fact that the present BH mass density agrees with the accreted (baryonic) mass density derived from the bolometric luminosity function of quasars (Soltan 1982; Small & Blandford 1992; Hopkins, Richards & Hernquist 2007) and with a negligible amount of accreted dark matter (Peirani & de Freitas Pacheco 2008). Seeds could be intermediate mass ($10^3-10^4)~M_\odot$ black holes formed during the collapse of primordial gas clouds (Haehnelt & Rees 1993; Eisenstein & Loeb 1995; Koushiappas, Bullock & Dekel 2004) or during the core collapse of relativistic star clusters formed in star-bursts, which may have occurred in the early evolution of galaxies (Shapiro 2004). Here, as it will be discussed later, seeds are assumed to be black holes with masses around 100-500 $M_\odot$ originated from the first generation of stars, supposed to be quite massive due to the absence of metals, which are the main contributors to the cooling of the gas.

The different correlations between the black hole mass and the dynamic or the photometric properties of the host galaxy suggest a gradual growth of the seed as the host galaxy itself is assembled. However this scenario seems to be inconsistent with the fact that up to now more than 40 bright 