Maximally-hard spectra from diffusive shock-acceleration in pulsar-wind nebulae

The processes leading to the exceptionally hard radio spectra of pulsar-wind nebulae (PWNe) are not yet understood. Radio photon spectral indices among $29$ PWNe from the literature show an approximately normal, $\alpha=0.2\pm0.2$ distribution. We present $\sim 3\sigma$ evidence for a distinct sub-population of PWNe, with a hard spectrum $\alpha=0.01\pm0.06$ near the termination shock and significantly softer elsewhere, possibly due to a recent evacuation of the shock surroundings. Such spectra, especially in the hard sub-population, suggest a Fermi process, such as diffusive shock acceleration, at its extreme, $\alpha=0$ limit. We show that this limit is approached for sufficiently anisotropic small-angle scattering, enhanced on either side of the shock for particles approaching the shock front. In the upstream, the spectral hardening is mostly associated with an enhanced energy gain, possibly driven by the same beamed particles crossing the shock. Downstream, the main effect is a diminished escape probability, but this lowers the acceleration efficiency to $\lesssim25\%$ for $\alpha=0.3$ and $\lesssim1\%$ for $\alpha=0.03$.