Cluster Radio Halos at the crossroads between astrophysics and cosmology in the SKA era

Giant Radio Halos (RH) are diffuse, Mpc-sized, synchrotron radio sources observed in a fraction of merging galaxy clusters. The current scenario for the origin of RHs assumes that turbulence generated during cluster mergers re-accelerates pre-existing fossil and/or secondary electrons in the intra-cluster-medium (ICM) to the energies necessary to produce the observed radio emission. Moreover, more relaxed clusters could host diffuse "off state" halos produced by secondary electrons. In this Chapter we use Monte Carlo simulations, that combine turbulent-acceleration physics and the generation of secondaries in the ICM, to calculate the occurrence of RHs in the Universe, their spectral properties and connection with properties of the hosting clusters. Predictions for SKA1 surveys are presented at low (100-300 MHz) and mid (1-2 GHz) frequencies assuming the expected sensitivities and spatial resolutions of SKA1. SKA1 will step into an unexplored territory allowing us to study the formation and evolution of RHs in a totally new range of cluster masses and redshift, allowing firm tests of the current theoretical hypothesis. In particular, the combination of SKA1-LOW and SUR will allow the discovery of ~1000 ultrasteep- spectrum halos and to detect for the very first time "off state" RHs. We expect that at least ~2500 giant RHs will be discovered by SKA1-LOW surveys up to z~0.6. Remarkably these surveys will be sensitive to RHs in a cluster mass range (down to ~10^14 solar masses) and redshifts (up to ~1) that are unexplored by current observations. SKA1 surveys will be highly competitive with present and future SZ-surveys in the detection of high-redshift massive objects