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

TL;DR

This work addresses whether the cosmic-ray knee arises from propagation by jointly analyzing knee-region energy spectra and anisotropy within a spatially dependent diffusion framework. It introduces a rigidity-dependent knee in the diffusion coefficient at $R_{knee}$ with index change $\delta_{knee}$, supplemented by a nearby local source to reproduce sub-100 TeV spectral features, and constrains these parameters using the LHAASO proton spectrum. Although the model can reproduce the knee in spectra, anisotropy data—after correcting for extragalactic contamination—disfavor a propagation-origin knee at 95% CL, highlighting hemispheric uncertainties and the need for further measurements. The results constrain knee-origin scenarios and motivate future high-precision, nucleus-resolved spectral and anisotropy observations to provide a definitive test.

Abstract

The origin mechanism of the cosmic-ray knee region remains an unresolved mystery, with acceleration, interaction, and propagation models drawing significant attention. The latest experimental observations of the PeV total spectrum, composition energy spectrum, and anisotropy-particularly the precise measurements of the proton spectrum by the LHAASO experiment-have provided crucial breakthroughs in uncovering its origin. Based on the latest LHAASO measurements of the proton energy spectrum, combined with cosmic-ray spectral and anisotropy data, this study proposes that the spectral index variation in the knee region arises from changes in the propagation coefficient. By introducing a knee position $\rm \mathcal{R}_{knee}$ and an index variation $\rm δ_{knee}$, we construct a rigidity-dependent double-power-law diffusion model to reproduce the knee-region spectral structure. Through modifications to the diffusion coefficient, we successfully replicate the observed knee-region spectral structure in the LHAASO proton spectrum and calculate the corresponding anisotropy. Under current data and model dependencies, a joint analysis of the energy spectrum and anisotropy does not support the propagation origin model of the cosmic-ray knee at a 95\% confidence level. We hope that future LHAASO experiments will provide precise measurements of the energy spectra and anisotropies of various nuclei in the knee region, thereby offering a definitive test of the propagation model as the origin mechanism of the knee-region spectral structure.

Figures (7)

• Figure 1: The calculated proton spectrum compared with observed data. The data points are taken from the AMS-02 2015PhRvL.114q1103A, DAMPE-2019 2019SciA....5.3793A, IceTop 2019PhRvD.100h2002A, and LHAASO 2025arXiv250514447T. The gray shaded area represents the $1~\sigma$ uncertainty band of the energy spectrum calculated based on the $1~\sigma$ errors of $\rm \mathcal{R}_{knee}$ and $\rm \delta_{knee}$.
• Figure 2: Sum flux of individual species, compared with all-particle spectrum data from experiments. All-particle spectra are from Tibet-III 2008ApJ...678.1165A, TALE 2018ApJ...865...74A, TA 2023APh...15102864A, NUCLEON 2019AdSpR..64.2546G, KASCADE 2024JCAP...05..125K, HAWC 2017PhRvD..96l2001A, KASCADE-Grande 2013APh....47...54A, TUNKA-133 2020APh...11702406B and LHAASO LHAASOspectrum2024prl.
• Figure 3: The energy dependence of the phases of the dipole anisotropies when adding all of the major elements together. The data points are taken from Norikura 1973ICRC....2.1058S, Ottawa 1981ICRC...10..246B, Bolivia 1985PSS...33.1069S, Budapest 1985PSS...33.1069S, Hobart 1985PSS...33.1069S, London 1985PSS...33.1069S, Misato 1985PSS...33.1069S, Socomo 1985PSS...33.1069S, Yakutsk 1985PSS...33.1069S, Liapootah 1995ICRC....4..639M, Matsushiro 1995ICRC....4..648M, Poatina 1995ICRC....4..635F, kamiokande1 1997PhRvD..56...23M, PeakMusala 1975ICRC....2..586G, Norikura 1989NCimC..12..695N, Macro 2003PhRvD..67d2002A, SuperK 2007PhRvD..75f2003G, EAS-TOP 1995ICRC....2..800A1996ApJ...470..501A2009ApJ...692L.130A, Baksan 1987ICRC....2...22A, Milagro 2009ApJ...698.2121A, K-Grande KASCADEAniso2015, IceCube 2010ApJ...718L.194A2012ApJ...746...33A2025ApJ...981..182A, IceTop 2013ApJ...765...55A, AS-$\gamma$2005ApJ...626L..29A2017ApJ...836..153A2015ICRC...34..355A, ARGO 2018ApJ...861...93B, AUGER 2024ApJ...976...48A.
• Figure 4: The same as Fig. \ref{['fig3']} but for the amplitudes. The gray shaded area represents the $1~\sigma$ uncertainty band of the amplitude of anisotropy calculated based on the $1~\sigma$ errors of $\rm \mathcal{R}_{knee}$ and $\rm \delta_{knee}$, which have been precisely determined by LHAASO proton energy spectrum 2025arXiv250514447T. The amplitude of anisotropy with energies below 10 PeV is shown in the upper panel, while the amplitude of anisotropy with energies above 1 PeV is plotted in the lower panel. In the lower panel, $\rm Modified-I$: Contamination correction applied using only the last three high-energy (blue) data points to adjust low-energy (black) data; $\rm Modified-II$: Comprehensive correction using all high-energy (blue) data points; MCS: the most conservative scenario, giving upper limits of GCR amplitude constrained by upper limits of EGCR amplitude, see Table \ref{['tab3']} in \ref{['appendixC']}.
• Figure 5: Fitting of extragalactic cosmic ray anisotropy observed by the AUGER experiment2024ApJ...976...48A. The red line represents the fitting result of the three highest energy data points, and the blue line represents the fitting result of all four data points.