The building up of the black hole-stellar mass relation

We derive the growth of supermassive black holes (SMBHs) relative to the stellar content of their host galaxy predicted under the assumption of black hole (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 SMBHs. We show that, within this framework, the ratio Gamma equivalent to (M(BH)/M(*))(z)/(M(BH)/M(*))(z = 0) between the BH 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 Gamma(z) increase with z, such an effect is larger for massive BHs, reaching values Gamma approximate to 5 for massive BHs (M >= 109 M(circle dot)) at z greater than or similar to 4, in agreement with recent observations of high-redshift quasi-stellar objects; this is due to the effectiveness of interactions in triggering BH accretion in high-density environments (where massive haloes form) at high redshifts. To investigate how different observations of Gamma(z) fit within our framework, we worked out specific predictions for subsamples of the simulated galaxies corresponding to the different observational samples for which measurements of Gamma have been obtained. We found that for broad-line active galactic nuclei at intermediate redshifts 1 less than or similar to z less than or similar to 2 values of Gamma approximate to 2 are expected, with a mild trend towards 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 < Gamma < 1.5, consistent with recent observational findings on samples of submillimitre galaxies; in the framework of our model, these objects end up at z = 0 in low-to-intermediate mass BHs (M < 109 M(circle dot)), and they do not represent typical paths leading to local massive galaxies. The latter have formed preferentially through paths (in the M(*)-M(BH) plane) passing above the local M(*)-M(BH) 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.