SIMBAD references

We develop a semi-analytic model to explore different prescriptions of supermassive black hole (SMBH) fuelling. This model utilizes a merger-triggered burst mode in concert with two possible implementations of a long-lived steady mode for assembling the mass of the black hole in a galactic nucleus. We improve modelling of the galaxy-halo connection in order to more realistically determine the evolution of a halo's velocity dispersion. We use four model variants to explore a suite of observables: the M_*-σ relation, mass functions of both the overall and broad-line quasar population, and luminosity functions as a function of redshift. We find that 'downsizing' is a natural consequence of our improved velocity dispersion mappings, and that high-mass SMBHs assemble earlier than low-mass SMBHs. The burst mode of fuelling is sufficient to explain the assembly of SMBHs to z = 2, but an additional steady mode is required to both assemble low-mass SMBHs and reproduce the low-redshift luminosity function. We discuss in detail the trade-offs in matching various observables and the interconnected modelling components that govern them. As a result, we demonstrate the utility as well as the limitations of these semi-analytic techniques.