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) $M_*^{\texttt {BCG}}$ and the satellite richness λ, defined as their correlation coefficient ρ_cc at fixed halo mass, using the abundance and weak lensing of SDSS clusters as functions of $M_*^{\texttt {BCG}}$ and λ. We detect a halo mass-dependent conformity as ρ_cc = 0.60 + 0.08ln (M_h/3 × 10¹⁴h^-1M_⊙). The strong conformity successfully resolves the 'halo mass equality' conundrum discovered in Zu et al. - when split by $M_*^{\texttt {BCG}}$ at fixed λ, the low- and high-$M_*^{\texttt {BCG}}$ clusters have the same average halo mass despite having a 0.34-dex discrepancy in average $M_*^{\texttt {BCG}}$. 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 ${\sim }20{{\ \rm per\ cent}}$ lower halo concentration c, the low-$M_*^{\texttt {BCG}}$ clusters are ${\sim }10{{\ \rm per\ cent}}$ more biased than the high-$M_*^{\texttt {BCG}}$ 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 $M_*^{\texttt {BCG}}$-c correlation, the late-time dry merger-induced BCG growth naturally produces the $M_*^{\texttt {BCG}}$-λ conformity despite the well-known anticorrelation between λ and c. Our method paves the path towards simultaneously constraining cosmology and cluster formation with future cluster surveys.