The basic census of the galaxy cluster population is the density of clusters of a given mass, the mass function, which tells us much about the fluctuation amplitude of the dark matter density distribution from which the clusters form (e.g. [158]). Observa-tionally the X-ray luminosity function is a close representation of the mass function. A binned representation of the luminosity function for the REFLEX Cluster Survey is shown in Fig. 23.25 (right). Because of the large sample size the statistical uncertainties are small. A Schechter function provides a reasonable fit to the data [14].

3 The NORAS cluster survey is based on the northern sky region of the RASS and complements the REFLEX survey in the northern hemisphere [11].

Fig. 23.25 Left: Histogram of the normalized KL eigenvalues used to analyze the large-scale structure statistics [132]; a fitted Gaussian function is superposed on the data. This distribution is effectively a cluster count in cells distribution, showing that the large-scale structure is approximately Gaussian on scales of ^50h~l Mpc as far as can be measured with the low number statistics of REFLEX. Right: X-ray luminosity function of the REFLEX cluster Survey. The filled and open data points refer to observed and corrected total luminosities, respectively. A Schechter function (solid line) is fitted to the data by an ML method including the flux uncertainties [14]

Fig. 23.25 Left: Histogram of the normalized KL eigenvalues used to analyze the large-scale structure statistics [132]; a fitted Gaussian function is superposed on the data. This distribution is effectively a cluster count in cells distribution, showing that the large-scale structure is approximately Gaussian on scales of ^50h~l Mpc as far as can be measured with the low number statistics of REFLEX. Right: X-ray luminosity function of the REFLEX cluster Survey. The filled and open data points refer to observed and corrected total luminosities, respectively. A Schechter function (solid line) is fitted to the data by an ML method including the flux uncertainties [14]

The spatial distribution of galaxy clusters is following the overall matter distribution in the Universe, the seeds for which were set in the early Universe. The relation of the amplitude of the cluster density and dark matter density fluctuations can be derived from statistical considerations based on the valid assumption that galaxy clusters form from the large-scale high amplitude peaks of the matter density fluctuation field, e.g. [102]. It turns out that the amplitude of the cluster density fluctuations is amplified compared to that of the matter density. This effect, called "biasing," helps us to trace the cosmic large-scale structure with increased sensitivity. Therefore, one of the very important goals of contiguous X-ray cluster surveys is the assessment of the statistics of the large-scale structure. The most fundamental statistical description of the spatial structure is based on the second moments of the distribution, characterized either by the two-point-correlation function or its Fourier transform, the density fluctuation power spectrum. The two-point correlation function of REFLEX shows a power law shaped function with a slope of 1.83, a correlation length of 18.8h1010Mpc, and a possible zero crossing at ~45hi010Mpc [38]. The density fluctuation power spectrum (Fig. 23.26), [129] is characterized by a power law at large values of the wave vector, k, with a slope of « k1 for k < 0.1h Mpc 1 and a maximum around k ~ 0.03h Mpc 1 (corresponding to a wavelength of about 200h_1 Mpc). This maximum reflects the size of the horizon when the Universe featured equal energy density in radiation and matter and is a sensitive measure of the mean density of the Universe, Qm, [129].

Fig. 23.26 Left: Power spectra of the density fluctuations in the REFLEX cluster sample together with predictions from various popular cosmological models taken from the literature. The shape of the power spectrum is best represented by the ACDM and OCDM models with a low matter density parameter (Qm ~ 0.3). For details see Schuecker et al. [129]. Right: Constraints on the cosmological density parameter, Qm, and the amplitude of the matter density fluctuations on a scale of 8h-1 Mpc, o8, obtained from the comparison of the density fluctuation power spectrum and the cluster abundance as a function of redshift in a Karhunen-Loeve statistical analysis of the REFLEX Survey data (Schuecker et al. [131]). The likelihood contours give the 1, 2, and 3 o limits

Fig. 23.26 Left: Power spectra of the density fluctuations in the REFLEX cluster sample together with predictions from various popular cosmological models taken from the literature. The shape of the power spectrum is best represented by the ACDM and OCDM models with a low matter density parameter (Qm ~ 0.3). For details see Schuecker et al. [129]. Right: Constraints on the cosmological density parameter, Qm, and the amplitude of the matter density fluctuations on a scale of 8h-1 Mpc, o8, obtained from the comparison of the density fluctuation power spectrum and the cluster abundance as a function of redshift in a Karhunen-Loeve statistical analysis of the REFLEX Survey data (Schuecker et al. [131]). The likelihood contours give the 1, 2, and 3 o limits

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