Resolving the Extragalactic $\gamma$-ray Background above 50 GeV with
  Fermi-LAT
  The Fermi Large Area Telescope (LAT) Collaboration has recently released a
catalog of 360 sources detected above 50 GeV (2FHL). This catalog was obtained
using 80 months of data re-processed with Pass 8, the newest event-level
analysis, which significantly improves the acceptance and angular resolution of
the instrument. Most of the 2FHL sources at high Galactic latitude are blazars.
Using detailed Monte Carlo simulations, we measure, for the first time, the
source count distribution, $dN/dS$, of extragalactic $\gamma$-ray sources at
$E>50$ GeV and find that it is compatible with a Euclidean distribution down to
the lowest measured source flux in the 2FHL ($\sim8\times 10^{-12}$ ph
cm$^{-2}$ s$^{-1}$). We employ a one-point photon fluctuation analysis to
constrain the behavior of $dN/dS$ below the source detection threshold. Overall
the source count distribution is constrained over three decades in flux and
found compatible with a broken power law with a break flux, $S_b$, in the range
$[8 \times 10^{-12},1.5 \times 10^{-11}]$ ph cm$^{-2}$ s$^{-1}$ and power-law
indices below and above the break of $\alpha_2 \in [1.60,1.75]$ and $\alpha_1 =
2.49 \pm 0.12$ respectively. Integration of $dN/dS$ shows that point sources
account for at least $86^{+16}_{-14}\%$ of the total extragalactic $\gamma$-ray
background. The simple form of the derived source count distribution is
consistent with a single population (i.e. blazars) dominating the source counts
to the minimum flux explored by this analysis. We estimate the density of
sources detectable in blind surveys that will be performed in the coming years
by the Cherenkov Telescope Array.
