Different physical and numerical sources of scatter in the $M_{\star}$-$M_{\mathrm{BH}}$ relation and their connection to galaxy evolution

Observations have established that the masses of supermassive black holes (SMBHs) correlate tightly with the stellar masses of their host galaxies, albeit with substantial scatter. The magnitude of this scatter as a function of galaxy mass and redshift contains valuable information about the origin of SMBHs and the physical nature of their co-evolution with galaxies. In this work, we highlight this connection by studying the scatter in the $M_{\rm BH}$-$M_\star$ relation for massive galaxies in the Illustris, TNG100, and EAGLE cosmological simulations. We find that TNG100 shows significantly lower scatter than Illustris and EAGLE, reflecting different BH feedback models. Using numerical experiments, we separate different contributions to the scatter, including an intrinsic component. At $z=0$, Illustris and EAGLE show $\sim 0.3$ dex intrinsic scatter dominated by BH accretion, while the smaller scatter in TNG100 is dominated by hierarchical merging, implying more tightly quenched massive galaxies. BH seed mass variations can add scatter, though their impact at $z=0$ depends on the feedback model. Without AGN feedback the scatter is much larger for low-mass galaxies ($\gtrsim 0.5$ dex for $\log M_\star < 10^{10.5},\mathrm{M_\odot}$ at $z=0-3$), underscoring the crucial role of feedback in SMBH-galaxy co-evolution. In contrast, hierarchical merging of quenched systems is the main factor reducing scatter for massive galaxies. Based on our results, we expect that the scatter in the $M_{\rm BH}$-$M_\star$ relation at high redshift could be particularly powerful in providing clues to the origin of SMBHs.