Constraints on the Cosmological parameters by means of the clusters mass function

We present constraints on the values of ${\mathrm{\Omega }}_m$, $n$, ${\sigma }_8$, obtained from measurements of the X-ray luminosity function of galaxy clusters as compiled in EMSS, RDCS and BCS galaxy cluster samples. The values obtained ${\mathrm{\Omega }}_m=0.25_{-0.05}^{+0.15}$, $n=-1_{-0.05}^{+0.05}$, and ${\sigma }_8=0.8_{-0.1}^{+0.1}$ are in agreement with WMAP data. We then put constraints on the quoted parameters and the dark-energy equation-of-state parameter, $w$, using Chandra observations of 37 clusters with $<z>$=0.55 derived from 400 deg${}^2$ ROSAT serendipitous survey and 49 brightest $z \simeq 0.05$clustersdetectedintheAll-SkySurvey\mathrm{.}Inthecaseof$\Lambda$CDM model, we get ${\mathrm{\Omega }}_m=0.25_{-0.08}^{+0.1}$ and ${\sigma }_8=0.75_{-0.05}^{+0.08}$, while in the case of the $w$CDM model, we get ${\mathrm{\Omega }}_m=0.28_{-0.10}^{+0.10}$, $w=-1.25_{-0.35}^{+0.30}$, and ${\sigma }_8=0.8_{-0.09}^{+0.09}$. Our results are consistent with those from recent analyses of type Ia supernovae, cosmic microwave background anisotropies, the X-ray gas mass fraction of relaxed galaxy clusters, baryon acoustic oscillations and cosmic shear. The improvement in data quality from EMSS, RDCS, and BCS to Chandra observations leads to an improvement in the constraints even if not of the same entity of the improvement in data.