tokens: of least massive black holes (shown as green points) in order to compare the properties of the descendants of the haloes with massive quasars with those of haloes whose black hole mass is much lower, around that of the seed black hole mass. While matching haloes in the BT hydrodynamic and BTMassTracer simulation, we note that the halo mass is different in the matched haloes due to baryonic effects and there can be differences, which are mostly within 20 per cent. Figure 4. Two-dimensional histogram of the logarithm of halo mass (log 10 ( M halo [M ⊙ ])) and the black hole mass (log 10 ( M BH [M ⊙ ])) in the BT Hydrodynamic simulation at z = 8. The 23 most massive black holes are shown by scatter points in purple, while the least massive blackholes are shown in green. We also show as cyan points, the blackholes in the progenitor haloes of the most massive haloes at z = 0. Note that, for the cyan points, the halo masses shown correspond to that of the progenitors at z = 8. In the left-hand panel of Fig. 5 , we plot the halo mass function of the dark matter haloes at z = 8, 5, 2.5, 1.2 and 0. We also plot as points on the halo mass function at z = 8 and 0 the hosts of the most massive black hole tracer particles (corresponding to the purple points in Fig. 4 ). We find that at redshift z = 0, the halo masses are distributed over a wide range, compared to that at z = 8. From the figure, we can therefore clearly see that the descendants of the most massive black holes at z = 8 are not all the most massive haloes at z = 0. However, the descendants are still relatively massive haloes with mass above M * = 5.0 × 10 12 h −1 M ⊙ at z = 0. The individual tracks for the halo masses of the four most massive black hole tracers are also shown in the figure. We can see that these objects approximately keep their rank order over all redshifts. Figure 5. Left-hand panel: halo Mass function at z = 8, 5, 2.5, 1.2 and 0, and the location of the tracked hosts of the four most massive BH's on the mass function. Right-hand panel: normalized histogram of the halo masses at z = 0 of the descendants of the most massive black holes at z = 8 compared with the histogram of most massive haloes at z = 0. To further understand the distribution of the halo masses, we plot the normalized histogram of the host halo masses of the descendants of the z = 8 massive black holes, compared with the 23 most massive haloes at z = 0 in the right-hand panel of Fig. 5 . While the most massive haloes are distributed above 3.2 × 10 14 h −1 M ⊙ with a peak in the distribution at ∼1.8 × 10 15 h −1 M ⊙ , the quasar descendants have a peak distribution in the mass at ∼2.24 × 10 14 h −1 M ⊙ . These 23 most massive haloes and 23 quasar descendants have a space density of 4 × 10 −7 h 3 Mpc −3 . The massive haloes are all Coma-like objects, whereas the quasar descendants have masses an order of magnitude smaller, comparable to galaxy groups with a space density ∼ 10 − 5 h 3 Mpc − 3 . We can
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