The systematic search for $z\gtrsim5$ active galactic nuclei in the
  $Chandra$ Deep Field South
  We investigate early black hole (BH) growth through the methodical search for
$z\gtrsim5$ AGN in the $Chandra$ Deep Field South. We base our search on the
$Chandra$ 4-Ms data with flux limits of $9.1\times\ 10^{-18}$ (soft, 0.5 - 2
keV) and $5.5\times\ 10^{-17}\ \mathrm{erg}\ \mathrm{s}^{-1}\ \mathrm{cm}^{-2}$
(hard, 2 - 8 keV). At $z\sim5$ this corresponds to luminosities as low as
$\sim10^{42}$ ($\sim10^{43}$) $\mathrm{erg}\ \mathrm{s}^{-1}$ in the soft
(hard) band and should allow us to detect Compton-thin AGN with
$M_\mathrm{BH}>10^7 M_{\odot}$ and Eddington ratios > 0.1. Our field
($0.03~\mathrm{deg}^2$) contains over 600 $z\sim5$ Lyman Break Galaxies. Based
on lower redshift relations we would expect $\sim20$ of them to host AGN. After
combining the $Chandra$ data with GOODS/ACS, CANDELS/WFC3 and $Spitzer$/IRAC
data, the sample consists of 58 high-redshift candidates. We run a photometric
redshift code, stack the GOODS/ACS data, apply colour criteria and the Lyman
Break Technique and use the X-ray Hardness Ratio. We combine our tests and
using additional data find that all sources are most likely at low redshift. We
also find five X-ray sources without a counterpart in the optical or infrared
which might be spurious detections. We conclude that our field does not contain
any convincing $z\gtrsim5$ AGN. Explanations for this result include a low BH
occupation fraction, a low AGN fraction, short, super-Eddington growth modes,
BH growth through BH-BH mergers or in optically faint galaxies. By searching
for $z\gtrsim5$ AGN we are setting the foundation for constraining early BH
growth and seed formation scenarios.
