BAYESIAN ANALYSIS OF COSMIC RAY PROPAGATION: EVIDENCE AGAINST HOMOGENEOUS DIFFUSION

Abstract

We present the results of the most complete scan of the parameter space for cosmic ray (CR) injection and propagation. We perform a Bayesian search of the main GALPROP parameters, using the MultiNest nested sampling algorithm, augmented by the BAMBI neural network machine-learning package. This is the first study to separate out low-mass isotopes (p, $\overline{p}$ , and He) from the usual light elements (Be, B, C, N, and O). We find that the propagation parameters that best-fit p, $\overline{p}$ , and He data are significantly different from those that fit light elements, including the B/C and ¹⁰Be/⁹Be secondary-to-primary ratios normally used to calibrate propagation parameters. This suggests that each set of species is probing a very different interstellar medium, and that the standard approach of calibrating propagation parameters using B/C can lead to incorrect results. We present posterior distributions and best-fit parameters for propagation of both sets of nuclei, as well as for the injection abundances of elements from H to Si. The input GALDEF files with these new parameters will be included in an upcoming public GALPROP update.

Keywords: Galaxy: general; ISM: general; astroparticle physics; cosmic rays; diffusion; methods: statistical.

Figure 1.

Sets of neural-network-assisted nested sampling scans that we perform in this work. We separate p, $\overline{p}$, and He (left) from the light elements (right) into two separate runs. For the light elements, we also vary the elemental abundances in separate, faster runs, which are performed iteratively with the propagation parameter scans. Since the GALPROP output is linear in the injection abundances, this allows extremely rapid convergence of the abundances. We will keep the same color code throughout the text: blue for p, $\overline{p}$, and He results, magenta for light elements, and orange for the abundances.

Figure 2.

1D marginalized posterior distributions, showing 1- and 2-sigma credible intervals, for the propagation parameters that were varied in the propagation scan. Light blue: the constraints from p, $\overline{p}$, and He scan, using PAMELA and CREAM data only; Purple: light element scan, fitting Be, B, C, N, and O data. (Given in Table 3.) While most of the propagation parameters overlap between runs, there is a clear (>2σ) separation seen in the Alfvén speed and in the low-energy injection break rigidity ρ_br. Differences in the D₀ − z_h plane can be clearly seen in Figure 3. The injection index for p, $\overline{p}$, and He is also consistently lower below the 220 GV break, suggesting a harder source injection spectrum.

Figure 3.

2D posterior distributions, showing 1- and 2-sigma credible intervals for the p, $\overline{p}$, and He scan (blue), and for the light element (Be–Si, magenta). The posterior mean in each case is shown as a dot and the best fit as a cross.

Figure 4.

Spectral fluxes with 68% and 95% posterior regions from the posteriors of our light element (Be–Si) scan, shown in magenta in Figure 1, and using the HEAO modulation posteriors. Data shown are HEAO (blue), CREAM (green), and TRACER (cyan). The best fit is shown as a black line, and the dashed lines correspond to the LIS (unmodulated) spectra.

Figure 5.

Secondary-to-primary ratio 68% and 95% posterior bands from our light element (Be–Si) scan, shown in magenta in Figure 1. The $\overline{p}∕p$ ratio is shown to indicate that using the same propagation parameters for hydrogen yields a very bad fit to the data. Data shown are HEAO (blue), CREAM (green), ACE (light blue), ISOMAX (black), and PAMELA (red). The best fit is shown as a black line, and the dashed lines correspond to the LIS (unmodulated) ratios. In the left-hand panel, we use the HEAO modulation posterior, and the solid line uses the HEAO best-fit modulation potential. The dash-dotted line is the modulated spectrum using the best fit to the ACE-CRIS modulation potential; for clarity we do not show the posterior intervals for this case. Correspondingly, the central plot uses the ACE modulation (BF in black), and we show the best fit using the ISOMAX best-fit modulation potential with a dash–dotted line.

Figure 6.

Spectra and $\overline{p}∕p$ ratio 68% and 95% posterior bands of our $\overline{p}$, p, He scan, shown in blue in Figure 2. The best fit is plotted in black, and the dashed lines correspond to the LIS (umodulated) spectra. PAMELA data are shown in red. We also show recent AMS-02 (Aguilar et al. 2015, blue) for the available proton and helium flux data, which were not available at the time of our analysis (and hence are not included in the likelihood).

Figure 7.

95% (light bars) and 68% (dark bars) posterior intervals from our final abundance study. Total elemental abundances are in orange, while individual isotopes are in green. We show the latest determination of the solar photospheric (blue dots) elemental abundances and errors from Asplund et al. (2009), with updated heavier (A ⩾ 23) elemental abundances from Scott et al. (2015). We also show previously used values from GALPROP (Moskalenko et al. 2008) with open black circles.

Figure 8.

Posterior distributions of the modulation parameters for each experiment used in the fit, with 1- and 2-sigma credible intervals.

Figure 9.

Posterior distributions of the τ rescaling parameters, with 1- and 2-sigma credible intervals.

Figure 10.

1D posterior distributions (with 68% and 95% credible intervals) for the different CR propagation parameters in a low-resolution, {p, $\overline{p}$, and He} propagation scenario using MultiNest as a sampler (no neural network speed up, magenta) and from BAMBI runs with two different values for the neural network input parameter σ. Light blue: σ = 0.5; Orange: σ = 0.8. All BAMBI chains have been post-processed in the same way as in our main paper runs.