Comparing gravitational redshifts of SDSS galaxy clusters with the
  magnification redshift enhancement of background BOSS galaxies
  A clean measurement of the evolution of the galaxy cluster mass function can
significantly improve our understanding of cosmology from the rapid growth of
cluster masses below z < 0.5. Here we examine the consistency of cluster
catalogues selected from the SDSS by applying two independent gravity-based
methods using all available spectroscopic redshifts from the DR10 release.
First, we detect a gravitational redshift related signal for 20,119 and 13,128
clusters with spectroscopic redshifts contained in the GMBCG and redMaPPer
catalogues, respectively, at a level of $\sim - 10$ km s$^{-1}$. This we show
is consistent with the magnitude expected using the richness-mass relations
provided by the literature and after applying recently clarified relativistic
and flux bias corrections. This signal is also consistent with the richest
clusters in the larger catalogue of Wen et al. (2012), corresponding to
$M_{200m} \gtrsim 2 \times 10^{14}\,\mathrm{M}_\odot\,h^{-1}$, however we find
no significant detection of gravitational redshift signal for less riched
clusters, which may be related to bulk motions from substructure and spurious
cluster detections. Second, we find all three catalogues generate
mass-dependent levels of lensing magnification bias, which enhances the mean
redshift of flux-selected background galaxies from the BOSS survey. The
magnitude of this lensing effect is generally consistent with the corresponding
richness-mass relations advocated for the surveys. We conclude that all
catalogues comprise a high proportion of reliable clusters, and that the GMBCG
and redMaPPer cluster finder algorithms favor more relaxed clusters with a
meaningful gravitational redshift signal, as anticipated by the red-sequence
colour selection of the GMBCG and redMaPPer samples.
