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Because early black holes (BHs) grew to $\sim10^{9} ~M_\odot$ in less than 1Gyr of cosmic time, BH seeding models face stringent constraints. Toefficiently constrain the parameter space of possible seeding criteria, wecombine the advantages of the cosmological IllustrisTNG (TNG) simulations withthe flexibility of semi-analytic modeling. We identify TNG galaxies as BHseeding sites based on various criteria including a minimum gas mass of$10^7$-$10^9~M_\odot$, total host mass of $10^{8.5}$-$10^{10.5}~M_\odot$, and amaximum gas metallicity of $0.01 - 0.1 ~Z_\odot$. Each potential host isassigned a BH seed with a probability of $0.01 - 1$; these BHs are then tracedthrough the TNG galaxy merger tree. This approach improves upon the predictivepower of the simple TNG BH seeding prescription, especially in the low-massregime at high redshift, and it is readily adaptable to other cosmologicalsimulations. Most of our seed models predict $z\lesssim4$ BH mass densitiesthat are consistent with empirical data as well as the TNG BHs. However,high-redshift BH number densities can differ by factors of $\sim$ 10 - 100between models. In most models, $\lesssim10^5~M_\odot$ BHs substantiallyoutnumber heavier BHs at high redshifts. Mergers between such BHs are primetargets for gravitational-wave detection with LISA. The $z=0$ BH mass densitiesin most models agree well with observations, but our strictest seeding criteriafail at high redshift. Our findings strongly motivate the need for betterempirical constraints on high-$z$ BHs, and they underscore the significance ofrecent AGN discoveries with JWST.

Building Semi-Analytic Black Hole Seeding Models Using IllustrisTNG Host Galaxies