Cosmology with massive neutrinos III: the halo mass function and an
  application to galaxy clusters
  We use a suite of N-body simulations that incorporate massive neutrinos as an
extra-set of particles to investigate their effect on the halo mass function.
We show that for cosmologies with massive neutrinos the mass function of dark
matter haloes selected using the spherical overdensity (SO) criterion is well
reproduced by the fitting formula of Tinker et al. (2008) once the cold dark
matter power spectrum is considered instead of the total matter power, as it is
usually done. The differences between the two implementations, i.e. using
$P_{\rm cdm}(k)$ instead of $P_{\rm m}(k)$, are more pronounced for large
values of the neutrino masses and in the high end of the halo mass function: in
particular, the number of massive haloes is higher when $P_{\rm cdm}(k)$ is
considered rather than $P_{\rm m}(k)$. As a quantitative application of our
findings we consider a Planck-like SZ-clusters survey and show that the
differences in predicted number counts can be as large as $30\%$ for $\sum
m_\nu = 0.4$ eV. Finally, we use the Planck-SZ clusters sample, with an
approximate likelihood calculation, to derive Planck-like constraints on
cosmological parameters. We find that, in a massive neutrino cosmology, our
correction to the halo mass function produces a shift in the
$\sigma_8(\Omega_{\rm m}/0.27)^\gamma$ relation which can be quantified as
$\Delta \gamma \sim 0.05$ and $\Delta \gamma \sim 0.14$ assuming one
($N_\nu=1$) or three ($N_\nu=3$) degenerate massive neutrino, respectively. The
shift results in a lower mean value of $\sigma_8$ with $\Delta \sigma_8 = 0.01$
for $N_\nu=1$ and $\Delta \sigma_8 = 0.02$ for $N_\nu=3$, respectively. Such
difference, in a cosmology with massive neutrinos, would increase the tension
between cluster abundance and Planck CMB measurements.
