How baryons can significantly bias cluster count cosmology

We quantify two main pathways through which baryonic physics biases cluster count cosmology. We create mock cluster samples that reproduce the baryon content inferred from X-ray observations. The clusters are linked to their counterparts in a dark matter-only universe, whose abundances can be predicted robustly, by assuming the shape of the dark matter density profile does not change significantly due to baryons. We derive halo masses from different weak lensing fitting methods and infer the best-fitting cosmological parameters $\Omega_\mathrm{m}$, $S_8=\sigma_8(\Omega_\mathrm{m}/0.3)^{0.2}$, and $w_0$ from the mock cluster sample. We find that because of the need to accommodate the change in the density profile due to the ejection of baryons, weak lensing mass calibrations are only unbiased if the concentration is left free when fitting the reduced shear with NFW profiles. However, even unbiased total mass estimates give rise to biased cosmological parameters if the measured mass functions are compared with predictions from dark matter-only simulations. This is the dominant bias for haloes with $m_\mathrm{500c} < 10^{14.5} \, h^{-1} \, \mathrm{M}_\odot$. For a stage IV-like cluster survey with area $\approx 15000 \, \mathrm{deg^2}$ and a constant mass cut of $m_\mathrm{200m,min} = 10^{14} \, h^{-1} \, \mathrm{M}_\odot$, the biases are $-11 \pm 1 \, \%$ in $\Omega_\mathrm{m}$, $-3.29 \pm 0.04 \, \%$ in $S_8$, and $9 \pm 1.5 \, \%$ in $w_0$. These systematic biases exceed the statistical uncertainties by factors of 11, 82, and 6, respectively. We suggest that rather than the total halo mass, the (re-scaled) dark matter mass inferred from the combination of weak lensing and observations of the hot gas, should be used for cluster count cosmology.