Gamma-ray and radio background constraints on cosmic rays in Milky Way circumgalactic medium

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), its anisotropy and radio continuum. We modify different models of CGM gas in hydrostatic equilibrium discussed in literature by including a cosmic ray population, parametrized by $\eta \equiv P_{\rm CR}/P_{\rm th}$. For the simplest isothermal model, while the IGRB intensity allows $\eta \lesssim 3$, the anisotropy resulting from the Solar system's off-center position in MW rules out all values of $\eta$. For the precipitation model, in which the cooling of the CGM gas is regulated with an optimum ratio of cooling time to free-fall time,while the observed IGRB intensity allows $\eta \lesssim 230$, the observed anisotropy allows only very large values of $\eta$, of order $\gtrsim 100$. 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.