and motivated choice for a harder injection spectrum [@nagaiando]. This choice is, in addition, consistent with the results of the simulations of Ref. [@Berrington:2002bh]. However, if cosmic rays in galaxy clusters are accelerated in [*weaker*]{} merger shocks the expectation is one of a softer injection spectral index. In this case, strong confinement of cosmic rays in clusters will likely not make the spectrum much harder. Numerical simulations, though, indicate a low efficiency for the acceleration of cosmic rays at weak merger shocks [@kangjones]. In summary, thus, we regard our benchmark value $\alpha_p=2.7$ as an example case rather than, strictly, as an upper limit to the cosmic ray injection spectrum.

       Model    $\alpha_p$   $\beta_p$
      -------- ------------ -----------
       CR\_HC     $2.1$       $-0.5$
       CR\_HF     $2.1$         $1$
       CR\_SC     $2.7$       $-0.5$
       CR\_SF     $2.7$         $1$

      : Summary of parameters for cosmic ray models used. The parameter $\alpha_p$ indicates the primary cosmic-ray injection spectral index, while the coefficient $\beta_p$ stands for the bias of the cosmic-ray source spatial distribution with respect to the cluster’s thermal gas spatial distribution (see eq. \[eq:spatial\]). For all models, we set the ratio of the energy density of cosmic rays compared to the thermal gas $X_p=0.1$. \[tab:cr\_models\]

-   The results of recent hydrodynamical simulations 