On the theoretical side SORs are discussed analytically in more detail in Bond et al. (1991), Bower (1991), Richstone et al. (1992), Lacey & Cole (1993, 1994), Kauffmann & White (1993), Kitayama & Suto (1996a,b), Tormen (1998), Percival & Miller (1999), Somerville et al. (2000), Cohn et al. (2001). A frequently applied formalism is the excursion set variant of the Press-Schechter prescription, i.e., the extended Press-Schechter (EPS) formalism which is expected to provide a full theoretical treatment of hierarchical structure formation.

The idea is to describe the hierarchical growth of structure as an abstract diffusion process in a density-contrast/mass space (see also Sarazin, this volume). Here the fraction of trajectories reaching specific density-contrast/mass points are used to derive analytic formulae for mass functions and merger rates. The basic conclusion of the investigations mentioned above is that numerical N-body simulations and EPS semi-analytic estimates of merger counts do roughly agree.

In order to illustrate the inferred results, one can use within the framework of EPS the simplified counting arguments of diffusion trajectories, first discussed in Bond et al. (1991) and Lacey & Cole (1993). The semi-analytic estimate of the conditional probability that a halo of mass observed at the cosmic time t2 has a parent in the mass range M2/2 < Mi < M2 at the time t\ < t2 can thus be used to estimate the fraction of clusters that are formed, i.e., assembled at least half of their mass on the past dynamical time scale of the cluster between t\ and ¿2 (today),where At = - t\ represents the time scale over which significant distortions of the cluster potential are detectable.

Assuming that all substructures detected in the observations mentioned in § 4.2 are major merger events in the sense described above (a quite crude assumption), one can compute for a typical cluster with M2 = 1015 Mq the SOR for different structure formation scenarios. The theoretical SORs appear to be consistent with the estimate (1) for a standard Cold Dark Matter (CDM) model with Q0 = Qm = 1 and dynamical time scales in the range The presently favoured CDM variant with a large cosmological constant (ACDM) and fim = 0.3 appears to be consistent with (1) and dynamical time scales > 2.5 Gyr. If we assume that significant distortions of X-ray emissi-tivity distribution of cluster merger are still seen at least 3.5 Gyr after first core passage as suggested by numerical experiments (e.g., Ricker & Sarazin 2001) one could conclude that the presently observed SORs support low-density structure formation models.

However, in addition to the problem that the observed merger events might not necessarily be of the same type as assumed by EPS, the question arises whether the application of EPS is appropriate in general. The similarity of the statistical results obtained with N-body simulations and EPS appears to be quite suprising since many assumptions of EPS are expected to be incorrect in detail (see also Conn et al. 2001): spherical collapse (Sheth et al. 2001), monotonic growth of halos (Tormen 1998), association of initial density peaks with final halos (Frenk et al. 1988; Carlberg 1990; Katz et al. 1993), application of sharp /c-space filters to frame the region of primordial material that ultimately collapses to form a virialzed halo (e.g., Schuecker et al. 2001a). It also appears questionable whether Markovian processes, used to derive EPS, provide the correct theoretical framework (White 1997, Schuecker et al. 2001a).

Note also that in contrast to the general agreement of statistical quantities of simulations and EPS predictions, on the halo-by-halo basis the mass assignment scheme of EPS clearly contradicts numerical experiments (see Fig. 8 in White 1996). Therefore, much work has been done in order to improve the original formalism (see, e.g., Lucchin & Matar-rese 1988; Lilje 1992; Cavaliere & Menci 1994; Jedamzik 1995; Monaco 1995, 1997a,b; Yano et al. 1996; Valageas & Schaeffer 1997; Lee & Shan-darin 1998; Gross et al. 1998; Sheth et al. 2001; Sheth &; Tormen 1999; Jenkins et al. 2001), but without performing the critical halo-by-halo test (see also Gottlober et al. 2001). More realistic comparisons of observation and theoretical expectation in the sense described in § 4.2 are clearly needed.

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