Since the characteristics and statistics of the galaxy cluster population can be predicted in the frame of cosmological and structure formation models with reasonable precision (at the state-of-the-art on the 10-20% level for clusters characterized by their mass) [80], the above results can be used for cosmological tests. The currently most critical aspect in this comparison is the uncertainty in the relation of the observable cluster parameters like X-ray luminosity and temperature and the inferred cluster mass [113]. Three different characteristics of the cluster populations provide independent constraints on cosmological models: (1) the cluster abundance at present, (2) the statistics of the spatial distribution of clusters, and (3) the evolution of the cluster abundance.

Taking only the results of (1) the most important constraints are on the matter density parameter, Qm, and on the amplitude of the matter density fluctuations, expressed usually in terms of the rms normalization of the fluctuations at a scale of 8^1010Mpc, a8. These constraints involve a strong degeneracy of these two parameters, which can be overcome if the shape of the mass function or its observable counterpart, the X-ray luminosity function, is very precisely known. For the bright better calibrated part of the REFLEX X-ray luminosity function we obtain constraints in the range Am = 0.23-0.36 and o8 = 0.73-0.84 [20] if we use the empirical mass-luminosity relation of Reiprich and Bohringer [123].

Including the information on the spatial distribution of the clusters greatly helps to break this degeneracy. This was exploited in a comprehensive approach to simultaneously compare the cluster abundance and spatial distribution of the REFLEX sample with cosmological predictions in a study by Schuecker et al. [131]. To fully exploit the information in the REFLEX survey the spatial distribution was not characterized by Fourier modes which are badly suited to the REFLEX survey geometry but by Karhunen-Loeve eigenmodes which by construction are ideally adapted to the survey characteristics. The constraints obtained for the parameters An = 0.27 - 0.43 and o8 = 0.55-0.83 (including systematic errors and uncertainties in various priors) are shown in Fig. 23.26 [131]. The parameter constraints are well consistent with the now preferred concordance cosmological model and they are also consistent with the findings from cluster evolution as shown in Fig. 23.27 [21,113]. Similar, consistent results from cluster evolution studies have also been derived from the observations of the cluster temperature function at different redshifts [74].

To derive constraints on other very important cosmological parameters, in particular on the A-parameter (now often termed the dark energy density Parameter), Aa , we have to include the study of cluster evolution with higher precision. This information can in particular not be obtained from studies at a single epoch. Thus this remains one of the goals for future X-ray cluster studies [75, 91]. However, important information can be obtained on this parameter by combining the cosmo-logical constraints from cluster studies with other observations. Figure 23.29 shows the constraints on the cosmological parameters Am and Aa for three different types of cosmological tests: (1) the study of the fluctuations in the cosmic microwave background with the WMAP satellite [139], (2) the measurements on the geometry of the Universe using supernovae type Ia as standard candles [124,147], and the statistics of the large scale structure from X-ray cluster observations [131,135]. The fact that all three constraints meet at the same small parameter region around the concordance cosmological model with parameters of Am ~ 0.3 and Aa ~ 0.7 is very comforting. Other measures of the large-scale structure, e.g., from the study of the large-scale galaxy distribution also provide similar constraints on the cosmological parameters as the clusters [111,145]. The combination of several cosmological tests can also be included in one statistical parameter constraint approach. This was, for example, done for the combination of the REFLEX cluster survey and the supernova studies by Schuecker et al. [135]. The results of this test are shown in Fig. 23.29 (right), where the constraints on the parameter Am and the equation of state parameter, w, of the Dark Energy are shown. For a value of w = -1 the Dark Energy model is equivalent to the classical A cosmological model. Figure 23.29 shows that the observations are fully consistent with the A model.

Fig. 23.29 Left: Constraints on the cosmological parameters Qm and Q^ from three different cos-mological tests: supernovae Ia as standard candles [124,147], analysis of the fluctuation spectrum of the cosmic microwave background [139], and abundance and spatial clustering observations of X-ray clusters of galaxies [131,132]. All the tests meet at the same parameter region encircling the concordance model. Right: Constraints on the cosmological density parameter, Qm, and the equation of state parameter of the dark energy, wx, for the REFLEX cluster data combined with literature results from distant supernovae obtained by Schuecker et al. [135]. The likelihood contours give the 1,2, and 3 a limits

Fig. 23.29 Left: Constraints on the cosmological parameters Qm and Q^ from three different cos-mological tests: supernovae Ia as standard candles [124,147], analysis of the fluctuation spectrum of the cosmic microwave background [139], and abundance and spatial clustering observations of X-ray clusters of galaxies [131,132]. All the tests meet at the same parameter region encircling the concordance model. Right: Constraints on the cosmological density parameter, Qm, and the equation of state parameter of the dark energy, wx, for the REFLEX cluster data combined with literature results from distant supernovae obtained by Schuecker et al. [135]. The likelihood contours give the 1,2, and 3 a limits

Still another way of obtaining cosmological constraints from galaxy cluster studies is based on the measure of the gas mass fraction in clusters, which was described in Sect. 23.2.2., e.g. [2,49,121]. Using the baryon mass fraction in clusters and the results of the cosmic nucleosynthesis, as shown in Table 23.1, one can conclude on a value of Qm around 0.3. Taking the baryon mass fraction for massive, relaxed galaxy clusters as a universal parameter independent of redshift, as suggested by simulations, e.g. [48], one can use this quantity for standard candle cosmological tests. This is based on the fact that the mass measurement depends on the cluster distance in a linear way through the conversion of the apparent to physical size of the cluster (see (1)), while the determination of the gas mass depends on the distance to the 5/2 power. The cosmological constraints derived this way are in very good agreement with the above results [2,49,121].

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