Intrinsic mass-richness relation of clusters from THE THREE HUNDRED hydrodynamic simulations

The main systematics in cluster cosmology is the uncertainty in the mass-observable relation. In this paper, we focus on the most direct cluster observable in optical surveys, i.e. richness, and constrain the intrinsic mass-richness (MR) relation of clusters in THE THREE HUNDRED hydrodynamic simulations with two runs: GIZMO-SIMBA and GADGET-X. We find that modeling the richness at fixed halo mass with a skewed Gaussian distribution yields a simpler and smaller scatter compared to the commonly used log-normal distribution. Additionally, we observe that baryon models have a significant impact on the scatter, while exhibiting no influence on the mass dependence and a slight effect on the amplitude in the MR relation. We select member galaxies based on both stellar mass $M_\star$ and absolute magnitude $\mathscr{M}$. We demonstrate that the MR relation obtained from these two selections can be converted to each other by using the $M_\star-\mathscr{M}$ relation. Finally, we provide a 7-parameter fitting result comprehensively capturing the dependence of the MR relation on both stellar mass cutoff and redshift.