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Second order Fermi reacceleration mechanisms and large scale synchrotron radio emission in intra-cluster bridges

Radio observations at low frequencies with the Low Frequency Array (LOFAR) start discovering gigantic radio bridges connecting pairs of massive galaxy clusters. These observations probe unexplored mechanisms of in situ particle acceleration that operate on volumes of several Mpc$^3$. Numerical simulations suggest that such bridges are dynamically complex and that weak shocks and super-Alfvénic turbulence can be driven across the entire volume of these regions. In this Letter we explore, for the first time, the role of second order Fermi mechanisms for the reacceleration of relativistic electrons interacting with turbulence in these peculiar regions. We assume the turbulent energy flux measured in simulations and adopt a scenario in which relativistic particles scatter with magnetic field lines diffusing in super-Alfvénic turbulence and magnetic fields are amplified by the same turbulence. We show that steep spectrum and volume filling synchrotron emission can be generated in the entire intra-cluster bridge region thus providing a natural explanation for radio bridges. Consequently, radio observations have the potential to probe the dissipation of energy on scales larger than galaxy clusters and second order Fermi mechanisms operating in physical regimes that are still poorly explored. This has a potential impact on several branches of astrophysics and cosmology.