The Suppression of Star Formation on the Smallest Scales: What Role Does
  Environment Play?
  The predominantly ancient stellar populations observed in the lowest-mass
galaxies (i.e. ultra-faint dwarfs) suggest that their star formation was
suppressed by reionization. Most of the well-studied ultra-faint dwarfs,
however, are within the central half of the Milky Way dark matter halo, such
that they are consistent with a population that was accreted at early times and
thus potentially quenched via environmental processes. To study the potential
role of environment in suppressing star formation on the smallest scales, we
utilize the Exploring the Local Volume in Simulations (ELVIS) suite of $N$-body
simulations to constrain the distribution of infall times for low-mass subhalos
likely to host the ultra-faint population. For the ultra-faint satellites of
the Milky Way with star-formation histories inferred from
$Hubble~Space~Telescope$ imaging, we find that environment is highly unlikely
to play a dominant role in quenching their star formation. Even when including
the potential effects of pre-processing, there is a $\lesssim 0.1\%$
probability that environmental processes quenched all of the known ultra-faint
dwarfs early enough to explain their observed star-formation histories.
Instead, we argue for a mass floor in the effectiveness of satellite quenching
at roughly $M_{\star} \sim 10^{5}~M_{\odot}$, below which star formation in
surviving galaxies is globally suppressed by reionization. We predict a large
population of quenched ultra-faint dwarfs in the Local Field ($1 <
R/R_{\rm{vir}} < 2$), with as many as $\sim250$ to be discovered by future
wide-field imaging surveys.
