Euclid preparation. CosmoPostProcess: A simulation calibrated framework for weak lensing selection bias in richness-selected galaxy clusters

We present $\mathtt{\text{CosmoPostProcess}}$, a simulation-based forward-modelling algorithm calibrated to reproduce Euclid optical cluster observables. Its main deliverable is a correction for stacked surface-density profiles, binned in richness and redshift, accounting for selection systematics in richness-selected samples relative to unbiased references. We focus on the Euclid richness definition foreseen for cosmological analyses, which does not apply a colour selection; red-sequence richness is not considered. The algorithm processes $N$-body simulations by painting galaxies with a halo-occupation model and emulating survey detection and richness assignment. We also implement a novel estimate of optical cluster centres from projected galaxy densities, validated against Euclid pipelines. Baryonic effects are included through a correction calibrated on hydrodynamical simulations; the baryon-corrected excess surface density agrees within $2\mathrm{\%}$ over $r\in [0.1,5]h^{-1}\mathrm{M}\mathrm{p}\mathrm{c}$. Selection-bias contributions are assessed by varying cosmology and the mass–richness relation. Projection-induced selection bias follows a robust pattern: correlated large-scale structure projected along the line of sight enhances the stacked profile near the one-halo to two-halo transition, peaking at about $1h^{-1}\mathrm{M}\mathrm{p}\mathrm{c}$ with an amplitude of $20\text{\hspace{0.17em}⁣}-\text{\hspace{0.17em}⁣}40\mathrm{\%}$, depending on richness and redshift. The effect is mild at low and intermediate redshift ($z\lesssim 0.7$), at the few-percent level, but becomes more relevant at higher redshift ($z\gtrsim 0.7$). Baryonic modifications remain sub-dominant outside the core, at about $2\mathrm{\%}$ beyond $r\gtrsim 0.3h^{-1}\mathrm{M}\mathrm{p}\mathrm{c}$. The framework delivers radial profile corrections with uncertainties, combining projection-induced selection bias, baryonic physics, and miscentring, to control systematics in Euclid DR1 cluster cosmology. (abridged)