On the Level of Cluster Assembly Bias in SDSS
  Recently, several studies have discovered a strong discrepancy between the
large-scale clustering biases of two subsamples of galaxy clusters at the same
halo mass, split by their average projected membership distances
$R_{\mathrm{mem}}$. The level of this discrepancy significantly exceeds the
maximum halo assembly bias signal predicted by LCDM. In this study, we explore
whether some of the clustering bias differences could be caused by biases in
$R_{\mathrm{mem}}$ due to projection effects from other systems along the
line-of-sight. We thoroughly investigate the halo assembly bias of the
photometrically-detected redMaPPer clusters in SDSS, by defining a new variant
of the average membership distance estimator $\tilde{R}_{\mathrm{mem}}$ that is
more robust against projection effects in the cluster membership
identification. Using the angular mark correlation functions of clusters, we
show that the large-scale bias differences when splitting by $R_{\mathrm{mem}}$
can be largely attributed to such projection effects. After splitting by
$\tilde{R}_{\mathrm{mem}}$, the anomalously large signal is reduced, giving a
ratio of $1.02\pm0.14$ between the two clustering biases as measured from weak
lensing. Using a realistic mock cluster catalog, we predict that the bias ratio
between two $\tilde{R}_{\mathrm{mem}}$-split subsamples should be $<1.10$,
which is at least 60% weaker than the maximum halo assembly bias signal (1.24)
when split by halo concentration. Therefore, our results demonstrate that the
level of halo assembly bias exhibited by redMaPPer clusters in SDSS is
consistent with the LCDM prediction. With a ten-fold increase in cluster
numbers, deeper ongoing surveys will enable a more robust detection of halo
assembly bias. Our findings also have important implications for how projection
effects and their impact on cluster cosmology can be quantified in photometric
cluster catalogs.
