We study the interaction of cosmic rays ( CRs ) with the diffuse circumgalactic gas of Milky Way ( MW ) galaxy that results in hadronic \gamma - ray emission and radio synchrotron emission .
We aim to constrain the CR population in our circumgalactic medium ( CGM ) with the help of observed isotropic \gamma -ray background ( IGRB ) and radio continuum .
We modify different models of CGM gas in hydrostatic equilibrium discussed in literature by including a cosmic ray population , parametrized by \eta \equiv P _ { CR } / P _ { th } .
For the simplest isothermal model , the IGRB only allows \eta \lesssim 1 .
On the other hand , the precipitation model , in which the cooling of the CGM gas is regulated with an optimum ratio of cooling time to free-fall time , the IGRB allows \eta \lesssim 230 .
There is a small forbidden range near \eta \sim 1 that can be circumvented by changing boundary conditions .
We also discuss the importance of anisotropy in \gamma -ray emission from CGM because of the Solar system ’ s off-center position in the MW , and this consideration may pose problems for models of CR dominated CGM even when flux limit is satisfied .
The radio continuum limits \eta \lesssim 400 for precipitation model and does not constrain isothermal model , however these constraints are mitigated by synchrotron loss time being comparable to CR diffusion time scales .
These bounds are relevant for current numerical simulations that indicate a significant CR population in CGM of galaxies of MW mass .