Characterizing the Best Cosmic Telescopes with the Millennium
  Simulations
  Certain configurations of massive structures projected along the line of
sight maximize the number of detections of gravitationally lensed $z\sim10$
galaxies. We characterize such lines of sight with the \'etendue $\sigma_\mu$,
the area in the source plane magnified over some threshold $\mu$. We use the
Millennium I and Millennium XXL cosmological simulations to determine the
frequency of high $\sigma_\mu$ beams on the sky, their properties, and
efficient selection criteria. We define the best beams as having
$\sigma_{\mu>3} >2000$ arcsec$^2$, for a $z\sim10$ source plane, and predict
$477 \pm 21$ such beams on the sky. The total mass in the beam and
$\sigma_{\mu>3}$ are strongly correlated. After controlling for total mass, we
find a significant residual correlation between $\sigma_{\mu>3}$ and the number
of cluster-scale halos ($>10^{14} M_\odot h^{-1}$) in the beam. Beams with
$\sigma_{\mu>3} >2000$ arcsec$^2$, which should be best at lensing $z\sim10$
galaxies, are ten times more likely to contain multiple cluster-scale halos
than a single cluster-scale halo. Beams containing an Abell 1689-like massive
cluster halo often have additional structures along the line of sight,
including at least one additional cluster-scale ($M_{200}>10^{14}M_\odot
h^{-1}$) halo 28% of the time. Selecting beams with multiple, massive
structures will lead to enhanced detection of the most distant and
intrinsically faint galaxies.
