Revisiting the role of the streaming instability for the cosmic-ray spectrum in the GeV to TeV range

TL;DR

This paper investigates whether self-generated turbulence from the CR streaming instability can explain the GeV–TeV spectral hardening observed in the Galactic CR spectrum. By solving a coupled set of CR transport and wave evolution equations in a 1D flux-tube framework and comparing two turbulence cascades (Kolmogorov and Kraichnan), the authors demonstrate that the streaming instability can produce a spectral break near a few hundred GeV. They perform a broad parameter exploration and perform a Bayesian MCMC fit to CALET data, finding correlated constraints on the turbulence level, injection strength, and Kolmogorov/Kraichnan constants, with Kolmogorov-like turbulence generally favored. The results suggest that self-consistent CR transport is a viable explanation for the observed hardening and provide quantitative priors on ISM properties, while highlighting the need to include damping and multi-dimensional effects in future work.

Abstract

A complete understanding of the cosmic-ray energy spectrum remains a challenge to theory that must be met by comprehensive modeling efforts. One of these is the subject of the present study, namely, an explanation of the recently discovered spectral hardening at $\sim 300$ GeV with self-consistently treated cosmic-ray diffusion, where self-generated waves resulting from the streaming instability impact the diffusion of high-energy particles. We revisit the corresponding model by Blasi et al. (2012), perform an extensive parameter study, and determine an optimal range of parameters that best fit the cosmic-ray data. We conclude that self-consistently treated cosmic-ray transport remains a competitive alternative to explain the spectral hardening of the cosmic-ray energy spectrum at a few hundred GeV.

Figures (8)

• Figure 1: Comparison of the expected break energies for different values of $A,\,\eta,\,c_K,\,\alpha,$ and $B_0$ in case of either Kolmogorov- or Kraichnan-type of wave cascade. The respective fixed values are chosen as in Fig. \ref{['fig:default_params']}.
• Figure 2: Comparison of the cosmic-ray fluxes (top) and wave spectra (bottom) obtained with the default parameter sets. In the Kolmogorov case, we choose $A=10^{36}\,\mathrm{cm}^{-5}\mathrm{g}^{-3}\mathrm{s}^2$, $\alpha=4.25$, $\eta=0.03$, $v_A=25\,\mathrm{km\, s}^{-1}$, $B_0=1.6\,\mu\mathrm{G}$, $H= 4\,\mathrm{kpc}, k_0^{-1}=50\,\mathrm{pc}$, whereas for Kraichnan it is $A=10^{34.5}\,\mathrm{cm}^{-5}\mathrm{g}^{-3}s^2$, $\alpha=4.08$, $\eta=0.04$, $v_A=1\,\mathrm{km\,s}^{-1}$, $B_0=1.5\,\mu$G, $H=4\,$kpc, $k_0^{-1}=50\,$pc.
• Figure 3: Kolmogorov. We vary the model parameters around the default parameter set (as specified in Fig. \ref{['fig:default_params']}), one in each row while keeping the rest fixed. Line styles are changing from dashed-dotted, dashed, dotted, solid with increasing parameter value. The energies for the right plots are derived as $E=(\sqrt{1+(qB_0/(kmc^2))^2}-1)mc^2$, except for the second last row, where we varied $B_0$. The expected breaks according to Eq. (\ref{['eq:p_break']}) are denoted by small vertical lines.
• Figure 4: Kraichnan. Starting from the default values, the parameters are changed in a similar manner as in Fig. \ref{['fig:param_kolm']}, with the line styles kept in the same changing order. Again, small vertical lines denote the expected breaks due to the streaming instability.
• Figure 5: Corner plot of our parameters for a Kolmogorov-type cascade. On the diagonal, the marginalized distributions of the Alfvén speed $v_A$, turbulence measure $\eta$, mean magnetic field $B_0$, cosmic-ray normalization constant $A$ and Kolmogorov constant $c_K$ are displayed with the dashed lines denoting the 68%-credible areas. The remaining panels show the marginalized two-dimensional distributions.