  Among the most important goals in cosmology is detecting and quantifying
small ($M_{\rm halo}\simeq10^{6-9}~\mathrm{M}_\odot$) dark matter (DM)
subhalos. Current probes around the Milky Way (MW) are most sensitive to such
substructure within $\sim20$ kpc of the halo center, where the galaxy
contributes significantly to the potential. We explore the effects of baryons
on subhalo populations in $\Lambda$CDM using cosmological zoom-in baryonic
simulations of MW-mass halos from the Latte simulation suite, part of the
Feedback In Realistic Environments (FIRE) project. Specifically, we compare
simulations of the same two halos run using (1) DM-only (DMO), (2) full
baryonic physics, and (3) DM with an embedded disk potential grown to match the
FIRE simulation. Relative to baryonic simulations, DMO simulations contain
$\sim2\times$ as many subhalos within 100 kpc of the halo center; this excess
is $\gtrsim5\times$ within 25 kpc. At $z=0$, the baryonic simulations are
completely devoid of subhalos down to $3\times10^6~\mathrm{M}_\odot$ within
$15$ kpc of the MW-mass galaxy, and fewer than 20 surviving subhalos have
orbital pericenters <20 kpc. Despite the complexities of baryonic physics, the
simple addition of an embedded central disk potential to DMO simulations
reproduces this subhalo depletion, including trends with radius, remarkably
well. Thus, the additional tidal field from the central galaxy is the primary
cause of subhalo depletion. Subhalos on radial orbits that pass close to the
central galaxy are preferentially destroyed, causing the surviving subhalo
population to have tangentially biased orbits compared to DMO predictions. Our
method of embedding a disk potential in DMO simulations provides a fast and
accurate alternative to full baryonic simulations, thus enabling suites of
cosmological simulations that can provide accurate and statistical predictions
of substructure populations.
