In order to have a reliable estimation of the mass density of the universe, it is vital to have the possibility to use global tests rather than local ones.
4.1. The evolution of the abundance of clusters
The abundance of clusters at high redshift has been used as a cosmological constraint more than ten years ago (Peebles et al., 1989; Evrard, 1989). Ten years ago, Oukbir and Blanchard (1992) emphasized that the evolution of the abundance of clusters with redshift was rather different in low and high density universe, offering a possible new cosmological test. The interest of this test is that it is global, not local, and therefore allows to actually constraint directly m. It is relatively insensitive to the cosmological constant. In principle, this test is relatively easy to apply, because the abundance at redshift ~ 1. is more than an order of magnitude less in a critical universe, while it is almost constant in a low density universe. Therefore the measurement of the temperature distribution function (TDF) even at z ~ 0.5 should provide a robust answer. In recent years, this test has received considerable attention (Borgani et al, 1999; Eke et al., 1998; Henry, 1997; Henry, 2000; Viana and Liddle, 1999, among others). The first practical application was by Donahue (1996) who emphasized that the properties of MS0451, with a temperature of around 10 keV at a redshift of 0.55, was already a serious piece of evidence in favor of a low density universe. This argument has been comforted by the discovery of a high temperature cluster at redshift z ~ 0.8, MS1054, which has a measured temperature of ~ 12 keV (Donahue et al, 2000). In the mean time, however, the redshift distribution of EMSS clusters was found to be well fitted by a high density universe under the assumption of a non evolving luminosity-temperature relation (Oukbir and Blanchard, 1997; Reichert et al., 1999), as seems to follow from the properties of distant X-ray clusters (Mushotsky, R.F. and Sharf, 1997; Sadat et al., 1998).
Application of this test is the purpose of the XMM program during the guaranty time phase (Bartlett et al., 2001). In principle, this test can also be applied by using other mass estimates, like velocity dispersion (Carlberg et al, 1997), Sunyaev-Zeldovich (Barbosa et al, 1996), or weak lensing. However, mass estimations based on X-ray temperatures is up to now the only method which can be applied at low and high redshift with relatively low systematic uncertainty. For instance, if velocity dispersions at high redshift (~ 0.5) are overestimated by 30%, the difference between low and high density universe is canceled. Weak lensing and SZ surveys of clusters to allow this test remain to be done.