Euclid preparation: XXXIX. The effect of baryons on the halo mass function

The Euclid photometric survey of galaxy clusters stands as a powerful cosmological tool, with the capacity to significantly propel our understanding
of the Universe. Despite being subdominant to dark matter and dark energy, the baryonic component of our Universe holds substantial influence
over the structure and mass of galaxy clusters. This paper presents a novel model that can be used to precisely quantify the impact of baryons
on the virial halo masses of galaxy clusters using the baryon fraction within a cluster as a proxy for their effect. Constructed on the premise of
quasi-adiabaticity, the model includes two parameters, which are calibrated using non-radiative cosmological hydrodynamical simulations, and a
single large-scale simulation from the Magneticum set, which includes the physical processes driving galaxy formation. As a main result of our
analysis, we demonstrate that this model delivers a remarkable 1% relative accuracy in determining the virial dark matter-only equivalent mass
of galaxy clusters starting from the corresponding total cluster mass and baryon fraction measured in hydrodynamical simulations. Furthermore,
we demonstrate that this result is robust against changes in cosmological parameters and against variation of the numerical implementation of the
subresolution physical processes included in the simulations. Our work substantiates previous claims regarding the impact of baryons on cluster
cosmology studies. In particular, we show how neglecting these effects would lead to biased cosmological constraints for a Euclid-like cluster
abundance analysis. Importantly, we demonstrate that uncertainties associated with our model arising from baryonic corrections to cluster masses
are subdominant when compared to the precision with which mass–observable (i.e. richness) relations will be calibrated using Euclid and to our
current understanding of the baryon fraction within galaxy clusters.