energy density in cosmic rays to the energy density of the thermal gas; this ratio is taken to be a power law with radius, parametrized by $\beta_p$ as $$\label{eq:spatial}
X_p(r) = X_p(R_{500})\left(\frac{r}{R_{500}}\right)^{\beta_p}$$ where $X_p$ is the ratio of the energy density of cosmic rays compared to the thermal gas, and $R_{500}$ is the radius of an enclosed spherical overdensity $500$ times the critical density of the universe at the source’s redshift [@nagaiando]. The parameter $\beta_p$ physically reflects the possibility that the spatial distribution of the sources of high-energy cosmic rays deviates, via a power-law as a function of radius, from the density profile of the thermal gas in the cluster. While a variation in the Mach number during the course of a cluster merger can drive different injection indexes at different locations [@Berrington:2002bh], we neglect here, for simplicity, any spatial variation associated to $\alpha_p$.

In summary, the simple model we use here depends on three orthogonal parameters: $\alpha_p$, that sets the spectral shape of both the injected primary cosmic rays and the resulting gamma rays, $X_p$ that sets the normalization of the gamma-ray flux, and $\beta_p$, that (together with the gas density distribution, that can be inferred e.g. from X-ray 