  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, such
as the Fermi and eROSITA bubbles. 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 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. Our derived gas density profile may correspond to the hot gas
distribution in the inner 10-kpc region of the eROSITA bubbles about $5$ Myr
ago.
