  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.
