  The nature of the first seeds of supermassive black holes (SMBHs) is
currently unknown, with postulated initial masses ranging from
$\sim10^5~M_{\odot}$ to as low as $\sim10^2~M_{\odot}$. However, most existing
cosmological simulations resolve BHs only down to $\sim10^5-10^6~M_{\odot}$. In
this work, we introduce a novel sub-grid BH seed model that is directly
calibrated from high resolution zoom simulations that can trace the formation
and growth of $\sim 10^3~M_{\odot}$ seeds forming in halos with pristine,
star-forming gas. We trace the BH growth along merger trees until their
descendants reach masses of $\sim10^4$ or $10^5~M_{\odot}$. The descendants
assemble in galaxies with a broad range of properties (e.g., halo masses
$\sim10^7-10^9~M_{\odot}$) that evolve with redshift and are sensitive to seed
parameters. The results are used to build a new stochastic seeding model that
directly seeds these descendants in lower resolution versions of our zoom
region. Remarkably, we find that by seeding the descendants simply based on
total galaxy mass, redshift and an environmental richness parameter, we can
reproduce the results of the detailed gas based seeding model. The baryonic
properties of the host galaxies are well reproduced by the mass-based seeding
criterion. The redshift-dependence of the mass-based criterion captures the
influence of halo growth, star formation and metal enrichment on seed
formation. The environment based seeding criterion seeds the descendants in
rich environments with higher numbers of neighboring galaxies. This accounts
for the impact of unresolved merger dominated growth of BHs, which produces
faster growth of descendants in richer environments with more extensive BH
merger history. Our new seed model will be useful for representing a variety of
low mass seeding channels within next generation larger volume uniform
cosmological simulations.
