Spectroscopic Quantification of Projection Effects in the SDSS redMaPPer
  Galaxy Cluster Catalogue
  Projection effects, whereby galaxies along the line-of-sight to a galaxy
cluster are mistakenly associated with the cluster halo, present a significant
challenge for optical cluster cosmology. We use statistically representative
spectral coverage of luminous galaxies to investigate how projection effects
impact the low-redshift limit of the Sloan Digital Sky Survey (SDSS) redMaPPer
galaxy cluster catalogue. Spectroscopic redshifts enable us to differentiate
true cluster members from false positives and determine the fraction of
candidate cluster members viewed in projection. Our main results can be
summarized as follows: first, we show that a simple double-Gaussian model can
be used to describe the distribution of line-of-sight velocities in the
redMaPPer sample; second, the incidence of projection effects is substantial,
accounting for $\sim 16$ per cent of the weighted richness for the lowest
richness objects; third, projection effects are a strong function of richness,
with the contribution in the highest richness bin being several times smaller
than for low-richness objects; fourth, our measurement has a similar amplitude
to state-of-the-art models, but finds a steeper dependence of projection
effects on richness than these models; and fifth, the slope of the observed
velocity dispersion--richness relation, corrected for projection effects,
implies an approximately linear relationship between the true,
three-dimensional halo mass and three-dimensional richness. Our results provide
a robust, empirical description of the impact of projection effects on the SDSS
redMaPPer cluster sample and exemplify the synergies between optical imaging
and spectroscopic data for studies of galaxy cluster astrophysics and
cosmology.
