The Mass-Richness Relation of MaxBCG Clusters from Quasar Lensing
  Magnification using Variability
  Accurate measurement of galaxy cluster masses is an essential component not
only in studies of cluster physics, but also for probes of cosmology. However,
different mass measurement techniques frequently yield discrepant results. The
SDSS MaxBCG catalog's mass-richness relation has previously been constrained
using weak lensing shear, Sunyaev-Zeldovich (SZ), and X-ray measurements. The
mass normalization of the clusters as measured by weak lensing shear is >~25%
higher than that measured using SZ and X-ray methods, a difference much larger
than the stated measurement errors in the analyses. We constrain the
mass-richness relation of the MaxBCG galaxy cluster catalog by measuring the
gravitational lensing magnification of type I quasars in the background of the
clusters. The magnification is determined using the quasars' variability and
the correlation between quasars' variability amplitude and intrinsic
luminosity. The mass-richness relation determined through magnification is in
agreement with that measured using shear, confirming that the lensing strength
of the clusters implies a high mass normalization, and that the discrepancy
with other methods is not due to a shear-related systematic measurement error.
We study the dependence of the measured mass normalization on the cluster halo
orientation. As expected, line-of-sight clusters yield a higher normalization;
however, this minority of haloes does not significantly bias the average
mass-richness relation of the catalog.
