On the evolution of the intrinsic scatter in black hole versus galaxy mass relations

We present results on the evolution of the intrinsic scatter of black hole masses considering different implementations of a model in which black holes only grow via mergers. We demonstrate how merger driven growth affects the correlations between black hole mass and host bulge mass. The simple case of an initially log-normal distributed scatter in black hole and bulge masses combined with random merging within the galaxy population results in a decreasing scatter with merging generation/number as predicted by the Central-limit theorem. In general we find that the decrease in scatter {\sigma} is well approximated by {\sigma}merg(m) = {\sigma}ini \times (m + 1)^(-a/2) with a = 0.42 for a range of mean number of mergers m < 50. For a large mean number of mergers (m > 100) we find a convergence to a = 0.61. This is valid for a wide range of different initial distributions, refill-scenarios or merger mass-ratios. Growth scenarios based on halo merger trees of a (100 Mpc)^3 dark matter LambdaCDM-simulation show a similar behaviour with a scatter decrease of a = 0.30 with typical number of mergers m < 50 consistent with random merging (best matching model: a = 0.34). Assuming a present day scatter of 0.3 dex in black hole mass and a mean number of mergers not exceeding m = 50 our results imply a scatter of 0.6 dex at z = 3 and thus a possible scenario in which overmassive (and undermassive) black holes at high redshift are a consequence of a larger intrinsic scatter in black hole mass. A simple toy model connecting the growth of black holes to the growth of LambdaCDM dark matter halos via mergers, neglecting any contribution from accretion, yields a consistent M\cdot -MBulge relation at z = 0 - if we assume the correct initial relation.