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

In this paper, we use large $\pppm$ N-body simulations to study thethree-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 anEinstein-de-Sitter model with its linear power spectrum obtained fromobservations. We found that the scaled function $Q(r,u,v)$, which is defined asthe 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-pointcorrelation function of clusters), depends weakly on $r$ and $u$, but verystrongly on $v$. $Q(r,u,v)$ is about 0.2 at $v=0.1$ and 1.8 at $v=0.9$. A modelof $Q(r,u,v)=\Theta 10^{1.3v^2}$ can fit the data of $\zeta$ very nicely with$\Theta\approx 0.14$. This model is found to be universal for the LCDM clustersand for the PIM clusters. Furthermore, $Q(r,u,v)$ is found to be insensitive tothe cluster richness. We have compared our N-body results with simpleanalytical theories of cluster formation, like the peak theories or the localmaxima theories. We found that these theories do not provide an adequatedescription for the three-point function of clusters. We have also examined theobservational data of $\zeta$ presently available, and have not found anycontradiction between the observations and our model predictions. The$v$-dependence of $q$ in a projected catalogue of clusters is shown to be muchweaker than the $v$-dependence of $Q$ found in the three-dimensional case. Itwould be important to search for the $v$-dependence of $Q$ in redshift samplesof rich clusters.Comment: 11 pages, uuencoded compressed tar ps-file. 2 tables and 8 figures included. The ps file (without uuencoding compressing) can also be obtained by anonymous ftp at ftp://ibm-3.mpa-garching.mpg.de/pub/jing/9506134.p