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Register a new userTime-dependent treatment of cosmic-ray spectral steepening due to turbulence driving
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Martin Pohl
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Cosmic-ray acceleration at non-relativistic shocks relies on scattering by turbulence that the cosmic rays drive upstream of the shock. We explore the rate of energy transfer from cosmic rays to non-resonant Bell modes and the spectral softening it implies. Accounting for the finite time available for turbulence driving at supernova-remnant shocks yields a smaller spectral impact than found earlier with steady-state considerations. Generally, for diffusion scaling with the Bohm rate by a factor $eta$, the change in spectral index is at most $eta$ divided by the Alfvenic Mach number of the thermal sub-shock. For $M_mathrm{A}lesssim 50$ it is well below this limit. Only for very fast shocks and very efficient cosmic-ray acceleration the change in spectral index may reach $0.1$. For standard SNR parameters it is negligible. Independent confirmation is derived by considering the synchrotron energy losses of electrons: if intense nonthermal multi-keV emission is produced, the energy loss, and hence the spectral steepening, is very small for hadronic cosmic rays that produce TeV-band gamma-ray emission.
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We show that the complex shape of the cosmic ray (CR) spectrum, as recently measured by PAMELA and inferred from Fermi-LAT gamma-ray observations of molecular clouds in the Gould belt, can be naturally understood in terms of basic plasma astrophysics phenomena. A break from a harder to a softer spectrum at blue rigidity Rsimeq 10 GV follows from a transition from transport dominated by advection of particles with Alfven waves to a regime where diffusion in the turbulence generated by the same CRs is dominant. A second break at Rsimeq 200 GV happens when the diffusive propagation is no longer determined by the self-generated turbulence, but rather by the cascading of externally generated turbulence (for instance due to supernova (SN) bubbles) from large spatial scales to smaller scales where CRs can resonate. Implications of this scenario for the cosmic ray spectrum, grammage and anisotropy are discussed.
The direct measurements of cosmic rays (CRs), after correction for the propagation effects in the interstellar medium, indicate that their source spectra are likely to be significantly steeper than the canonical $E^{-2}$ spectrum predicted by the standard Diffusive Shock Acceleration (DSA) mechanism. The DSA has long been held responsible for the production of galactic CRs in supernova remnant (SNR) shocks. The $gamma$-ray probes of the acceleration spectra of CRs on-the-spot, inside of the SNRs, lead to the same conclusion. We show that the steep acceleration spectrum can be attributed to the $combination$ of (i) spherical expansion, (ii) tilting of the magnetic field along the shock surface and (iii) shock deceleration. Because of (i) and (ii), the DSA is efficient only on two ``polar caps of a spherical shock where the local magnetic field is within $simeq45^{circ}$ to its normal. The shock-produced spectrum observed edge-on steepens with the particle energy because the number of freshly accelerated particles with lower energies continually adds up to a growing acceleration region. We demonstrate the steepening effect by obtaining an exact self-similar solution for the particle acceleration at expanding shock surface with an arbitrary energy dependence of particle diffusivity $kappa$. We show that its increase toward higher energy steepens the spectrum, which deeply contrasts with the standard DSA spectrum where $kappa$ cancels out.
Understanding the transport of energetic cosmic rays belongs to the most challenging topics in astrophysics. Diffusion due to scattering by electromagnetic fluctuations is a key process in cosmic-ray transport. The transition from a ballistic to a diffusive-propagation regime is presented in direct numerical calculations of diffusion coefficients for homogeneous magnetic field lines subject to turbulent perturbations. Simulation results are compared with theoretical derivations of the parallel diffusion coefficients dependencies on the energy and the fluctuation amplitudes in the limit of weak turbulence. The present study shows that the widely-used extrapolation of the energy scaling for the parallel diffusion coefficient to high turbulence levels predicted by quasi-linear theory does not provide a universally accurate description in the resonant-scattering regime. It is highlighted here that the numerically calculated diffusion coefficients can be polluted for low energies due to missing resonant interaction possibilities of the particles with the turbulence. Five reduced-rigidity regimes are established, which are separated by analytical boundaries derived in the present work. Consequently, a proper description of cosmic-ray propagation can only be achieved by using a turbulence-level-dependent diffusion coefficient and can contribute to solving the Galactic cosmic-ray gradient problem.
Cosmic-ray transport in astrophysical environments is often dominated by the diffusion of particles in a magnetic field composed of both a turbulent and a mean component. This process needs to be understood in order to properly model cosmic-ray signatures. One of the most important aspects in the modeling of cosmic-ray diffusion is that fully resonant scattering, the most effective such process, is only possible if the wave spectrum covers the entire range of propagation angles. By taking the wave spectrum boundaries into account, we quantify cosmic-ray diffusion parallel and perpendicular to the guide field direction at turbulence levels above 5% of the total magnetic field. We apply our results of the parallel and perpendicular diffusion coefficient to the Milky Way. We show that simple purely diffusive transport is in conflict with observations of the inner Galaxy, but that just by taking a Galactic wind into account, data can be matched in the central 5 kpc zone. Further comparison shows that the outer Galaxy at $>5$ kpc, on the other hand, should be dominated by perpendicular diffusion, likely changing to parallel diffusion at the outermost radii of the Milky Way.
Blazars are potential candidates of cosmic-ray acceleration up to ultrahigh energies ($Egtrsim10^{18}$ eV). For an efficient cosmic-ray injection from blazars, $pgamma$ collisions with the extragalactic background light (EBL) and cosmic microwave background (CMB) can produce neutrino spectrum peaks near PeV and EeV energies, respectively. We analyze the contribution of these neutrinos to the diffuse background measured by the IceCube neutrino observatory. The fraction of neutrino luminosity originating from individual redshift ranges is calculated using the distribution of BL Lacs and FSRQs provided in the textit{Fermi}-LAT 4LAC catalog. Furthermore, we use a luminosity dependent density evolution to find the neutrino flux from unresolved blazars. The results obtained in our model indicate that as much as $approx10%$ of the flux upper bound at a few PeV energies can arise from cosmic-ray interactions on EBL. The same interactions will also produce secondary electrons and photons, initiating electromagnetic cascades. The resultant photon spectrum is limited by the isotropic diffuse $gamma$-ray flux measured between 100 MeV and 820 GeV. The latter, together with the observed cosmic-ray flux at $E>10^{16.5}$ eV, can constrain the baryonic loading factor depending on the maximum cosmic-ray acceleration energy.
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