The Three Hundred Project hydrodynamical simulations: Hydrodynamical weak-lensing cluster mass biases and richnesses using different hydro models

The mass of galaxy clusters estimated from weak-lensing observations is affected by projection effects, leading to a systematic underestimation compared to the true cluster mass, varying with both mass and redshift. The magnitude depends on the criteria used to select clusters and the spatial scale over which their mass is measured. We leverage hydrodynamical simulations of galaxy clusters carried out with GadgetX and GIZMO-SIMBA as part of the Three Hundred project. We used them to quantify weak-lensing mass biases with respect also to the results from dark matter-only simulations. We also investigate how the biases propagate into the richness-mass relation. We aim to shed light on the effect of the presence of baryons on the weak-lensing mass bias and also whether this bias depends on the galaxy formation recipe; we seek to model the richness-mass relation that can be used as guidelines for observational experiments for cluster cosmology. We produced weak-lensing simulations of random projections to model the expected excess surface mass density profile of clusters up to redshift $z=1$. We then estimated the observed richness by counting the number of galaxies in a cylinder and correcting by projected contaminants. We derived the weak-lensing mass-richness relation and found consistency across hydrodynamical simulations. The intercept parameter of the relation is independent of redshift but varies with the minimum of the stellar mass to define the richness. At the same time, the slope is relatively constant up to $z=0.55$. The scatter in observed richness at a fixed weak-lensing mass increases linearly with redshift at a fixed stellar mass cut. As expected, we observed that the scatter in richness at a given true mass is smaller than at a given weak-lensing mass. Our results for the weak-lensing mass-richness relation align well with SDSS redMaPPer cluster analyses. [Abridged]