CFHTLenS and RCSLenS: Testing Photometric Redshift Distributions Using
  Angular Cross-Correlations with Spectroscopic Galaxy Surveys
  We determine the accuracy of galaxy redshift distributions as estimated from
photometric redshift probability distributions $p(z)$. Our method utilises
measurements of the angular cross-correlation between photometric galaxies and
an overlapping sample of galaxies with spectroscopic redshifts. We describe the
redshift leakage from a galaxy photometric redshift bin $j$ into a
spectroscopic redshift bin $i$ using the sum of the $p(z)$ for the galaxies
residing in bin $j$. We can then predict the angular cross-correlation between
photometric and spectroscopic galaxies due to intrinsic galaxy clustering when
$i \neq j$ as a function of the measured angular cross-correlation when $i=j$.
We also account for enhanced clustering arising from lensing magnification
using a halo model. The comparison of this prediction with the measured signal
provides a consistency check on the validity of using the summed $p(z)$ to
determine galaxy redshift distributions in cosmological analyses, as advocated
by the Canada-France-Hawaii Telescope Lensing Survey (CFHTLenS). We present an
analysis of the photometric redshifts measured by CFHTLenS, which overlaps the
Baryon Oscillation Spectroscopic Survey (BOSS). We also analyse the
Red-sequence Cluster Lensing Survey (RCSLenS), which overlaps both BOSS and the
WiggleZ Dark Energy Survey. We find that the summed $p(z)$ from both surveys
are generally biased with respect to the true underlying distributions. If
unaccounted for, this bias would lead to errors in cosmological parameter
estimation from CFHTLenS by less than $\sim 4\%$. For photometric redshift bins
which spatially overlap in 3-D with our spectroscopic sample, we determine
redshift bias corrections which can be used in future cosmological analyses
that rely on accurate galaxy redshift distributions.
