The supermassive black hole population from seeding via collisions in Nuclear Star Clusters

The coexistence of nuclear star clusters (NSCs) and supermassive black holes (SMBHs) in galaxies with stellar masses $\sim 10^{10}~$M$_\odot$, the scaling relations between their properties and properties of the host galaxy (e.g., $M_{NSC}^{stellar}-M_{galaxy}^{stellar}$, $M_{BH}-M_{galaxy}^{stellar}$), and the fact that NSCs seem to take on the role of SMBHs in less massive galaxies and vice versa in the more massive ones, suggest that the origin of NSCs and SMBHs is related. In this study, we implement an 'in-situ' NSC formation scenario, where NSCs are formed in the center of galaxies due to star formation in the accumulated gas. We explore the impact of the free parameter $A_{res}$ which regulates the amount of gas transferred to the NSC reservoir, playing a crucial role in shaping the cluster's growth. Simultaneously, we include a BH seed formation recipe based on stellar collisions within NSCs in the Semi-Analytical Model (SAM) Galacticus to explore the resulting population of SMBHs. We determine the parameter space of the NSCs that form a BH seed and find that in initially more compact NSCs the formation of these BH seeds is more favorable, leading to the formation of light, medium and heavy BH seeds which finally reach masses up to $\sim 10^9$~M$_\odot$ and is comparable with the observed SMBH mass function at masses above $10^8$~M$_\odot$. Additionally, we compare the resulting population of NSCs with a derived NSC mass function from the stellar mass function of galaxies from the GAMA survey at $z