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In a CDM universe, the formation of cosmic structures is governed by gravitational processes. At large scales the influence of baryons is negligible. Starting from a Gaussian density fluctuation field with a given processed power spectrum, the gravitational clustering of the fluctuations is followed to the present epoch by means of cosmological N-body simulations, and the results are confronted with observations (for recent numerical results see e.g., Jenkins et al. 1998; Evrard et al. 2002). The completion of large surveys of galaxies and clusters of galaxies, the measurements of sub-degree anisotropies in the microwave background radiation, and the detection of cosmological SNe improved significantly our understanding of the large-scale structure and the mass-energy composition of the Universe (for a recent review see Guzzo 2002 and the references therein). The agreement between observations and the CDM predictions is remarkable (see Frenk, this volume). The "concordance" LambdaCDM cosmological model (Bahcall et al. 1999) emerges as the favorite one. For this model, the universe is flat with the following approximate values of the cosmological parameters: Omegabar = 0.04, OmegaCDM = 0.26, OmegaLambda = 0.7, sigma8 = 0.9, and H0 = 65 km s-1 Mpc-1. In this review, with the generic term CDM we will refer to the LambdaCDM model.

At the scales of galaxies and clusters of galaxies, where high resolution is required in the simulations, an extensive work on formation, mass function, structure, and evolution of the CDM matter halos has been done in the last decade. Analytical and semi-analytical approaches, but mainly numerical N-body simulations, were used. The dark halos are the backbone of the galaxy formation models. Following, we discuss some results which appear relevant for the properties of galaxies:

i) The shape of the mass function of CDM halos is approximately similar to that of the observed Schechter luminosity function of galaxies (e.g., Press & Schechter 1974; Lacey & Cole 1993). The semi-analytical models show that the main problem is at the faint end of the luminosity function (e.g., Kauffmann et al. 1993; Cole et al. 2000; Somerville & Primack 1999); however, reionization and feedback may possibly solve the conflict (Benson et al. 2002a, see also this volume).

ii) The average density profile of CDM halos is described typically by a universal two-parameter profile, both parameters depending ultimately only on the halo mass (Navarro, Frenk, & White 1997, hereafter NFW). Less massive halos tend to be more concentrated than the more massive ones. However, for a given mass, the halo density profiles show a scatter around the NFW profile. This scatter correlates with the halo mass aggregation history (MAH), in the sense that halos assembled earlier are more concentrated (Avila-Reese et al. 1998, 1999; Wechsler et al. 2002). Some dependence on the environment has also been reported (but see Lemson & Kauffmann 1999). The CDM halos are too concentrated and their inner density profiles are cuspy, in apparent disagreement with observations, mainly the inner rotation curves of dwarf and LSB galaxies (Moore 1994; Burkert 1995; see also Bosma, de Blok, and Colín et al. in this volume). At galaxy-cluster scales, the inferred halo inner density profiles, under the uncertainties, typically are fitted by both the NFW and the pseudo-isothermal profiles. Observations seem to show that, from dwarf to galaxy-cluster scales, the central halo density is poorly dependent on mass, and the core radius increases roughly proportional to the maximum circular velocity Vmax (Firmani et al. 2000, 2001). Figure 1 presents the approximate range of values of the halo central density and core radius vs. Vmax inferred from observations. Another potential problem of the CDM halos is that the number of subhalos within MW-sized halos overwhelms the number of observed satellite galaxies by a large factor (Kauffmann et al. 1993; Klypin et al. 1999; Moore et al. 1999). Besides, the large population of satellite subhalos could have a dramatic effect on the dynamics of the galaxy disks (Moore et al. 1999; Colpi, Mayer & Governato 1999). Owing to the success of the CDM model at large scales, only minor modifications to the model have been proposed in order to solve these potential problems. For example, in a Warm Dark Matter (WDM) scenario with particle masses of ~ 0.6 - 1 KeV, the satellite velocity function in MW-sized halos is well reproduced, while at larger scales the predictions are similar to CDM (Colín et al 2000). However, the inner density profiles of galaxy-sized and larger halos are similar to their CDM counterparts; even the small subhalos show density profiles well fitted by the NFW profile, although with concentrations lower than predicted by CDM (Avila-Reese et al. 2001; Bode, Ostriker & Turok 2001). Conversely, self-interacting dark matter (SIDM) with a velocity-dependent cross sections, sigmaDM = 0.5 - 1.0 (100 km s-1 / Vmax) cm2 / gr, produces inner halo density profiles in agreement with observations at all scales (Fig. 1), but the substructure remains similar to CDM (Firmani et al. 2001; Colín et al. 2002, see also this volume). The substructure problem may be actually alleviated by the reionization, which certainly has to be taken into account in the formation of dwarf galaxies (Bullock et al. 2000; Benson et al. 2002a).

