Joint constraint on the propagation origin of the cosmic-ray spectral knee from energy spectrum and anisotropy observations

1. Problem Statement

The origin of the "knee" in the cosmic-ray (CR) energy spectrum (a steepening of the spectrum around a few PeV) remains one of the most significant unsolved mysteries in particle astrophysics. Three primary theoretical frameworks compete to explain this feature:

1. Acceleration Limits: The maximum energy achievable by Galactic sources (e.g., supernova remnants), leading to a rigidity-dependent ($R$) cutoff.
2. Propagation Effects: Changes in the diffusion properties of CRs as they traverse the Galaxy (e.g., variations in the diffusion coefficient), also leading to a rigidity-dependent feature.
3. Interaction Scenarios: Interactions with background particles or new physics, leading to a mass-number ($A$) dependent structure.

Distinguishing between acceleration and propagation origins is challenging because both predict similar rigidity-dependent spectral breaks. While spectral data alone cannot differentiate them, the anisotropy (directional distribution) of cosmic rays is sensitive to the diffusion process. If the knee arises from propagation changes, a specific correlation between the spectral break and anisotropy amplitude is expected. This study aims to test the propagation origin hypothesis by jointly analyzing the latest high-precision energy spectrum and anisotropy data.

2. Methodology

The authors constructed a comprehensive model to simulate cosmic-ray propagation and compared the results with observational data from LHAASO, AUGER, and other experiments.

• Propagation Model:

  • The study utilizes the DRAGON numerical code to solve the cosmic-ray propagation equation.
  • Source Distribution: Modeled using Supernova Remnants (SNRs) with a spatial distribution following a radial and vertical profile.
  • Injection Spectrum: A broken power-law spectrum is used. To account for low-energy spectral features (hardening at ~200 GV and softening at ~14 TeV), a nearby source component (located at $l=161^\circ, b=9^\circ$) is included.
  • Diffusion Coefficient ($D$): The core of the propagation-origin test is a rigidity-dependent double-power-law diffusion model. The diffusion coefficient is modified at the knee rigidity ($R_{knee}$):
    $$D(R) = D_0 \beta \epsilon \left(\frac{R}{R_0}\right)^\delta \times \begin{cases} 1 & R < R_{knee} \\ \left(\frac{R}{R_{knee}}\right)^{\delta_{knee}} & R \ge R_{knee} \end{cases}$$
    Here, $\delta_{knee}$ represents the change in the diffusion index across the knee.
  • Parameters: The values for $R_{knee}$ and $\delta_{knee}$ were fixed based on precise LHAASO proton spectrum measurements ($R_{knee} = 3.3 \pm 0.6$ PV, $\delta_{knee} = 0.99 \pm 0.29$).

• Anisotropy Calculation & Correction:

  • The model calculates the expected Galactic CR (GCR) anisotropy amplitude and phase.
  • Extragalactic Contamination: At energies above the knee (specifically in the EeV range observed by the Pierre Auger Observatory), the data contains a mix of Galactic and extragalactic cosmic rays (EGCRs). Since the model only describes GCRs, the authors developed a method to extract the pure GCR anisotropy component ($\delta_G$) from the total observed anisotropy ($\delta_{tot}$) using the relation:
    $$\delta_G = \frac{\delta_{tot} + \eta \delta_E}{1 - \eta}$$
    where $\eta$ is the fraction of extragalactic flux and $\delta_E$ is the extragalactic anisotropy.
  • Three correction scenarios were tested: Modified-I (using the last 3 high-energy points), Modified-II (using all high-energy points), and a Most Conservative Scenario (MCS) (assuming upper limits for EGCR contamination).

• Statistical Test:

  • A Kolmogorov-Smirnov (K-S) test was employed to compare the distribution of residuals between the observed anisotropy amplitudes and the model predictions. The null hypothesis assumes the knee originates from propagation effects.

3. Key Contributions

• Joint Analysis Framework: The paper provides a rigorous framework for testing propagation models by combining spectral constraints (to fix propagation parameters) with anisotropy predictions (to test the physical consistency of those parameters).
• Extragalactic Decontamination: It introduces a robust method to isolate the Galactic anisotropy signal from Auger data in the transition region (PeV to EeV), addressing a major systematic uncertainty in previous studies.
• Definitive Constraint: It moves beyond qualitative arguments to provide a quantitative statistical rejection of the propagation origin model based on current data.

4. Results

• Spectral Reproduction: The model successfully reproduces the LHAASO proton spectrum and the all-particle spectrum (including the first and second knees) by adjusting the diffusion coefficient at $R_{knee}$. The spectral fit is excellent.
• Anisotropy Discrepancy:
  • Low Energy (< 10 PeV): The model predictions for anisotropy amplitude and phase align well with observational data (e.g., IceCube, AS$\gamma$).
  • High Energy (> 1 PeV): A significant deviation emerges. The propagation model predicts a sharp increase in anisotropy amplitude near the knee energy due to the change in the diffusion coefficient. However, the observed data (after correcting for extragalactic contamination) shows a much flatter or lower amplitude than predicted.
• Statistical Rejection:
  • The K-S test was performed on the anisotropy amplitude distribution from 2.7 PeV to 2.48 EeV.
  • The test statistic ($D_{12}$) exceeded the critical value (0.375) for all three scenarios (Modified-I, Modified-II, and MCS).
  • Conclusion: The null hypothesis (that the knee is caused by propagation effects) is rejected at the 95% confidence level (CL).

5. Significance and Implications

• Ruling Out Propagation Origin: The study provides strong evidence that the spectral knee is not primarily caused by a change in the Galactic diffusion coefficient. If the knee were a propagation feature, the anisotropy would be significantly higher than what is observed.
• Implications for Origin Theories: Since the propagation model is disfavored, the origin of the knee is more likely to be:
  • Acceleration-limited: The maximum energy limit of Galactic sources (e.g., SNRs).
  • Interaction-based: Though less likely given the rigidity dependence, it remains a possibility.
• Future Directions: The authors emphasize that future high-precision measurements of the energy spectra and anisotropies of various nuclei (protons, helium, iron) in the knee region by LHAASO and other experiments are crucial. These will provide definitive tests to distinguish between acceleration limits and other potential mechanisms, and to refine the understanding of the transition from Galactic to extragalactic cosmic rays.

Caveats: The authors acknowledge a potential systematic uncertainty: the assumption that the extragalactic anisotropy measured in the Southern Hemisphere (by Auger) is representative of the Northern Hemisphere (where LHAASO is located). If this extrapolation is invalid, the 95% CL conclusion might need revision, though the authors argue the result is robust even under conservative assumptions.