Scaling Relations for Galaxies Prior to Reionization
  The first galaxies in the Universe are the building blocks of all observed
galaxies. We present scaling relations for galaxies forming at redshifts $z \ge
15$ when reionization is just beginning. We utilize the ``Rarepeak'
cosmological radiation hydrodynamics simulation that captures the complete star
formation history in over 3,300 galaxies, starting with massive Population III
stars that form in dark matter halos as small as ~$10^6 M_\odot$. We make
various correlations between the bulk halo quantities, such as virial, gas, and
stellar masses and metallicities and their respective accretion rates,
quantifying a variety of properties of the first galaxies up to halo masses of
$10^9 M_\odot$. Galaxy formation is not solely relegated to atomic cooling
halos with virial temperatures greater than $10^4$ K, where we find a dichotomy
in galaxy properties between halos above and below this critical mass scale.
Halos below the atomic cooling limit have a stellar mass -- halo mass
relationship $\log M_\star \simeq 3.5 + 1.3\log(M_{\rm vir} / 10^7 M_\odot)$.
We find a non-monotonic relationship between metallicity and halo mass for the
smallest galaxies. Their initial star formation events enrich the interstellar
medium and subsequent star formation to a median of $10^{-2} Z_\odot$ and
$10^{-1.5} Z_\odot$, respectively, in halos of total mass $10^7 M_\odot$ that
is then diluted by metal-poor inflows, well beyond Population III
pre-enrichment levels of $10^{-3.5} Z_\odot$. The scaling relations presented
here can be employed in models of reionization, galaxy formation and chemical
evolution in order to consider these galaxies forming prior to reionization.
