Three-point correlation function of galaxy clusters in cosmological models: a strong dependence on triangle shape

In this paper, we use large $\pppm$ N-body simulations to study the three-point correlation function $\zeta$ of clusters in two theoretical models. The first model (LCDM) is a low-density flat model of $\Omega_0=0.3$, $\Lambda_0=0.7$ and $h=0.75$, and the second model (PIM) is an Einstein-de-Sitter model with its linear power spectrum obtained from observations. We found that the scaled function $Q(r,u,v)$, which is defined as the ratio of $\zeta (r, ru, ru+rv)$ to the hierarchical sum $\xi (r)\xi (ru)+ \xi (ru) \xi (ru+rv) +\xi (ru+rv)\xi (r)$ (where $\xi$ is the two-point correlation function of clusters), depends weakly on $r$ and $u$, but very strongly on $v$. $Q(r,u,v)$ is about 0.2 at $v=0.1$ and 1.8 at $v=0.9$. A model of $Q(r,u,v)=\Theta 10^{1.3v^2}$ can fit the data of $\zeta$ very nicely with $\Theta\approx 0.14$. This model is found to be universal for the LCDM clusters and for the PIM clusters. Furthermore, $Q(r,u,v)$ is found to be insensitive to the cluster richness. We have compared our N-body results with simple analytical theories of cluster formation, like the peak theories or the local maxima theories. We found that these theories do not provide an adequate description for the three-point function of clusters. We have also examined the observational data of $\zeta$ presently available, and have not found any contradiction between the observations and our model predictions. The $v$-dependence of $q$ in a projected catalogue of clusters is shown to be much weaker than the $v$-dependence of $Q$ found in the three-dimensional case. It would be important to search for the $v$-dependence of $Q$ in redshift samples of rich clusters.