Figure 1

Figure 1. Central density and core radius in this plot are defined as the density and radius where the halo profile slope becomes steeper than -1. The dotted lines encompass roughly the 2 sigma dispersion of the observational inferences of rhoc,-1 and rc,-1 vs. Vmax for dwarf and LSB galaxies and several clusters of galaxies (see Colín et al. 2002 for the source references). Squeletal and open triangles are high-resolution halo simulations for the LambdaCDM and LambdaSIDM models; the latter one is for a cross section inversely proportional to the relative velocity.

iii) The angular momentum distribution in most of CDM halos seems to be well parametrized by a universal function, and the disks formed within them, assuming detailed angular momentum conservation, are roughly exponential (Bullock et al. 2001b). The global spin parameter lambda has a lognormal distribution and is approximately independent of the cosmology, mass, and environment (e.g. Catelan & Theuns 1996). Two mechanisms for the origin of the halo angular momentum seem to compete: linear tidal torques and orbital angular momentum transfer of merging satellites (e.g., Peebles 1969; Vitvistkaya et al. 2002; Maller, Dekel & Somerville 2002). The role of angular momentum in formation of galaxies is crucial. More work should be done, for example, on the symmetry and degree of alignment of the angular momentum distribution in the halos (see van den Bosch et al. 2002 for recent results) and on the angular momentum dependence on environment.

iv) The hierarchical mass aggregation history (MAH) of CDM halos scatters around its average, because they emerge from a stochastic density field. On average, the less massive halos tend to assemble a given fraction of their mass at earlier epochs than the more massive halos (Avila-Reese et al. 1998). The halos may grow by a variety of merging regimes going from smooth mass accretion to violent major mergers (e.g., Salvador-Solé et al. 1998). This variety of merging regimes is relevant to the morphology of the galaxies formed within the halos. Cosmological simulations show that (i) most of the z = 0 galaxy-sized halos are not contained within larger halos nor have close massive companions, and that (ii) most of the mass of these halos has been aggregated by smooth accretion (Avila-Reese et al. 1999; Somerville & Kolatt 1999; Gottlöber et al. 2001). Nevertheless, the pair abundance and the major merger rate increase with z, and this increase is faster for cluster halos than for field halos (Gottlöber et al. 2001). The predicted major merging rates agree with those inferred from accurate statistics of galaxy pairs. From the fraction of normal galaxies in close companions (with separations less than 50 kpch-1) inferred from observations at z = 0 and z = 0.3 (Patton et al. 2000, 2002), and assuming an average merging time of ~ 1 Gyr, we estimate that the major merging rate at the present epoch is ~ 0.01 Gyr-1 for halos in the range of 0.1 - 2.0 1012 Modot, while at z = 0.3 the rate increased to ~ 0.018 Gyr-1. These values are only slightly lower than predictions for the LambdaCDM model, suggesting that the model can be used to predict the increasing merging rate at earlier epochs. The fact that at high redshifts the major merging rate is much higher than at the present epoch, may be at the basis of early spheroid formation, as well as energetic phenomena like QSOs and submillimeter sources (see Sections 5 and 6).

Using cosmological N-body simulations, we have learnt much about the clustering and properties of collapsed CDM structures. Nevertheless, important details about the innermost, less resolved regions of dark halos, where just luminous galaxies form, as well as about the angular momentum, the MAHs and their dependences on the environment, remain still unexplored. On the other hand, the potential conflicts of CDM at small scales are still open. More observational and theoretical studies are necessary to understand the depth of these conflicts.

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