In the gravitational instability scenario structure in the Universe forms from the collapse of initially small perturbations. The power spectrum of these perturbation depends on the background cosmological model and on the kind of (dark, baryonic, cosmological constant) matter-energy present. In the most popular case in which the matter component is dominated by an unknown "cold" and "dark" particle (Cold Dark Matter, CDM), the first collapsed structrures form at high redshift (z~30) and larger objects thus form from the aggregation of smaller subunits. This scenario is known as "Hierarchical Clustering".
Gravitational collapse finally leads to the formation of relaxed halos. While the mass of these halos is dominated by dark matter, the gas included is shocked to its virial temperature, which is of order of a million degrees for a galactic-sized halo. The following cooling and fragmentation of the gas leads to the formation of stars.
The process of galaxy formation is triggered by the gravitational collapse of dark matter. Then, predicting accurately the statistical properties of the dark matter halos is a fundamental step in the modeling of galaxy formation and evolution.
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The evolution of perturbations in the highly non-linear regime (when the density contrast reaches values much larger than unity) cannot be solved for general initial conditions. It is then customary to resort to N-body simulations to describe this non-linear evolution. The degree of developement in such numerical techniques has reached such a level that it is possible today to run N-body simulation with number of particles approaching one billion. However, such simulations are obviously very expensive in terms of computing time. |
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Besides, a number of approximations to the gravitational evolution of perturbations have been devised to predict the statistics of the dark matter halos withouth running a simulation. These studies also have the merit of providing insight into the physical problem. The simplest approximation is linear theory, which is valid when the perturbations are still very small, and which allows a complete analytical treatment of the gravitational problem. Although linear theory cannot describe the highly non-linear evolution of perturbations, its extension to this regime has often been used to give order-of-magnitude estimates of quantities related to the halos. The most famous approach of this kind is the one proposed by Press & Schechter (1974), which is able to predict the spectrum of masses of dark matter halos, called mass function, that are present at a given time. The prediction of Press & Schechter was found to agree with the results of N-body simulations (Efstathiou et al. 1988). This led to a great effort to improve and extend this approach; this is reviewed in Monaco (1998). In particular, the extended Press & Schechter model (Bond et al. 1991; Lacey & Cole 1993) was found to reproduce not only the mass function of the halos but also their merging history. This method is nowadays widely used to generate Monte Carlo catalogues of dark matter halos with known mass and merger history, which have been used to model the formation and evolution of galaxies (Kauffmann, White & Guiderdoni 1993; Cole at al. 1994; Somerville & Primack 1999). However, no spatial information is available for the objects generated with this model. Moreover, comparison with bigger, more recent N-body simulations has revealed significant discrepancies between Press & Schechter (be it extended or not) and the numerical experiment (Governato et al. 1999; Jenkins et al. 2001).
A powerful tool for predicting the gravitational evolution of pertubations is the Zel'dovich (1970) approximation. According to it the particles that belong to a perturbed matter distribution travel along straight lines, and with a constant velocity (in the comoving space). This very simple approximation gives a very good description of the evolution in the so-called mildly non-linear regime, when the density contrast is of order one and the streams of particles have not crossed each other. Indeed, when the orbits of two particles cross, they are heavily bent due to their mutual gravity, and remain bound in the halo that forms at the crossing point of the streams. On the other hand, the simple Zel'dovich approximation does not reproduce this "confinement", thus breaking after orbit crossing (Shandarin & Zel'dovich 1989 ).
It was found later that the Zel'dovich approximation is just the first term of a perturbative approach, known as "Lagrangian perturbation theory" (Buchert 1996; Bouchet 1996; Catelan 1995). This powerful approach can be used to give prediction that are far more accurate than the simple linear theory. However, the approach breaks at the same "orbit crossing" instant as the Zel'dovich approximation; in this case the trajectories are not straight lines any more, but they fly away from the forming halo even faster than Zel'dovich. It is then clear that to describe the collapsed halos with Lagrangian perturbation theory it is necessary somehow to halt the evolution at the orbit crossing time.
The use of Lagrangian perturbation theory can give accurate estimates of the collapse time and peculiar velocity for each particle in a Monte Carlo realization of a perturbed matter field (Monaco 1997). This opens the possibility of constructing an accurate approximation to the gravitational problem, highly competitive both to the standard N-body approach, in terms of computing speed, and to the extended Press and Schechter approach, in terms of accuracy and wealth of information available. PINOCCHIO is a practical implementation of these ideas: to predict the hierarchical formation of dark matter halos using Largangian perturbation theory applied to a Monte Carlo realization of a perturbed initial density field, much similar to the initial conditions commonly used to run N-body simulations.