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

In this paper, we use large P³M N-body simulations to study the three-point correlation function ζ of clusters in two theoretical models. The first model (LCDM) is a low-density flat model of Ω₀ = 0.3, Ʌ₀ = 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 find that the scaled function Q(r, u, υ), which is defined as the ratio of ζ(r, ru, ru+rυ) to the hierarchical sum $$\xi(r)\xi(ru)+\xi(ru)\xi(ru+rv)+\xi(ru+rv)\xi(r)$$ (where ξ is the two-point correlation function of clusters), depends weakly on r and u, but very strongly on υ. Q(r, u, υ) is about 0.2 at υ = 0.1 and 1.8 at υ = 0.9. A model of Q(r, u, υ) = Θ10^1.3υ2 can fit the data of ζ very nicely with Θ ≈ 0.14. This model is found to be universal for the LCDM clusters and for the PIM clusters. Furthermore, Q(r,u,υ) is found to be insensitive to the cluster richness. We compare our N-body results with simple analytical theories of cluster formation, like the peak theories or the local maxima theories. We find that these theories do not provide an adequate description for the three-point function of clusters.We also examine the observational data of ζ presently available, and do not find any contradiction between the observations and our model predictions. The υ-dependence of q in a projected catalogue of clusters is shown to be much weaker than the υ-dependence of Q found in the three-dimensional case. This is probably the reason why the υ-dependence of Q has not been found in an angular correlation function analysis of the Abell catalogue. The υ-dependence found in this paper might be an important feature of clusters formed in the Gaussian gravitational instability theories. Therefore it would be important to search for the υ-dependence of Q in redshift samples of rich clusters.