Acceleration of primary and secondary particles in galaxy clusters by compressible MHD turbulence: from radio haloes to gamma-rays

Radio observations discovered large-scale non-thermal sources in the central Mpc regions of dynamically disturbed galaxy clusters (radio haloes). The morphological and spectral properties of these sources suggest that the emitting electrons are accelerated by spatially distributed and gentle mechanisms, providing some indirect evidence for turbulent acceleration in the intergalactic medium (IGM). Only deep upper limits to the energy associated with relativistic protons in the IGM have been recently obtained through gamma and radio observations. Yet these protons should be (theoretically) the main non-thermal particle component in the IGM implying the unavoidable production, at some level, of secondary particles that may have a deep impact on the gamma-ray and radio properties of galaxy clusters. Following Brunetti & Lazarian, in this paper we consider the advances in the theory of magnetohydrodynamics (MHD) turbulence to develop a comprehensive picture of turbulence in the IGM and extend our previous calculations of particle acceleration by compressible MHD turbulence by considering self-consistently the re-acceleration of both primary and secondary particles. Under these conditions we expect that radio to gamma-ray emission is generated from galaxy clusters with a complex spectrum that depends on the dynamics of the thermal gas and dark matter. The non-thermal emission results in very good agreement with radio observations and with present constraints from hard X-ray and gamma-ray observations. In our model giant radio haloes are generated in merging (turbulent) clusters only. However, in case secondaries dominate the electron component in the IGM, we expect that the level of the Mpc-scale synchrotron emission in more relaxed clusters is already close to that of the radio upper limits derived by present observations of clusters without radio haloes. Important constraints on cluster physics from future observations with present and future telescopes are also discussed.