Galaxy clusters are a recent cosmological probe. The precision and accuracy of the cosmological parameters inferred from these objects are affected by the knowledge of cluster physics, entering the analysis through the mass-observable scaling relations, and the theoretical description of their mass and redshift distribution, modelled by the mass function. In this work, we forecast the impact of different modelling of these ingredients for clusters detected by future optical and near-IR surveys. We consider the standard cosmological scenario and the case with a time-dependent equation of state for dark energy. We analyse the effect of increasing accuracy on the scaling relation calibration, finding improved constraints on the cosmological parameters. This higher accuracy exposes the impact of the mass function evaluation, which is a subdominant source of systematics for current data. We compare two different evaluations for the mass function. In both cosmological scenarios, the use of different mass functions leads to biases in the parameter constraints. For the $Lambda$CDM model, we find a $1.6 , sigma$ shift in the $(Omega_m,sigma_8)$ parameter plane and a discrepancy of $sim 7 , sigma$ for the redshift evolution of the scatter of the scaling relations. For the scenario with a time-evolving dark energy equation of state, the assumption of different mass functions results in a $sim 8 , sigma$ tension in the $w_0$ parameter. These results show the impact, and the necessity for a precise modelling, of the interplay between the redshift evolution of the mass function and of the scaling relations in the cosmological analysis of galaxy clusters.
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