The building up of the black hole–stellar mass relation

We derive the growth of SMBHs relative to the stellar content of their host galaxy predicted under the assumption of BH accretion triggered by galaxy encounters occurring during their merging histories. The latter are described through Monte Carlo realizations, and are connected to gas processes, star formation and BH accretion using a semi-analytic model of galaxy formation in a cosmological framework. This allows us to connect the star formation process in the host galaxies to the growth of Supermassive Black Holes. We show that, within this framework, the ratio ! ! (MBH/M" )(z)/(MBH/M" )(z = 0) between the Black Hole mass and the galactic stellar mass (normalized to the local value) depends on both BH mass and redshift. While the average value and the spread of ! (z) increase with z, such an e" ect is larger for massive BHs, reaching values ! # 5 for massive Black Holes (M $ 10 9 M% ) at z ! 4, in agreement with recent observations of high-redshift QSOs; this is due to the the e" ectiveness of interactions in triggering BH accretion in high-density environments (where massive haloes form) at high redshifts. To investigate how di" erent observations of ! (z) fit within our framework, we worked out specific predictions for sub-samples of the simulated galaxies corresponding to the di" erent observational samples for which measurements of ! have been obtained. We found that for Broad Line AGNs at intermediate redshifts 1 " z " 2 values of! # 2 are expected, with a mild trend toward larger value for increasing BH mass. Instead, when we select from our Monte Carlo simulations only extremely gas rich, rapidly star forming galaxies at the epoch of peak in the cosmic star formation (2 & z & 3), we find low values 0.3 & ! & 1.5, consistent with recent observational findings on samples of sub-mm galaxies; in the framework of our model, these objects end up at z = 0 in low-to-intermediate mass BHs (M & 10 9 M% ), and they do not represent typical paths leading to local massive galaxies. The latter have formed preferentially through paths (in the M" ' MBH plane) passing above the local M" ' MBH relation. We discuss how the global picture emerging from the model is consistent with a downsizing scenario, where massive BHs accrete a larger fraction of their final mass at high redshifts z $ 4