  While rich clusters are powerful sources of X-rays, gamma-ray emission from
these large cosmic structures has not been detected yet. X-ray radiative energy
losses in the central regions of relaxed galaxy clusters are so strong that one
needs to consider special sources of energy, likely AGN feedback, to suppress
catastrophic cooling of the gas. We consider a model of AGN feedback that
postulates that the AGN supplies the energy to the gas by inflating bubbles of
relativistic plasma, whose energy content is dominated by cosmic-ray (CR)
hadrons. If most of these hadrons can quickly escape the bubbles, then
collisions of CRs with thermal protons in the intracluster medium (ICM) should
lead to strong gamma-ray emission, unless fast diffusion of CRs removes them
from the cluster. Therefore, the lack of detections with modern gamma-ray
telescopes sets limits on the confinement time of CR hadrons in bubbles and CR
diffusive propagation in the ICM.
