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The classical way to use clusters to constraint the average matter density in to try to obtain a direct measurement of the local density. This is the principle of M/L test. Because the matter content of the universe is essentially in a dark form, we do not have direct measurement of the mass content even at the local level. This is the reason why in practice we rely on a two-steps procedure: first the average luminosity density of the universe is estimated from galaxy samples, this quantity is now relatively well known thanks to the large redshift surveys like the SLOAN or the 2dF (although difference of the order of 50% might still exist); the second ingredient is the value of the M/L ratio obtained from data on clusters (total luminosity and mass estimations). There might be a factor of two of uncertainty in this quantity. For instance Roussel et al. (2000) found that the average M/L could be as large as 750h when the mass-temperature relation for clusters is normalized from numerical simulations of Bryan and Norman (1998), while values twice smaller are currently obtained by other technics. The basic principle for estimating the average density of the Universe is then to write:

Equation 4 (4)

However, it should be realized that the volume occupied by clusters is a tiny fraction of the total volume of universe, of the order of 10-5. The application of the M/L relies therefore on an extrapolation over 105 in volume!

Figure 4

Figure 4. These plots illustrate the power of the cosmological test of the evolution of the abundance of X-ray clusters: the TDF (temperature distribution function) has been normalized to present day abundance (blue - dark grey - lines). The abundance of local clusters is given by the blue (dark grey) symbols (Blanchard et al., 2000). Present abundance allows one to set the normalization and the slope of the spectrum of primordial fluctuations on clusters scale (which is Omegam dependant). The evolution with redshift is much faster in a high matter density universe (left panel, Omegam = 0.89) than in a low density universe (right panel, Omegam = 0.3): z = 0.33 (yellow - light) the difference is already of the order of 3 or larger. It is relatively insensitive to the cosmological constant. We also give our estimate of the local TDF (blue symbols) derived by Blanchard et al. (2000), as well as our estimate of the TDF at z = 0.33 (yellow symbols- light grey). Also are given for comparison data (Henry 2000) and models predictions at z = 0.38 (red - dark grey - symbols and lines). On the left panel, the best model is obtained by fitting simultaneously local clusters and clusters at z = 0.33 leading to a best value of Omegam of 0.89 (flat universe). The right panel illustrates the fact that an flat low density universe Omegam = 0.3 which fits well local data does not fit the high redshift data properly at all.

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