A two-phase model of galaxy formation: IV. Seeding and growing supermassive black holes in dark matter halos

We present a theoretical framework for seeding and growing supermassive black holes (SMBHs) in dark matter halos along their assembly histories. Seeds are bred out of Pop-III stars formed during the first collapse of pristine gas in mini-halos that have reached the $\rm H_2$-cooling limit, modulated by UV radiation from star formation and dynamical heating from fast halo assembly. Such breeding persists until the enrichment of the intergalactic medium (IGM) enables Pop-II stars to form. Post-seeding growth of black holes (BHs) is driven by distinct channels, starting with episodic super-Eddington accretion associated with nuclear bursts induced by global disturbances of galaxies, followed by sustained sub-Eddington accretion via capturing sub-clouds formed in self-gravitating gas clouds (SGCs) in halos of fast assembly, and ending with merger-dominated, quiescent growth. We implement the model in subhalo merger trees to build a coherent framework to follow SMBH-galaxy-halo co-evolution across the whole history and structural hierarchy. BH seeds are bred with a broad mass spectrum of $M_{\rm BH} = 10 - 10^5\,{\rm M}_\odot$ at $z \approx 20 - 30$ in mini-halos with masses of $10^5 - 10^8\,{\rm M}_\odot$. Nuclear bursts provide the key condition for seeds to grow into SMBHs. The $M_{\rm BH}$-$M_*$ relation is a multi-piece, redshift-dependent function shaped by the interplay among different growth channels. Our model predictions are broadly consistent with existing observations; especially, a population of BHs reminiscent of 'little red dots' (LRDs) discovered by JWST naturally results from the seeding and growing processes. Potential future tests of the model are discussed.