We introduce an extension of the ELVIS project to account for the effects of the Milky Way galaxy on its subhalo population. Our simulation suite, Phat ELVIS, consists of twelve high-resolution cosmological dark matter-only (DMO) zoom simulations of Milky Way-size $Lambda$CDM~ haloes ($M_{rm v} = 0.7-2 times 10^{12} ,mathrm{M}_odot$) along with twelve re-runs with embedded galaxy potentials grown to match the observed Milky Way disk and bulge today. The central galaxy potential destroys subhalos on orbits with small pericenters in every halo, regardless of the ratio of galaxy mass to halo mass. This has several important implications. 1) Most of the $mathtt{Disk}$ runs have no subhaloes larger than $V_{rm max} = 4.5$ km s$^{-1}$ within $20$ kpc and a significant lack of substructure going back $sim 8$ Gyr, suggesting that local stream-heating signals from dark substructure will be rare. 2) The pericenter distributions of Milky Way satellites derived from $mathit{Gaia}$ data are remarkably similar to the pericenter distributions of subhaloes in the $mathtt{Disk}$ runs, while the DMO runs drastically over-predict galaxies with pericenters smaller than 20 kpc. 3) The enhanced destruction produces a tension opposite to that of the classic `missing satellites problem: in order to account for ultra-faint galaxies known within $30$ kpc of the Galaxy, we must populate haloes with $V_mathrm{peak} simeq 7$ km s$^{-1}$ ($M simeq 3 times 10^{7} ,mathrm{M}_odot$ at infall), well below the atomic cooling limit of $V_mathrm{peak} simeq 16$ km s$^{-1}$ ($M simeq 5 times 10^{8} ,mathrm{M}_odot$ at infall). 4) If such tiny haloes do host ultra-faint dwarfs, this implies the existence of $sim 1000$ satellite galaxies within 300 kpc of the Milky Way.
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