Statistics of giant radio haloes from electron reacceleration models

The most important evidence of non-thermal phenomena in galaxy clusters comes from giant radio haloes (GRHs), spectacular synchrotron radio sources extended over >= Mpc scales, detected in the central regions of a growing number of massive galaxy clusters. A promising possibility to explain these sources is given by in situ stochastic reacceleration of relativistic electrons by turbulence generated in the cluster volume during merger events. Cassano and Brunetti have recently shown that the expected fraction of clusters with radio haloes and the increase of such a fraction with cluster mass can be reconciled with present observations provided that a fraction of 20-30 per cent of the turbulence in clusters is in the form of compressible modes. In this work, we extend the above-mentioned analysis by including a scaling of the magnetic field strength with cluster mass. We show that, in the framework of the reacceleration model, the observed correlations between the synchrotron radio power of a sample of 17 GRHs and the X-ray properties of the hosting clusters are consistent with, and actually predicted by a magnetic field dependence on the virial mass of the form B proportional to M-v(b), with b greater than or similar to 0.5 and typical mu G strengths of the average B intensity. The occurrence of GRHs as a function of both cluster mass and redshift is obtained: the evolution of such a probability depends on the interplay between synchrotron and inverse Compton losses in the emitting volume, and it is maximized in clusters for which the two losses are comparable. The most relevant findings are that the predicted luminosity functions of GRHs are peaked around a power P-1.4 GHz similar to 10(24) W Hz(-1), and severely cut off at low radio powers due to the decrease of the electron reacceleration in smaller galaxy clusters, and that the occurrence of GRHs at 1.4 GHz beyond a redshift z similar to 0.7 appears to be negligible. As a related check, we also show that the predicted integral radio source counts within a limited volume (z <= 0.2) are consistent with present observational constraints. Extending the source counts beyond z = 0.2, we estimate that the total number of GRHs to be discovered at similar to mJy radio fluxes could be similar to 100 at 1.4 GHz. Finally, the occurrence of GRHs and their number counts at 150 MHz are estimated in view of the forthcoming operation of low-frequency observatories (LOFAR, LWA) and compared with those at higher radio frequencies.