### 4.2.3 Recent developments

After these important papers in which the basic scenario was outlined and the general assumptions justified, it was necessary to develop more detailed models, mainly numerical and N-body simulations due to the high complexity of the various physical processes involved.

A schematic list of models and reviews is now given, which should be completed with references therein: Cole (1991), White and Frenk (1991), Navarro and White (1993), Kauffmann, Guidernoni and White (1994), Cole et al. (1994), Navarro, Frenk and White (1996, 1997), Lacey and Cole (1993), Avila-Reese, Firmani and Hernandez (1998), Avila-Reese et al. (1999), Sensui, Funato and Makino (1999), Salvador-Sole, Solanes and Manrique (1998), Baugh et al. (1999), Subramanian, Cen and Ostriker (1999), Steinmetz (1999), van den Bosch (1999) and a large series of papers, reflecting the importance of the topic.

Models can be classified as "semi-analytical" (in which some processes are given a simplified treatment assuming simple recipes, based on either previous numerical calculation or on theoretical ideas), numerical simulations (e.g. hydrodynamical simulations, collision-less simulations) N-body simulations (the most widely used) and even analytical. Some hybrid models are difficult to classify in this scheme.

It is first necessary to adopt a cosmological model, the most popular one being the "standard" CDM (with = 1, h = 0.5, for instance) or the CDM (more in consonance with current values, = 0.3, = 0.7, h= 0.65). A primordial fluctuation spectrum must often be adopted, usually a power law P(k) kn, with n ranging from 0 to -1.5 (for example), where k is the wave number. Another important parameter used by most models is . In general, the variance is defined as < > 1/2; then is the present value for a scale-length of 8 Mpc. This parameter is adopted "a priori" taking into account the present large-scale structure, rather than considering a real free parameter. Usual values adopted are 0.6 for the standard CDM and 1 for the CDM.

Other parameters characterize the calculation methods. For instance the initial redshift, the number of particles in N-body simulations and the box in Mpc3 in which the calculations are performed. The so called "Virgo consortium" (Jenkins et al. 1997) is able to handle 2563 particles and a large volume of the order of 60 Mpc. Parameters controlling the resolution of the simulation and the efficiency with which gas cools have a higher influence on the results (Kay et al. 1999).

These models not only deal with the formation of halos, but also with the ability of gas to form stars, with matter and energy outputs, mainly due to supernova explosions, the evolution of the baryonic component, the explanation of the Hubble Sequence, how spirals merge to produce spirals and so on. From our point of view, the rotation of galaxies strongly depends on the structure of the halos, which is determined in the first stage of the computations. The latest evolution of visible galaxies is, paradoxically, the most difficult to understand and to model. For instance, the Initial Mass Function (IMF) is largely unknown and yet is decisive in galactic evolution.

The hierarchical process of merging, the formation and internal structure of dark matter halos is said to be the best known process. This could be due, in part, to the relative simplicity of the process, but also to the evident fact that it is easier to make predictions about the unobservable. In general, even if some observable facts remain insufficiently explained, these families of theoretical models provide a very satisfactory basis to interpret any evolutionary and morphological problem.