Galaxy cluster number count data constraints on cosmological parameters

[Abridged] We use data on massive galaxy clusters ($M_{\rm cluster} > 8 \times 10^{14} h^{-1} M_\odot$ within a comoving radius of $R_{\rm cluster} = 1.5 h^{-1}\Mpc$) in the redshift range $0.05 \lesssim z \lesssim 0.83$ to place constraints, simultaneously, on the nonrelativistic matter density parameter $\Omega_m$, on the amplitude of mass fluctuations $\sigma_8$, on the index $n$ of the power-law spectrum of the density perturbations, and on the Hubble constant $H_0$, as well as on the equation-of-state parameters $(w_0,w_a)$ of a smooth dark energy component. For the first time, we properly take into account the dependence on redshift and cosmology of the quantities related to cluster physics: the critical density contrast, the growth factor, the mass conversion factor, the virial overdensity, the virial radius and, most importantly, the cluster number count derived from the observational temperature data. We show that, contrary to previous analyses, cluster data alone prefer low values of the amplitude of mass fluctuations, $\sigma_8 \leq 0.69 (1\sigma C.L.)$, and large amounts of nonrelativistic matter, $\Omega_m \geq 0.38 (1\sigma C.L.)$, in slight tension with the $\Lambda$CDM concordance cosmological model, though the results are compatible with $\Lambda$CDM at $2\sigma$. In addition, we derive a $\sigma_8$ normalization relation, $\sigma_8 \Omega_m^{1/3} = 0.49 \pm 0.06 (2\sigma C.L.)$.