John, Alessandro Cuoco, Mattia Di Mauro

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Cosmic Rays and the Solar "Force Field"

Imagine the universe is filled with a constant rain of tiny, high-speed particles called cosmic rays. These are like cosmic hailstones made of protons, helium, and other atoms, zooming through space at nearly the speed of light.

Our Sun, however, is a bit of a bully. It constantly blows a wind (the solar wind) and has a giant magnetic bubble around it (the heliosphere). When cosmic rays try to enter our solar system to reach Earth, they have to fight their way through this wind and magnetic field.

This fight slows them down and changes their energy. Scientists call this process Solar Modulation. It's like trying to run through a crowded, windy hallway to get to a party; by the time you arrive, you're tired, and your speed has changed.

The Problem: We Don't Know the Rules of the Hallway

Scientists want to know what the cosmic rays looked like before they hit the Sun's hallway. This "original" version is called the Local Interstellar Spectrum (LIS). If we can figure out the original speed and direction of the cosmic rays, we can solve mysteries about:

Where they came from (supernovas?).
If they are hiding Dark Matter.
How they travel through the galaxy.

But to get the "original" picture, we have to mathematically reverse the Sun's effects. The problem is, we don't have a perfect map of the hallway.

The Sun's behavior changes over an 11-year cycle:

Solar Minimum: The hallway is quiet and orderly.
Solar Maximum: The hallway is chaotic, with magnetic storms and turbulence.

The paper argues that our current math models for "reversing" the Sun's effects are too simple. They work okay when the Sun is calm, but they fall apart when the Sun is wild. This introduces a systematic uncertainty (a built-in error) of about 10–15% in our calculations. Since modern telescopes (like AMS-02) can measure things with 1% precision, this 15% error is like trying to measure a human hair with a ruler that has a 1-inch gap between the marks. It ruins the precision.

The Experiment: Testing Three Different Maps

The authors used data from the AMS-02 experiment (a super-precise detector on the International Space Station) that has been watching cosmic rays for a full solar cycle (about 11 years). They tested three different ways to model the Sun's "hallway":

The Force-Field Approximation (The "One-Size-Fits-All" Hat):
- Analogy: Imagine you assume the hallway wind is always the same strength, no matter the time of day or the weather. You just use one number to represent the wind.
- Result: This works pretty well for positive particles (like protons) when the Sun is calm. But it fails miserably when the Sun is active, and it completely fails for antiprotons (the "anti-matter" twins of protons).
The Extended Force-Field (The "Two-Speed" Hat):
- Analogy: This model admits the wind changes depending on how fast you are running. It uses two different numbers: one for slow particles and one for fast ones.
- Result: This is better, but it still struggles to explain why antiprotons behave so differently from protons.
HelMod (The "Full Simulation" Video Game):
- Analogy: Instead of a simple hat, this is a full 3D video game simulation of the hallway, accounting for every twist, turn, and magnetic storm.
- Result: It's the most realistic, but it's also very hard to use because it requires knowing the "original" cosmic ray speed to begin with (which is what we are trying to find!). It creates a bit of a "chicken and egg" problem.

The Twist: The "Anti-Matter" Mystery

One of the most interesting findings involves antiprotons.

Protons are like regular balls rolling down a hill.
Antiprotons are like anti-balls.

The Sun treats them differently based on their electric charge. The paper found that our simple math models (the "One-Size-Fits-All" hat) cannot explain the behavior of antiprotons at all. The models predict they should act one way, but the data shows they act completely differently.

The authors suspect this is because antiprotons are "second-hand" particles (created when protons smash into gas), giving them a weird energy shape that our simple models just can't handle.

The Conclusion: We Need Better Tools

The main takeaway is this: We are hitting a wall in our understanding of the universe because our model of the Sun is too rough.

Even though our telescopes are incredibly precise, our math for the Sun's magnetic field is the weak link. The authors conclude that:

Current models introduce a 10–15% error in our maps of the galaxy.
We need to wait for a full 22-year cycle (two sun cycles) to see how the Sun behaves when it flips its magnetic polarity.
We need more complex, time-dependent models that can handle the chaos of a "Solar Maximum" and the weird behavior of antimatter.

In short: We have the best camera ever (AMS-02), but we are trying to develop the photo using a blurry lens (Solar Modulation models). Until we fix the lens, we can't see the universe as clearly as we'd like.

1. Problem Statement

Cosmic-ray (CR) propagation models rely on accurately reconstructing the Local Interstellar Spectrum (LIS) from data measured near Earth. However, CRs below rigidities of $\sim30\text{--}40$ GV are significantly suppressed and distorted by solar modulation—the interaction of CRs with the solar wind and the heliospheric magnetic field (HMF).

The Core Issue: Current solar modulation models introduce systematic uncertainties that are comparable to or larger than the statistical precision of modern experiments like AMS-02.
Specific Challenges:
- Time Dependence: Solar activity follows an 11-year cycle, altering modulation strength and charge-sign dependence (drift effects) over time.
- Model Limitations: Simple phenomenological models (e.g., the Force-Field approximation) often fail to capture complex physics like charge-sign dependence, rigidity breaks, and time-varying drifts, especially during solar maximum.
- Degeneracies: There is a strong degeneracy between low-energy propagation effects (like re-acceleration) and solar modulation, making it difficult to disentangle the two without robust modeling.

2. Methodology

The authors performed a comprehensive Bayesian analysis using the GALPROP propagation code and the MultiNest sampler to fit cosmic-ray data across different solar activity phases.

