The investigation of cluster properties within galaxy formation theories can provide fundamental informations on various aspects of the models. OBB have shown that a reliable modeling of X-ray clusters can be built and therefore, that the inferred constraints are robust. As other authors, they investigated the consequences of this modeling for flat models and found, in agreement with other analysis, that the spectrum on clusters scales ( ) has to be flatter than that given by the standard cold dark matter model (see for instance Peacock and Dodds, 1994). In the present paper we have carried out a similar analysis in a low density universe. Because of the degeneracy between the amplitude of the fluctuations and , the local data does not allow one to distinguish between an open model and the critical universe. However, it provides an interesting result: in the unbiased case, though the derivation is drawn from a different analysis, the estimate of the mean density of the universe is consistent with dynamical estimates and has to be greater than 0.15. In the second part of this paper we have analysed a new way to estimate the mean density of the universe. As structure formation occurs earlier in a low density universe than in the critical case, the evolutionary properties are different and this leads to important observational differences: we have shown that the redshift distribution of clusters selected on the basis of their apparent temperature is primarly sensible to the density parameter. Since we have also shown that this distribution is almost completely independent of the power spectrum, this test might provide an unambiguous determination of .
We have attempted to apply our test to the Einstein satellite X-ray survey. As the temperature information of these clusters is not available, our test cannot be fully carried out. Still, assuming a non-evolving relation between luminosities and temperatures at high redshifts, this first piece of information seems to favor an Einstein-de Sitter universe. However, a strong negative evolution of the luminosity-temperature relation might mimic the observed redshift distribution in a low density universe. Since present day data are clearly inconsistent with the simplest self-similar model, theoretical modeling should be taken with some caution. Indeed, it seems difficult to solve this problem from the theoretical point of view: the self-similar model implies that the luminosity, for a given temperature should be higher at higher redshifts, but scaling arguments do not allow to properly reproduce present day data. On the other hand, as structure formation occurs early in low density model and late evolution is a rather passive one, mainly due to the dilution of the universe, we believe that a strong negative evolution is rather unlikely. This gives support to the idea that the observed redshift distribution is not in agreement with an open model. However, such a conclusion can only be considered as tentative. Important progress on this question is expected from detailed observations of high redshift clusters: from our bootstrap analysis we have shown that temperature measurements of about ten high redshift ( ) clusters would allow us to fix the normalization of the correlation, and thus to distinguish between an open model and the flat model. Consequently, very sensitive instruments like AXAF and XMM could give a definitive answer to the question concerning the mean density of the universe.
© European Southern Observatory (ESO) 1997