preparation

The Euclid mission, designed to map the geometry of the dark Universe, presents an unprecedented opportunity for advancing our understanding of the cosmos through its photometric galaxy cluster survey. Central to this endeavor is the accurate calibration of the mass- and redshift-dependent halo bias (HB), which is the focus of this paper. Our aim is to enhance the precision of HB predictions, which is crucial for deriving cosmological constraints from the clustering of galaxy clusters. Our study is based on the peak-background split (PBS) model linked to the halo mass function (HMF), and it extends it with a parametric correction to precisely align with results from an extended set of N-body simulations carried out with the <ns0:monospace>OpenGADGET3</ns0:monospace> code. Employing simulations with fixed and paired initial conditions, we meticulously analyzed the matter-halo cross-spectrum and modeled its covariance using a large number of mock catalogs generated with Lagrangian perturbation theory simulations with the <ns0:monospace>PINOCCHIO</ns0:monospace> code. This ensures a comprehensive understanding of the uncertainties in our HB calibration. Our findings indicate that the calibrated HB model is remarkably resilient against changes in cosmological parameters, including those involving massive neutrinos. The robustness and adaptability of our calibrated HB model provide an important contribution to the cosmological exploitation of the cluster surveys to be provided by the Euclid mission. This study highlights the necessity of continuously refining the calibration of cosmological tools such as the HB to match the advancing quality of observational data. As we project the impact of our calibrated model on cosmological constraints, we find that given the sensitivity of the Euclid survey, a miscalibration of the HB could introduce biases in cluster cosmology analysis. Our work fills this critical gap, ensuring the HB calibration matches the expected precision of the Euclid survey.