Diffusive acceleration in relativistic shocks: particle feedback

The spectral index $s$ of particles diffusively accelerated in a relativistic shock depends on the unknown angular diffusion function $\mathcal{D}$, which itself depends on the particle distribution function $f$ if acceleration is efficient. We develop a relaxation code to compute $s$ and $f$ for an arbitrary functional $\mathcal{D}$ that depends on $f$. A local $\mathcal{D}(f)$ dependence is motivated and shown, when rising (falling) upstream, to soften (harden) $s$ with respect to the isotropic case, shift the angular distribution towards upstream (downstream) directions, and strengthen (weaken) the particle confinement to the shock; an opposite effect on $s$ is found downstream. However, variations in $s$ remain modest even when $\mathcal{D}$ is a strong function of $f$, so the standard, isotropic-diffusion results remain approximately applicable unless $\mathcal{D}$ is both highly anisotropic and not a local function of $f$. A mild, $\sim 0.1$ softening of $s$, in both 2D and 3D, when $\mathcal{D}(f)$ rises sufficiently fast, may be indicated by ab-initio simulations.