On the Evolution of High-Redshift Active Galactic Nuclei
  We build a simple physical model to study the high-redshift active galactic
Nucleus (AGN) evolution within the co-evolution framework of central black
holes (BHs) and their host galaxies. The correlation between the circular
velocity of a dark halo $V_c$ and the velocity dispersion of a galaxy $\sigma$
is used to link the dark matter halo mass and BH mass. The dark matter halo
mass function is converted to the BH mass function for any given redshift. The
high-redshift optical AGN luminosity functions (LFs) are constructed. At $z\sim
4$, the flattening feature is not shown at the faint end of the optical AGN LF.
This is consistent with observational results. If the optical AGN LF at $z\sim
6$ can be reproduced in the case in which central BHs have the
Eddington-limited accretion, it is possible for the AGN lifetime to have a
small value of $2\times 10^5$ yrs. The X-ray AGN LFs and X-ray AGN number
counts are also calculated at $2.0<z<5.0$ and $z>3$, respectively, using the
same parameters adopted in the calculation for the optical AGN LF at $z\sim 4$.
It is estimated that about 30 AGNs per $\rm{deg}^2$ at $z>6$ can be detected
with a flux limit of $3\times 10^{-17}~\rm{erg~cm^{-2}~s^{-1}}$ in the $0.5-2$
keV band. Additionally, the cosmic reionization is also investigated. The
ultraviolet photons emitted from the high-redshift AGNs mainly contribute to
the cosmic reionization, and the central BHs of the high-redshift AGNs have a
mass range of $10^6-10^8M_\odot$. We also discuss some uncertainties in both
the AGN LFs and AGN number counts originating from the $M_{\rm{BH}}-\sigma$
relation, Eddington ratio, AGN lifetime, and X-ray attenuation in our model.
