Strong conformity and assembly bias: towards a physical understanding of the galaxy-halo connection in SDSS clusters

Understanding the physical connection between cluster galaxies and massive haloes is key to mitigating systematic uncertainties in next-generation cluster cosmology. We develop a novel method to infer the level of conformity between the stellar mass of the bright central galaxies (BCGs) M-*(BCG) and the satellite richness lambda, defined as their correlation coefficient rho(cc) at fixed halo mass, using the abundance and weak lensing of SDSS clusters as functions of M-*(BCG) and lambda. We detect a halo mass-dependent conformity as rho(cc) = 0.60 + 0.08ln (M-h/3 x 10(14)h(-1)M(circle dot)). The strong conformity successfully resolves the 'halo mass equality' conundrum discovered in Zu et al. - when split by M-*(BCG) at fixed lambda, the low- and high-M-*(BCG) clusters have the same average halo mass despite having a 0.34-dex discrepancy in average M-*(BCG). On top of the best-fitting conformity model, we develop a cluster assembly bias (AB) prescription calibrated against the CosmicGrowth simulation and build a conformity + AB model for the cluster weak lensing measurements. Our model predicts that with an similar to 20 per cent lower halo concentration c, the low-M-*(BCG) clusters are similar to 10 per cent more biased than the high-M-*(BCG) systems, in good agreement with the observations. We also show that the observed conformity and assembly bias are unlikely due to projection effects. Finally, we build a toy model to argue that while the early-time BCG-halo co-evolution drives the M-*(BCG)-c correlation, the late-time dry merger-induced BCG growth naturally produces the M-*(BCG)-lambda conformity despite the well-known anticorrelation between lambda and c. Our method paves the path towards simultaneously constraining cosmology and cluster formation with future cluster surveys.