2 Cluster sample in numerical simulations

We utilize the AP3M code of Couchman (1991) to follow the dynamics of 256³ particles in a box of 500 h^-1 Mpc with periodic boundary conditions. We employ a cold dark matter model with a cosmological constant $\Omega_{\Lambda}=0.7$, a matter density $\Omega_{\rm m}=1-\Omega_{\Lambda}$, and a Hubble constant H₀=100  h km s^-1 Mpc^-1 with h=0.7. The age of the universe in this model is $\approx$13.5 Gyrs. The normalization, given by the linear mass variance of dark matter on 8 h^-1 Mpc scale, $\sigma_8=0.87$, is in accordance with the four year COBE DMR observations as well as the observed abundance of galaxy clusters. The code uses glass-like initial conditions, cp. Knebe & Müller (1999). The initial power spectrum was calculated with the CMBFAST code (Seljak & Zaldarriaga 1996). We start the simulations at an initial redshift z=25. Up to this time the Zeldovich approximation provides accurate results on the scales considered here. We employ a comoving softening length of 100 h^-1 kpc, and 1000 integration steps that are enough to avoid strong gravitational scattering effects on small scales, cp. Knebe et al. (2000). With this softening length, the inner cores or break radii of the cluster sized halos are resolved. For our statistical analysis the big simulation volume provides us with a sufficient number of clusters embedded into the large scale structure network. In particular an accurate representation of the large scale tidal field is important for getting stable orientation results. The particle mass in the simulation is $6.2 \times10^{11} ~{h^{-1}~M_\odot}$, comparable to the mass of one single galaxy. For reliable cluster orientations and angular momenta, the number of particles per cluster should not drop below a minimum of a few hundred of particles.

We search for bound systems using a friends-of-friends algorithm. In our $\Lambda$CDM model the virialization overdensity $\rho/\rho_{\rm mean}$ at z=0 is $\simeq$330(Kitayama & Suto 1996) which corresponds (for spherical isothermal systems) to a linking length of 0.17  times the mean inter particle distance. These bound systems are our clusters of galaxies, i.e. our cluster sized halos. The 3000 most massive ones are the constituents of our mock cluster sample. The mean distance between clusters in this sample is 34.7 h^-1 Mpc, which is in agreement with the mean distance of clusters in the REFLEX cluster survey (Böhringer et al. 1998). The most massive cluster has a mass of $2.3\times10^{15}~{h^{-1}~M_\odot}$, resolved with 3700  particles, the lightest still has $1.4\times10^{14}~{h^{-1}~M_\odot}$ which corresponds to 224  particles.

The two-point correlation function $\xi(r)$ of the cluster sized halos in redshift space shows the expected behavior (Fig. 1) compatible with the correlation function of the REFLEX clusters determined in Collins et al. (2000). The correlation length is 16 h^-1 Mpc, quantifying the amplitude of the correlation function. At scales of about 70 h^-1 Mpc $\xi(r)$ becomes negative.

Figure 1: The two-point correlation function of cluster sized halos in redshift space (solid line). The dashed line has a slope of -1.8. The data points are taken from Collins et al. (2000).

The principal axes of a cluster are determined by means of the eigenvectors of the inertial tensor using all halo mass points. Assuming an ellipsoidal shape these vectors can be uniquely transformed into the three axes of the underlying mass ellipsoid. Subsequently the three axes are assumed to be ordered by size as $a\ge b\ge c$. In addition to the mass ellipsoid, the angular momentum vector ${\vec{L}}$and the spin parameter $\lambda$ can be utilized to characterize the dynamical state of a cluster sized halo. ${\vec{L}}$ is calculated by summing up the angular momenta of all mass particles of the respective halo. The spin parameter is defined by $\lambda = \omega/\omega_{\rm sup}$, where $\omega$ denotes the actual angular velocity and $\omega_{\rm sup}$ the angular velocity needed for the system to be rotationally supported against gravity. This ratio can be expressed as

$$\lambda = L/(G^{1/2}M^{3/2}R^{1/2}),$$ (1)

where L is the absolute value of the angular momentum ${\vec{L}}$, M the total mass of the system and R its radius, respectively (e.g. Padmanabhan 1993). The radius R is taken as the radius of a sphere with the same volume as the halo (estimated on a fine grid).