Impact of baryons on the cluster mass function and cosmological
  parameter determination
  Recent results by the Planck collaboration have shown that cosmological
parameters derived from the cosmic microwave background anisotropies and
cluster number counts are in tension, with the latter preferring lower values
of the matter density parameter, $\Omega_\mathrm{m}$, and power spectrum
amplitude, $\sigma_8$. Motivated by this, we investigate the extent to which
the tension may be ameliorated once the effect of baryonic depletion on the
cluster mass function is taken into account. We use the large-volume Millennium
Gas simulations in our study, including one where the gas is pre-heated at high
redshift and one where the gas is heated by stars and active galactic nuclei
(in the latter, the self-gravity of the baryons and radiative cooling are
omitted). In both cases, the cluster baryon fractions are in reasonably good
agreement with the data at low redshift, showing significant depletion of
baryons with respect to the cosmic mean. As a result, it is found that the
cluster abundance in these simulations is around 15 per cent lower than the
commonly-adopted fit to dark matter simulations by Tinker et al (2008) for the
mass range $10^{14}-10^{14.5}h^{-1} \mathrm{M}_\odot$. Ignoring this effect
produces a significant artificial shift in cosmological parameters which can be
expressed as $\Delta[\sigma_8(\Omega_\mathrm{m}/0.27)^{0.38}]\simeq -0.03$ at
$z=0.17$ (the median redshift of the $\mathit{Planck}$ cluster sample) for the
feedback model. While this shift is not sufficient to fully explain the
$\mathit{Planck}$ discrepancy, it is clear that such an effect cannot be
ignored in future precision measurements of cosmological parameters with
clusters. Finally, we outline a simple, model-independent procedure that
attempts to correct for the effect of baryonic depletion and show that it works
if the baryon-dark matter back-reaction is negligible.
