  The hot halo gas distribution in the inner Milky Way (MW) contains key fossil
records of the past energetic feedback processes in the Galactic center. Here
we adopt a variety of spherical and disk-like MW halo gas models as initial
conditions in a series of simulations to investigate the formation of the Fermi
bubbles in the jet-shock scenario. The simulation results are compared directly
with relevant X-ray and gamma-ray observations of the Fermi bubbles to
constrain the halo gas distribution in the inner Galaxy before the Fermi bubble
event. Our best-fit gas density distribution can be described by a power law in
radius $n_{\rm e}(r)=0.01(r/1 \text{~kpc})^{-1.5}$ cm$^{-3}$. Our study can not
determine if there is an inner density core, which if exists, should be very
small with size $r_{c} \lesssim 0.5$ kpc. When extrapolating to large radii
$r\sim 50-90$ kpc, our derived density distribution lies appreciably below the
recently estimated gas densities from ram-pressure stripping calculations,
suggesting that the halo gas density profile either flattens out or has one or
more discontinuities within $10 \lesssim r \lesssim 50$ kpc. Some of these
discontinuities may be related to the eROSITA bubbles, and our derived gas
density profile may correspond to the hot gas distribution in the inner eROSITA
bubbles about $5$ Myr ago.
