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Appendix A: Contribution from secondary electrons

We briefly report here on the contribution from secondary electrons injected by hadronic proton-proton collision (Blasi & Colafrancesco 1999; Dolag & Enßlin 2000) to the total radio emission. The injection rate of secondary electrons is computed from the pool of CR-protons simulated at run-time by our ENZO cosmological simulation using PPM and a two-fluid model for modelling cosmic rays, as in Vazza et al. (2014b). For this we used the formalism by Dolag & Enßlin (2000), assuming a constant energy spectrum of α = 1.1 (N(E) ∝ E^{− α}) for cosmic-ray protons. The final budget of cosmic rays in the simulation follows from the assumed acceleration efficiency at shocks, which we took from Kang & Ryu (2013) and renormalised downwards by a factor 10 to be consistent with the upper limits from the Fermi satellite (Ackermann et al. 2014). Here we used the result from a (300 Mpc)³ volume simulated with 2048³ cells and DM-particles. This run did not use MHD, and therefore we assumed the magnetic field entirely in post-processing. Similar to the main article, we computed the gas energy in all cells and computed the magnetic field in a HA and LA model.

Figure A.1 shows the outcome of these two acceleration mechanisms for the HA model: the contribution from secondary electrons to the diffuse cosmic web is negligible everywhere, that is, it is only a few percent of the primary contribution. The emission increases in the centre of the clusters owing to the increase of the gas density and only forms rather low-power radio halos (e.g. Dolag & Enßlin 2000). Outside the central cluster regions, however, this signal usually is much weaker than the primary emission from strong accretion shocks. The distribution function of pixels in the maps (Fig. A.2) shows that the primary contribution by far dominates the high end of the emission tail in both magnetic field models. However, the secondary contribution becomes dominant in the low-brightness end of the emission distribution in an LA scenario because of the decrease

in primary emission from the WHIM outside of halos and because the secondary emission is nearly unaltered in the LA scenario since it mostly comes from well within the halos. The confusion caused by the secondary emission from galaxy clusters will probably become stronger at high redshift, where massive bright clusters can dominate the low-power background emission compared to the filaments in the LA scenario. Overall, we conclude that regardless of the assumed amplification model, the brightest emission from the cosmic web is expected to originate from primary electrons accelerated by cosmic shocks, and that the signal from secondary particles can only be relevant at very low surface brightness values and at high redshift in a low magnetisation scenario of the WHIM.

	Fig. A.1 Distribution of radio emission from the HA and the LA model in a cosmological run with CR-protons and including the effect of “secondary” radio emission from electrons injected by hadronic collision.
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	Fig. A.2 Total radio emission from primary electrons injected by shocks (right) and from secondary electrons injected by hadronic collisions (left) in a subvolume of our 20483 simulation with CR-physics (Vazza et al. 2014b). The colour bar gives the emission in units of [log 10 W/Hz].
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© ESO, 2015