Clustering, Cosmology and a New Era of Black Hole Demographics -- I. The
  Conditional Luminosity Function of Active Galactic Nuclei
  Deep X-ray surveys have provided a comprehensive and largely unbiased view of
active galactic nuclei (AGN) evolution stretching back to $z \sim 5$. However,
it has been challenging to use the survey results to connect this evolution to
the cosmological environment that AGNs inhabit. Exploring this connection will
be crucial to understanding the triggering mechanisms of AGNs and how these
processes manifest in observations at all wavelengths. In anticipation of
upcoming wide-field X-ray surveys that will allow quantitative analysis of AGN
environments, this paper presents a method to observationally constrain the
Conditional Luminosity Function (CLF) of AGNs at a specific $z$. Once measured,
the CLF allows the calculation of the AGN bias, mean dark matter halo mass, AGN
lifetime, halo occupation number, and AGN correlation function -- all as a
function of luminosity. The CLF can be constrained using a measurement of the
X-ray luminosity function and the correlation length at different luminosities.
The method is illustrated at $z \approx 0$ and $0.9$ using the limited data
that is currently available, and a clear luminosity dependence in the AGN bias
and mean halo mass is predicted at both $z$, supporting the idea that there are
at least two different modes of AGN triggering. In addition, the CLF predicts
that $z\approx 0.9$ quasars may be commonly hosted by haloes with
$M_{\mathrm{h}} \sim 10^{14}$ M$_{\odot}$. These `young cluster' environments
may provide the necessary interactions between gas-rich galaxies to fuel
luminous accretion. The results derived from this method will be useful to
populate AGNs of different luminosities in cosmological simulations.
