Cluster mergers and non-thermal phenomena: a statistical magneto-turbulent model

There is now firm evidence that the intracluster medium (ICM) consists of a mixture of hot plasma, magnetic fields and relativistic particles. The most important evidence for non-thermal phenomena in galaxy clusters comes from the spectacular synchrotron radio emission diffused over Mpc scales observed in a growing number of massive clusters and, more recently, in the hard X-ray tails detected in a few cases in excess of the thermal bremsstrahlung spectrum. A promising possibility to explain giant radio haloes is given by the presence of relativistic electrons reaccelerated by some kind of turbulence generated in the cluster volume during merger events. With the aim of investigating the connection between thermal and non-thermal properties of the ICM, in this paper we develop a statistical magneto-turbulent model which describes in a self-consistent way the evolution of the thermal ICM and that of the non-thermal emission from clusters. Making use of the extended Press-Schechter formalism, we follow cluster mergers and estimate the injection rate of the fluid turbulence generated during these energetic events. We then calculate the evolution of the spectrum of the relativistic electrons in the ICM during the cluster life by taking into account both the electron acceleration due to the merger-driven turbulence and the relevant energy losses of the electrons. We end up with a synthetic population of galaxy clusters for which the evolution of the ICM and of the non-thermal spectrum emitted by the accelerated electrons is calculated. The generation of detectable non-thermal radio and hard X-ray emission in the simulated clusters is found to be possible during major merger events for reliable values of the model parameters. In addition the occurrence of radio haloes as a function of the mass of the parent clusters is calculated and compared with observations. In this case it is found that the model expectations are in good agreement with observations: radio haloes are found in about 30 per cent of the more massive clusters in our synthetic population (M greater than or similar to 1.8 x 10(15) M-circle dot) and in about 4 per cent of the intermediate massive clusters (9 x 10(14) < M < 1.8 x 10(15) M-circle dot), while the radio halo phenomenon is found to be extremely rare in the case of the smaller clusters.