MASAJE EN MUJERES EMBARAZADAS

Although the linear theory represents an extremely useful tool to describe the early growth of structure and the growth on scales that are so large to be still in the linear regime, it obviously cannot provide a complete description of the highly asymmetric processes occurring in hierarchical galaxy formation. In that case, numerical simu- lations are usually employed. In N–body simulations the phase–space distribution function f(r,v) is replaced by a set of N particles that are subsequently evolved under their self–gravity. Each particle can essentially be viewed as a delta–function in the phase–space. Therefore, if the number of particles is large enough, the system provides a fair approximation of the phase–space distribution function.

N-body numerical simulations have nowadays evolved into a widely used tool in cosmology thanks to the rapidly growing computer performance and, at the same time, to the development of more sophisticated numerical algorithms.

Early simulations of structure formation (White 1976; Aarseth et al. 1979; Miller 1983) employed a direct summation method for the computation of the gravita- tional force. This method requires O(N2) operations for N particles and is there- fore prohibitive when the number of particles increases. Different techniques have been subsequently developed, that substantially reduce the computational time (see Sellwood 1987, for an excellent, although out–of–date, review of the topic). In the so–called PM (Particle Mesh) codes, for example, the density field produced by the particles is computed on a Cartesian grid and the Poisson equation is solved on the grid points. This can be done within relatively short computational time using Fast

1.4 N–body simulations

Fourier Transform Techniques (FFT), that provide directly the values of the gravi- tational potential on the grid. It is possible to show that the computational effort of the method is O(Ngridlog(Ngrid)). Further refinements of the PM code are given by the P3_{M (Particle–Particle–Particle–Mesh) and APM (Adaptive Particle Mesh)} codes. In these, the force computation is supplemented with a direct summation on scales below the mesh size (P3M), or mesh refinements are placed on highly clustered regions (APM).

Substantial progress in the field has arisen from the introduction of so–called treecodes, where particles are arranged in a hierarchy of groups and the gravita- tional field at each point is computed by a summation over the multipole expansion of the gravitational field of these groups.

The simulations I will use in this thesis have been carried out using the treecode

GADGET (Springel et al. 2001b), that is publicly available, and a particular re– simulation technique that is usually referred to asmass refinementorzoomtechnique (Tormen et al. 1997; Katz & White 1993). First, a cosmological simulation of a large region is used to select a suitable target cluster. The particles in the final cluster and its surroundings (usually all the particles within two times the virial radius of the selected halo) are then traced back to their initial Lagrangian region and are replaced by a larger number of lower mass particles. These are perturbed using the same fluc- tuation distribution as in the parent simulation, but now extended to smaller scales to account for the increase in resolution. This resampling of the initial conditions of the Lagrangian region of the cluster thus allows a localised increase in mass and force resolution. Outside thehigh-resolution region, particles of variable mass, increasing with distance, are used so that the computational effort is concentrated on the cluster of interest, while still maintaining a faithful representation of the large-scale density and velocity field of the parent simulation.

In the ‘boundary region’ a spherical grid is used, whose spacing grows with dis- tance from the high–resolution region and that extends to the box size of the parent simulation. Outside the boundary region, vacuum boundary conditions are used, i.e. a vanishing density fluctuation field.

The effect of increasing mass resolution in the high–resolution region, at the centre of the re–simulation box, is clearly visible in Fig. 1.2 that shows a zoom–in on the dark matter distribution for a ‘typical’ region of the Universe at redshift zero. The maps represent slices of 10 Mpc along the z–direction and the dimensions of the box decrease going from the top left to the bottom right panel.

Note that direct numerical simulations of a set ofN particles represent a challeng- ing technical problem also for other reasons. Despite the significant improvement achieved in the last years,N–body simulations still usually employ a number of par- ticles that is significantly smaller than the actual number of particles in the real system that is being modelled. In order to prevent excessive two–body relaxation, one is forced to use a numerical trick that is called softening of the gravitational

Figure 1.2: A zoom–in on the dark matter distribution for a ‘typical’ region of the Universe at redshift zero. The maps represent slices of 10 Mpc along the z–direction and the size of the box varies from 479 (top left), 380 (top right), 95 (bottom left) to 52 Mpc (bottom right). The figure clearly shows the increasing mass resolution in the high–resolution region at the centre of the re–simulation box.

force, i.e. at small separation the gravitational force is reduced below the Newtonian value in some smooth way.

A nice historical review of the increase rate of N in N-body simulations over the last thirty years can be found in Moore (2000). An important achievement of

1.4 N–body simulations

Figure 1.3: Density map of a high resolution cluster re–simulation at redshift zero.

From Springel et al. (2001a). The cluster has a virial mass of 8.4×1014_h−1_M

¯.

The high resolution region of the simulation contains 66 million particles.

the latest numerical simulations has been the solution of the ‘overmerging problem’ (Klypin et al. 1999a, see also Chapter 2), i.e. the possibility to resolve self-bound substructures in the smooth dark matter background of simulated haloes. This allows closer link to be made between simulated data and observational results.

As an example of the performance achieved by numerical N–body simulations in the last years, I reproduce in Fig. 1.3 the density map of one of the highest resolution simulations of a cluster carried–out so far (Springel et al. 2001a). The progress is impressive when compared to the state–of–art of numerical simulations of only two decades ago, which is reproduced in Fig. 1.4.

Figure 1.4: Projected distribution of a 700–body system with mass comparable to the virial mass of the Coma cluster. From White (1976).