We use a set of hydrodynamical (Hydro) and dark matter only (DMonly) simulations to calibrate the halo mass function (HMF). We explore the impact of baryons, propose an improved parametrization for spherical overdensity masses and identify differences between our DMonly HMF and previously published HMFs. We use the textit{Magneticum} simulations, which are well suited because of their accurate treatment of baryons, high resolution, and large cosmological volumes of up to $(3818~textrm{Mpc})^3$. Baryonic effects globally decrease the masses of galaxy clusters, which, at a given mass, results in a decrease of their number density. This effect vanishes at high redshift $zsim2$ and for high masses $M_{200textrm m}gtrsim10^{14}Modot$. We perform cosmological analyses of three idealized approximations to the cluster surveys by the South Pole Telescope (SPT), textit{Planck}, and eROSITA. We pursue two main questions: (1) What is the impact of baryons? -- For the SPT-like and the textit{Planck}-like samples, the impact of baryons on cosmological results is negligible. In the eROSITA-like case, however, neglecting the baryonic impact leads to an underestimate of $Omega_textrm m$ by about $0.01$, which is comparable to the expected uncertainty from eROSITA. (2) How does our DMonly HMF compare with previous work? -- For the textit{Planck}-like sample, results obtained using our DMonly HMF are shifted by $Delta(sigma_8)simeqDelta(sigma_8(Omega_textrm m/0.27)^{0.3})simeq0.02$ with respect to results obtained using the Tinker et al. (2008) fit. This suggests that using our HMF would shift results from textit{Planck} clusters toward better agreement with CMB anisotropy measurements. Finally, we discuss biases that can be introduced through inadequate HMF parametrizations that introduce false cosmological sensitivity.
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