Tracing the first stars and galaxies of the Milky Way
  We use 30 high-resolution dark matter halos of the $Caterpillar$ simulation
suite to probe the first stars and galaxies of Milky Way-mass systems. We
quantify the environment of the high-$z$ progenitors of the Milky Way and
connect them to the properties of the host and satellites today. We identify
the formation sites of the first generation of Population III (Pop III) stars
($z$ ~ 25) and first galaxies ($z$ ~ 22) with several different models based on
a minimum halo mass including a simple model for Lyman-Werner feedback. Through
this method we find approximately 23,000 $\pm$ 5,000 Pop III potentially
star-forming sites per Milky Way-mass host, though this number is drastically
reduced to ~550 star-forming sites when Lyman-Werner feedback is included, as
it has critical effects at these length scales. The majority of these halos
identified form in isolation (96% at $z$ = 15) and are not subject to external
enrichment by neighboring halos (median separation ~1 pkpc at $z$ = 15), though
half merge with a system larger than themselves within 1.5 Gyrs. Approximately
55% of the entire population has merged into the host halo by $z$ = 0. Using
particle tagging, we additionally trace the Pop III remnant population to $z$ =
0 and find an order of magnitude scatter in their number density at small (i.e.
r $<$ 5 kpc) and large (i.e. r $>$ 50 kpc) galactocentric radii at $z$ = 0.
Using our large number of realizations, we provide fitting functions for
determining the number of progenitor minihalo and atomic cooling halo systems
that present-day dwarf galaxies and the Magellanic cloud system might have
accreted since their formation. We demonstrate that observed dwarf galaxies
with stellar masses below 10$^{4.6}$ M$_{\odot}$ are unlikely to have merged
with any other star-forming systems.
