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Going, going, gone dark: Quantifying the scatter in the faintest dwarf galaxies

We predict the stellar mass-halo mass (SMHM) relationship for dwarf galaxies and their satellites residing in halos down to M$_{halo} =$ 10$^7$ M$_{\odot}$ with 10$^4$ M$_{\odot} <$ M$_{star}$($z=0$) $< 10^8$ M$_{\odot}$, and quantify the predicted scatter in the relation at the low mass end, using cosmological simulations. The galaxies were drawn from a cosmological simulation of dwarf galaxies, run with the N-body + SPH code, ChaNGA, at a high resolution of 60 pc. For M$_{halo} > 10^9$ M$_{\odot}$, the simulated SMHM relationship agrees with literature determinations, including exhibiting a small scatter. However, the scatter in the SMHM relation increases dramatically for lower-mass halos. We find that some of this scatter is due to {\em dark dwarfs}, halos devoid of stars. However, even when only considering well-resolved halos that contain a stellar population, the scatter in stellar mass reaches nearly 1 dex for M$_{halo}$($z=0$) 10$^7$ M$_{\odot}$. Much of this scatter is due to including satellites of the dwarf galaxies that have had their halo masses reduced through tidal stripping. The fraction of dark dwarfs (those that contain no stars) increases substantially with decreasing halo mass. When these dark halos are considered, the true scatter in the SMHM at low masses is even larger. At the faintest end of the SMHM relation probed by our simulations, a galaxy cannot be assigned a unique halo mass based solely on its luminosity. We provide a formula to stochastically populate low-mass halos following our results. Our predicted large scatter at low halo masses increases the slope of the resulting stellar mass function on the ultra-faint dwarf galaxy scales currently being probed by such surveys as the Dark Energy Survey or the Hyper-Suprime Cam Subaru Strategic Program, and in the future by the Large Synoptic Survey Telescope.