Understanding the radio spectral indices of galaxy cluster relics by superdiffusive shock acceleration

Galaxy cluster merger shocks are the likely source of relativistic electrons, but many observations do not fit into the standard acceleration models. In particular, there is a long-standing discrepancy between the radio derived Mach numbers M-radio and the Mach numbers derived from X-ray measurements, M-X. Here, we show how superdiffusive electron transport and superdiffusive shock acceleration (SSA) can help to solve this problem. We present a heuristic derivation of the superlinear time growth of the mean square displacement of particles, <Delta x(2)> proportional to t(beta), and of the particle energy spectral index in the framework of SSA. The resulting expression for the radio spectral index alpha is then used to determine the superdiffusive exponent beta from the observed values of alpha and of the compression ratio for a number of radio relics. Therefore, the fact that M-radio > M-X can be explained by SSA without the need to make assumptions on the energy spectrum of the seed electrons to be re-accelerated. We also consider the acceleration times obtained in the diffusive case, based both on the Bohm diffusion coefficient and on the quasi-linear diffusion coefficient. While in the latter case the acceleration time is consistent with the estimated electron energy loss time, in the former case it is much shorter.