to the cosmic ray proton energy density is limited by a few constraints [@Reimer:2004ac]:

    -   The gamma rays produced in inelastic collisions yielding neutral pions, which subsequently decay into two photons, must be consistent with the EGRET upper limits [@reimer]

    -   The inverse Compton up-scattering of microwave photons by the secondary $e^{\pm}$ population produced by charged pion decay (as for the neutral pions these particles are produced in collisions of the primary cosmic ray protons with nuclei in the ICM) will give rise to non-thermal hard X-ray and soft gamma-ray emission. The intensity of this radiation must also be consistent with observational data.

    -   Lastly, the mentioned secondary $e^\pm$ will also radiate at radio frequencies via synchrotron emission, providing additional constraints from radio data.

    Assuming $\alpha_p\simeq2.4$, Ref. [@Reimer:2004ac] shows that in the case of Coma the radio data are at the level predicted for $X_p\sim0.2$, with appropriate assumptions on the magnetic field distribution. Tighter constraints come from high frequency radio data, and depend quite sensitively on $\alpha_p$ and, more importantly, on the value of the average magnetic field. EGRET data, again for the case of Coma, put milder constraints ($X_p\lesssim0.45$ for $\alpha_p=2.1$ and $X_p\lesssim0.25$ for $\alpha_p=2.5$ [@pfrommer2004]). While we consider 