Annu. Rev. Astron. Astrophys. 2002. 40: 539-577 Copyright © 2002 by Annual Reviews. All rights reserved

4.2. The Cluster Abundance at Higher Redshifts and Its Evolution

A first analysis of the EMSS cluster sample (Gioia et al. 1990a) revealed negative evolution of the XLF - a steepening of the high-end of XLF indicating a dearth of high luminosity clusters at z > 0.3. This result was confirmed by Henry et al. (1992) using the complete EMSS sample with an appropriate sky coverage function. Edge et al. (1990) found evidence of a strong negative evolution already at redshifts < 0.2 using a HEAO-1 based cluster sample (see Section 3.2). The very limited redshift baseline made this result somewhat controversial, until it was later ruled out by the analysis of the first RASS samples (Ebeling et al. 1997). The ROSAT deep surveys extended the EMSS study on cluster evolution. Early results (Castander et al. 1995) seemed to confirm and even to reinforce the evidence of negative evolution. This claim, based on a sample of 12 clusters, was later recognized to be the result of sample incompleteness and an overestimate of the solid angle covered at low fluxes and its corresponding search volume (Burke et al. 1997, Rosati et al. 1998, Jones et al. 1998).

If cluster redshifts are not available, X-ray flux-limited samples can be used to trace the surface density of clusters at varying fluxes. In Figure 7, we show several determinations of the cumulative cluster number counts stretching over five decades in flux. This comparison shows a good agreement (at the 2 level) among independent determinations (see also Gioia et al. 2001). The slope at bright fluxes is very close to the Euclidean value of 1.5 (as expected for an homogeneous distribution of objects over large scales), whereas it flattens to 1 at faint fluxes. The slope of the LogN-LogS is mainly determined by the faint-to-moderate part of the XLF, but it is rather insensitive to the abundance of the most luminous, rare systems. The fact that the observed counts are consistent with no-evolution predictions, obtained by integrating the local XLF, can be interpreted as an indication that a significant fraction of the cluster population does not evolve with redshift (Rosati et al. 1995, 1998, Jones et al. 1998, Vikhlinin et al. 1998a). We have included the recent data from the Chandra Deep Fields North (Bauer et al. 2002) and South (Giacconi et al 2002), which have extended the number counts by two decades. Note that cosmic variance may be significant because these are only two, albeit deep, pencil beam fields ( 0.1 deg²). Serendipitous surveys with Chandra and XMM (see Figure 4) will fill the gap between these measurements and the ROSAT surveys. The no evolution curves in Figure 7 are computed by integrating the BCS local XLF (Ebeling et al. 1997) according to the evolutionary model in Figure 9.

Figure 7. The cluster cumulative number counts as a function of X-ray flux (log N - log S) as measured from different surveys.

A much improved picture of the evolution of the cluster abundance emerged when, with the completion of spectroscopic follow-up studies, several cluster samples were used to compute the XLF out to z 0.8. These first measurements are summarized in Figure 8. Although binned representations of the XLF are not straightforward to compare, it is evident that within the error bars there is little, if any, evolution of the cluster space density at L_X([0.5 - 2] keV) 3 × 10⁴⁴ erg s^-1 L^*_X out to redshift z 0.8. These results (Burke et al. 1997, Rosati et al. 1998, Jones et al. 1998, Vikhlinin et al. 1998a, Nichols et al. 1999) extended the original study of EMSS to fainter luminosities and larger redshifts, and essentially confirmed the EMSS findings in the overlapping X-ray luminosity range. The ability of all these surveys to adequately study the bright end of the XLF is rather limited, since there is not enough volume to detect rare systems with L_X > L^*_X. The 160 deg² survey by Vikhlinin et al. (1998a), with its large area, did however confirm the negative evolution at L_X 4 × 10⁴⁴ erg s^-1. Further analyses of these datasets have confirmed this trend, i.e. an apparent drop of super-L^*_X clusters at z 0.5 (Nichol et al. 1999 from the Bright-SHARC survey; Rosati et al. 2000 from the RDCS, Gioia et al. 2001 from the NEP survey). These findings, however, were not confirmed by Ebeling et al. (2000) in an analysis of the WARPS sample.

Figure 8. The X-ray Luminosity Function of distant clusters out to z 0.8 compiled from various sources and compared with local XLFs (an Einstein-de-Sitter universe with H0 = 50 km s-1 Mpc-1 is adopted). Numbers in parenthesis give the median redshift and number of clusters in each redshift bin.

The evolution of the bright end of the XLF has remained a hotly debated subject for several years. The crucial issue in this debate is to properly quantify the statistical significance of any claimed evolutionary effect. The binned representation of the XLF in Figure 8 can be misleading and can even lead to biases (Page & Carrera 2000). The full information contained in any flux-limited cluster sample can be more readily recovered by analyzing the unbinned (L_X, z) distribution with a maximum-likelihood approach, which compares the observed cluster distribution on the (L_X, z) plane with that expected from a given XLF model. Rosati et al. (2000) used this method by modeling the cluster XLF as an evolving Schechter function: (L) = ₀(1 + z)^A L^- exp(- L / L^*), with L^* = L^*₀(1 + z)^B; where A and B are two evolutionary parameters for density and luminosity; ₀ and L^*₀ the local XLF values (Equation 7). Figure 9 shows an application of this method to the RDCS and EMSS sample, and indicates that the no-evolution case (A = B = 0) is excluded at more than 3 levels in both samples when the most luminous systems are included in the analysis. However, the same analysis confined to clusters with L_X < 3 × 10⁴⁴ erg s^-1 yields an XLF consistent with no evolution. In Figure 9 we also report the latest determinations of the XLF out to z ~ 1.

Figure 9. (Left) the latest compilation of distant XLFs (RDCS: Rosati et al. 2000; NEP: Gioia et al. 2001; WARPS: Jones et al. 2000; an Einstein-de-Sitter universe with H0 = 50 km s-1 Mpc-1 is adopted). Right panel: Maximum-likelihood contours (1, 2 and 3 confidence level) for the parameters A and B defining the XLF evolution for the RDCS and EMSS samples (for two different cosmologies): * = 0(1 + z)A, L* = L*0(1 + z)B (see Equation 7).

The convergence of the results from several independent studies illustrates remarkable observational progress in determining the abundance of galaxy clusters out to z ~ 1. At the beginning of the ROSAT era, until the mid nineties, controversy surrounded the usefulness of X-ray surveys of distant galaxy clusters and many believed that clusters were absent at z ~ 1. This prejudice arose from an over-interpretation of the early results of the EMSS survey. Gioia et al. (1990a) did point out that the evolution of the XLF was limited only to the very luminous systems but this important caveat was often overlooked. The original controversy concerning cluster evolution inferred from optical and X-ray data finds an explanation in light of the ROSAT results. Optical surveys (Couch et al. 1991, Postman et al. 1996) have shown no dramatic decline in the comoving volume density of rich clusters out to z 0.5. This was considered to be in contrast with the EMSS findings. However, these optical searches covered limited solid angles (much smaller than the EMSS) and therefore did not probe adequately the seemingly evolving high end of the cluster mass function.