Constraints on the Cosmological parameters by means of the clusters mass
  function
  We present constraints on the values of $\Omega_m$, $n$, $\sigma_8$, 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
$\Omega_m=0.25^{+0.15}_{-0.05}$, $n=-1^{+0.05}_{-0.05}$, 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 $z \simeq
0.05$ clusters detected in the All-Sky Survey. In the case of $\Lambda$CDM
model, we get $\Omega_m=0.25^{+0.1}_{-0.08}$ and
$\sigma_8=0.75^{+0.08}_{-0.05}$, while in the case of the $w$CDM model, we get
$\Omega_m=0.28^{+0.10}_{-0.10}$, $w=-1.25^{+0.30}_{-0.35}$, 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.
