Cosmic ray heating in cool core clusters - II. Self-regulation cycle and
  non-thermal emission
  Self-regulated feedback by active galactic nuclei (AGNs) appears to be
critical in balancing radiative cooling of the low-entropy gas at the centres
of galaxy clusters and in regulating star formation in central galaxies. In a
companion paper, we found steady-state solutions of the hydrodynamic equations
that are coupled to the CR energy equation for a large cluster sample. In those
solutions, radiative cooling in the central region is balanced by streaming CRs
through the generation and dissipation of resonantly generated Alfv{\'e}n waves
and by thermal conduction at large radii. Here we demonstrate that the
predicted non-thermal emission resulting from hadronic CR interactions in the
intra-cluster medium exceeds observational radio (and gamma-ray) data in a
subsample of clusters that host radio mini halos (RMHs). In contrast, the
predicted non-thermal emission is well below observational data in cooling
galaxy clusters without RMHs. These are characterised by exceptionally large
AGN radio fluxes, indicating high CR yields and associated CR heating rates. We
suggest a self-regulation cycle of AGN feedback in which non-RMH clusters are
heated by streaming CRs homogeneously throughout the central cooling region. We
predict {\em radio micro halos} surrounding the AGNs of these CR-heated
clusters in which the primary emission may predominate the hadronically
generated emission. Once the CR population has streamed sufficiently far and
lost enough energy, the cooling rate increases, which explains the increased
star formation rates in clusters hosting RMHs. Those could be powered
hadronically by CRs that have previously heated the cluster core.
