Evolution of the mass relation between supermassive black holes and dark matter halos across the cosmic time

A positive observational proof suggests that most galaxies contain a central supermassive black hole (SMBH) with mass in the range of $10^6M_\odot$ $-$ $10^{10}M_\odot$. It is suggested that the mass of SMBHs is proportionally related to that of the dark matter (DM) halo, even at $z= 6$. This implies that these SMBHs could coevolve with the host DM halos. Here, we investigate the mass evolution of SMBHs in a hierarchical structure formation by building halo merger trees using the extended Press-Schechter formalism. An SMBH with a mass that follows various power-law relations with the DM halo mass is assigned as an initial condition. Assuming that the mass growth of all black holes is due to mergers, we obtain the relation between SMBH and DM halo masses at the present epoch. By requiring that the mass of the SMBHs at $z=0$ should not be greater than the currently observed SMBH-DM halo mass relation, a lower bound on the mass of the DM halo that can contain a SMBH can be imposed at $z= 6$ as $M_{\rm lim} > 3.6 \times 10^{10} M_\odot\times (1.4-n)^{2.3}$, where $n$ is the power-law index of the SMBH-DM halo mass relation at $z=6$. Because we only consider mergers for the mass evolution of SMBHs, this model is simplistic and should underestimate the mass of SMBHs relative to the mass of the DM halo at the present epoch. Our work aims to constrain the SMBH-DM mass relation at z=6 and not build a new model to explain the observations