Figure 7: (a) Mass function of seed black holes predicted by the model based on (14) and (15). The black solid line refers to , while the red line refers to . The long-dashed line shows the effect of reducing from 2 to 1.5. The short dashed line shows the effect of not including the possibility of fragmentation (more details can be found in ). (b) The integrated density of black holes predicted from a merger tree evolution of the black hole seed population. The three solid lines refer to three different choices of the parameters (see  for details). The dashed area indicates the observationally permitted region from estimates of the black hole density at low redshifts. At high redshift (), the density reflects the one attained after the seed formation phase, while the rise at lower redshifts indicates the growth through AGN activity.