We report on the relation between the mass of supermassive black holes ( SMBHs ; M _ { BH } ) and that of hosting dark matter halos ( M _ { h } ) for 49 z \sim 6 quasi-stellar objects ( QSOs ) with [ C II ] 158 \mu m velocity-width measurements .
Here , we estimate M _ { h } assuming that the rotation velocity from { FWHM _ { CII } } is equal to the circular velocity of the halo ; we have tested this procedure using z \sim 3 QSOs that also have clustering-based M _ { h } estimates .
We find that a vast majority of the z \sim 6 SMBHs are more massive than expected from the local M _ { BH } – M _ { h } relation , with one-third of the sample by factors \gtrsim 10 ^ { 2 } .
The median mass ratio of the sample , M _ { BH } / M _ { h } = 6 \times 10 ^ { -4 } , means that 0.4 \% of the baryons in halos are locked up in SMBHs .
The mass growth rates of our SMBHs amount to \sim 10 \% of the star formation rates ( SFRs ) , or \sim 1 \% of the mean baryon accretion rates , of the hosting galaxies .
A large fraction of the hosting galaxies are consistent with average galaxies in terms of SFR and perhaps of stellar mass and size .
Our study indicates that the growth of SMBHs ( M _ { BH } \sim 10 ^ { 8 - 10 } M _ { \odot } ) in luminous z \sim 6 QSOs greatly precedes that of hosting halos owing to efficient gas accretion even under normal star formation activities , although we can not rule out the possibility that undetected SMBHs have local M _ { BH } / M _ { h } ratios .
This preceding growth is in contrast to much milder evolution of the stellar-to-halo mass ratio .