Halo mass function: Baryon impact, fitting formulae and implications for
  cluster cosmology
  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 $z\sim2$ and for
high masses $M_{200\textrm m}\gtrsim10^{14}M\odot$. 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
$\Delta(\sigma_8)\simeq\Delta(\sigma_8(\Omega_\textrm m/0.27)^{0.3})\simeq0.02$
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.
