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 se
lected 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 $sum m_ u = 0.4$ 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 ($N_ u=1$) or three ($N_ u=3$) degenerate massive neutrino, respectively. The shift results in a lower mean value of $sigma_8$ with $Delta sigma_8 = 0.01$ for $N_ u=1$ and $Delta sigma_8 = 0.02$ for $N_ u=3$, respectively. Such difference, in a cosmology with massive neutrinos, would increase the tension between cluster abundance and Planck CMB measurements.
We perform a forecast analysis on how well a Euclid-like photometric galaxy cluster survey will constrain the total neutrino mass and effective number of neutrino species. We base our analysis on the Monte Carlo Markov Chains technique by combining i
nformation from cluster number counts and cluster power spectrum. We find that combining cluster data with CMB measurements from Planck improves by more than an order of magnitude the constraint on neutrino masses compared to each probe used independently. For the LCDM+m_nu model the 2 sigma upper limit on total neutrino mass shifts from M_nu < 0.35 eV using cluster data alone to M_nu < 0.031 eV when combined with CMB data. When a non-standard model with N_eff number of neutrino species is considered, we estimate N_eff<3.14 (95% CL), while the bounds on neutrino mass are relaxed to M_nu < 0.040 eV. This accuracy would be sufficient for a 2 sigma detection of neutrino mass even in the minimal normal hierarchy scenario. We also consider scenarios with a constant dark energy equation of state and a non-vanishing curvature. When these models are considered the error on M_nu is only slightly affected, while there is a larger impact of the order of ~ 15 % and ~ 20% respectively on the 2 sigma error bar of N_eff with respect to the standard case. We also treat the LCDM+m_nu+N_eff case with free nuisance parameters, which parameterize the uncertainties on the cluster mass determination. In this case, the upper bounds on M_nu are relaxed by a factor larger than two, M_nu < 0.083 eV (95% CL), hence compromising the possibility of detecting the total neutrino mass with good significance. We thus confirm the potential that a large optical/near-IR cluster survey, like that to be carried out by Euclid, could have in constraining neutrino properties [abridged].