Data Source: Time-dependent flux measurements from the AMS-02 experiment covering a full solar cycle (May 2011 – April 2021).
- Species: Protons, Helium, Carbon, Nitrogen, Oxygen, Boron, and crucially, Antiprotons (the only negatively charged CR nucleus).
- Time Periods: The data was divided into three distinct phases to test model consistency:
  1. preMAX: Before solar maximum (2011–2012).
  2. MAX: During solar maximum (2012–2015).
  3. postMAX: After solar maximum (2015–2021), featuring a flipped HMF polarity.
- Supplementary Data: 7-year time-integrated AMS-02 data, CALET, DAMPE, and Voyager (for unmodulated LIS below 1 GV).
Propagation Models: Two benchmark scenarios were tested:
1. Convection-dominated (conv): Includes Galactic wind convection; no re-acceleration.
2. Re-acceleration-dominated (reacc): Includes Alfvénic re-acceleration; no convection.
Solar Modulation Models: Three distinct approaches were compared:
1. Force-Field (FF): A single potential parameter ( $\phi$ ) assuming steady-state, spherically symmetric propagation.
2. Extended Force-Field (Ext): Introduces a rigidity-dependent potential with a break ( $R_{br}$ ) and separate potentials for low/high rigidity and positive/negative charges.
3. HelMod: A full numerical solution of the Parker transport equation (convection, diffusion, drifts, adiabatic losses). Note: In this study, HelMod matrices were used with a fixed LIS assumption rather than a fully self-consistent fit.

3. Key Contributions

Charge-Sign Sensitivity: The study uniquely leverages antiproton data to probe charge-sign dependent modulation effects, which are critical for testing drift models and HMF polarity effects.
Time-Resolved Systematics: Instead of a single time-integrated fit, the authors performed independent fits for preMAX, MAX, and postMAX periods. This allows for a direct test of whether the reconstructed LIS and propagation parameters remain consistent across different solar configurations (a requirement for a valid physical model).
Quantification of Uncertainties: The paper quantifies the systematic error introduced specifically by solar modulation modeling limitations, separating it from statistical measurement errors.

4. Key Results

A. Performance of Solar Modulation Models

Force-Field (FF) Limitations:
- Positively Charged Nuclei: The FF approximation works reasonably well during low/moderate solar activity (preMAX, postMAX) in convection-dominated models but fails significantly during solar maximum (MAX).
- Antiprotons: The FF model fails to describe antiproton fluxes at all times. The residuals show poor fits below $\sim10$ GV. The authors attribute this to the unique, peaked shape of the antiproton LIS (due to production thresholds) which the simple FF potential cannot model, combined with the model's inability to handle charge-sign dependence correctly.
Extended Model (Ext):
- Provides a better fit than FF, particularly during solar maximum and for antiprotons.
- However, in the re-acceleration (reacc) scenario, the Ext model yields unphysical results. The fit returns a LIS with a "kink" at $\sim4$ GV and modulation potentials that contradict theoretical expectations (e.g., antiprotons being more modulated than protons during periods where drifts should suppress protons more). This suggests a strong degeneracy between re-acceleration and modulation parameters that the fit cannot resolve.
HelMod:
- Provides a detailed physical description but, when used with a fixed LIS (as done here), yields results similar to the other models in terms of LIS reconstruction. The authors note that a fully self-consistent HelMod fit (re-deriving the LIS simultaneously) is necessary for future work.

B. Systematic Uncertainties

LIS Inconsistency: When using the Force-Field model, the reconstructed LIS fluxes for protons differ by 10–15% across the three time periods (preMAX, MAX, postMAX). Since the true LIS should be time-invariant on these scales, this discrepancy represents the systematic uncertainty floor.
Parameter Degeneracy:
- In the conv model, the proton injection spectral index ( $\gamma_{1,p}$ ) varies by $\sim4\sigma$ between MAX and postMAX when using the FF model.
- In the reacc model, the Alfvén velocity ( $v_A$ ) shows similar large variations, indicating that the model cannot distinguish between energy gains from re-acceleration and energy losses from solar modulation.

C. Antiproton Specifics

The antiproton LIS is more stable across time periods than the proton LIS, suggesting it is less sensitive to the specific choice of modulation model (likely because antiprotons are secondary particles produced at higher energies where modulation is weaker).
However, the persistent failure of the FF model for antiprotons suggests that the standard approximation is fundamentally inadequate for describing the modulation of secondary, peaked-spectrum species.

5. Significance and Conclusions

Precision Limit: The current systematic uncertainty in solar modulation modeling is 10–15%. This is a major bottleneck for cosmic-ray physics, as it exceeds the percent-level statistical precision of AMS-02 data. This limits the ability to search for exotic components (e.g., dark matter) or precisely determine propagation parameters.
Model Recommendations:
- The simple Force-Field approximation is insufficient for high-precision studies, particularly during solar maximum and for antiprotons.
- Extended models improve fits but may introduce unphysical degeneracies when combined with complex propagation physics (like re-acceleration).
- Full numerical codes (like HelMod) are necessary but require self-consistent coupling with propagation models.
Future Outlook: The authors emphasize that future data spanning a full 22-year solar cycle (covering both magnetic polarity epochs) is essential to break the degeneracies between solar modulation and low-energy propagation processes. Without this, the reconstruction of the LIS and the interpretation of cosmic-ray anomalies will remain limited by solar modulation systematics.