MODERN COSMOLOGY

Cluster surveys 337

figure 11.12 shows indeed less structure atz=3intheCDM model when
compared with OCDM).
One of the fundamental quantities that a CDM model predicts is thecluster
mass function,N(M,z), i.e. the number of virialized clusters per unit volume
and mass, at different epochs. This can be derived by applying cluster-finding
algorithms directly on simulations, as in figure 11.12. A very simple and
powerful method proposed by Press and Schechter (1974) is, however, often
used to computeN(M,z). This analytical approach is found to be in remarkable
agreement withN-body simulations, although slight refinements have recently
been proposed (Sheth and Tormen 1999). We refer the reader to the original
papers or the aforementioned textbooks for a derivation of the Press–Schechter
method.

11.4.2 Cluster search methods

The cluster mass is not a direct observable, although several methods exist to
estimate the total gravitational mass of clusters. In order to derive the cluster
mass function at varying redshifts, one needs three essential tools:

(1) an efficient method to find clusters at least out toz1;

(2) an estimator (observable),Mˆ, of the cluster mass; and

(3) a simple method to compute the selection function, i.e. the comoving volume

within which clusters are found.

We can summarize the methods of finding distant clusters as follows:

• Galaxy overdensities in optical/IR images: this is the traditional way
  which was successfully used by Abell to compile his milestone cluster
  catalogue. At high redshifts, chance superpositions of unvirialized systems
  and strongK-corrections for cluster galaxies make optical searches very
  inefficient. Near-IR searches, supported by some colour information,
  improve substantially the effectivness of this method. In general, however,
  the estimate of the survey volume is ill defined and model dependent. In
  addition, the optical luminosity is poorly correlated with the cluster mass.


• X-ray selected searches: arguably, the most efficient method used so far to
  construct distant cluster samples and to estimate the mass function. The x-
  ray luminosty is well correlated with the mass and the selection function
  is straightforward, since it is the one of a (x-ray) flux-limited sample.
  Possible biases, similar to galaxy searches, are connected to possible surface
  brightness limits.


• Search for galaxy overdensities around high-z radio galaxies or AGN:
  searches are conducted in near-IR or narrow-band filters. This method has
  provided so far the only examples of possibly virialized systems atz> 1. 5
  (e.g. Pentericciet al2000).


• Sunyaev–Zeldovich (SZ) effect: distortion of the CMB spectrum due to
  the cluster hot intra-cluster medium. Being a detection in absorption,