Nonlinear diffusive shock acceleration with upstream escape reproduces DAMPE observations

TL;DR

The study addresses how nonlinear diffusive shock acceleration (DSA) with cosmic-ray backreaction and upstream escape can reproduce DAMPE's proton spectrum features—gradual hardening from hundreds of GeV to multi-TeV energies followed by an exponential cutoff. It develops a framework where CR pressure creates an extended upstream precursor and an escape term with rate $\tau_{\mathrm{esc}}^{-1}$, producing a momentum-dependent cutoff that replaces an artificial hard cutoff. A self-consistent injection mechanism links downstream thermal leakage to nonlinear shock modification via an injection parameter $\eta$, with CR heating providing negative feedback that stabilizes the solution. Applied in a conventional one-zone diffusion model with $D(p) \propto p^{1/3}$ for young SNR-like parameters, the model reproduces DAMPE data, showing spectral hardening and a cutoff produced by upstream escape and nonlinear feedback between injection and CR heating.

Abstract

We develop a self-consistent nonlinear extension of diffusive shock acceleration that incorporates cosmic ray (CR) backreaction on the shock precursor together with a physically motivated upstream-escape mechanism that produces an exponential high energy cutoff. The CR pressure gradient decelerates the upstream flow facing the shock wave, generating an extended precursor in which higher rigidity particles sample a larger cumulative velocity gradient and thereby acquire a progressively harder spectrum. Finite-size/escape effects are modeled by a momentum-dependent loss term, which naturally terminates acceleration and steepens the spectrum near the cutoff. The precursor compression ratio is not imposed as a closure condition but is determined dynamically by requiring consistency between the injection rate inferred from thermal leakage at the subshock and the injection strength demanded by the nonlinear shock modification, with CR-driven wave heating providing stabilizing negative feedback. Applying the model to young supernova-remnant-like parameters and standard one-zone Galactic diffusion, we reproduce the main features of the latest DAMPE proton spectrum: gradual hardening from hundreds of GeV to multi-TeV energies and a subsequent exponential cutoff at tens of TeV. The resulting spectral evolution follows directly from the competition between precursor-mediated nonlinear feedback and upstream escape.

Figures (3)

• Figure 1: Injection-rate response curves: the thermal-injection constraint (red, Eq. \ref{['eq40']}) and the system's self-consistent nonlinear response $\eta(R)$ (blue, Eq. \ref{['eq41']}). Their intersection determines the precursor compression ratio $R$.
• Figure 2: Comparison between the theoretical model (black curve) and the latest DAMPE-2025 DAMPE2025 proton spectrum, together with CALET-2022 Adriani2022 and ISS-CREAM Choi2022 data.
• Figure 3: Cumulative normalized velocity gradient $U$ (blue) and spectral index $q$ (red) as functions of kinetic energy, computed for the same parameters as in Fig. \ref{['fig:1']}.