4. Conclusion and further discussion
In this paper, we have carefully studied the three-point correlation function in the quasilinear regime both with the second-order Eulerian perturbation theory and with a large set of N-body simulations. With the perturbation theory, we showed that the normalized three-point correlation function Q sensitively depends on the shape of the linear power spectrum. For scale-free power spectra with index , is an increasing function of v, and the larger n (i.e. the flatter the spectrum), the more rapidly Q increases with v. Because of differences among the effective slopes of the SCDM, LCDM and MDM spectra in the quasilinear regime, for a fixed configuration of a triangle, the SCDM model shows the strongest and the MDM the weakest variation of Q with v. With these dependences, even the two popular models, LCDM and MDM, could perhaps be discriminated by measuring Q for a large galaxy survey. Motivated by this potential importance, we analyzed the three-point correlation functions for a large set of N-body simulations. Our N-body results show that in the quasilinear regime (), Q increases with v in all three models. Furthermore Q has the least increase with v in the MDM model and has the most increase in the SCDM model. These two points qualitatively agree with the prediction of the perturbation theory. However, quantitatively, the N-body results show less variations of Q with v in the SCDM model and more variations in the other two models than the prediction of the perturbation theory. Thus the three-point correlation function is less powerful as a discriminator between the popular models than the perturbation theory originally suggests.
On the other hand, the robust dependence of Q on v predicted by our N-body simulations should exist in the distributions of galaxies if the galaxies really trace the underlying mass in spatial distribution. Search for this dependence will eventually put constraints on the bias. We note that Groth and Peebles (1977) have found a weak dependence of q (the normalized three-point correlation function in angular distribution) on v for the Lick catalogue. However, in their analysis, they have mixed (averaged) q of large size and small size triangles. Since q is approximately a constant for small triangles where clustering is strongly non-linear (Efstathiou et al. 1988, Matsubara & Suto 1994; we have confirmed this result in our simulations), their results cannot be directly compared with our N-body results here. We also note that a similar dependence of Q on v has been found for clusters of galaxies in N-body simulations (Jing et al. 1995) which means that the spatial distributions of clusters are closely related to the distribution of the underlying mass.
Besides many investigations which have tested with N-body simulations the high-order correlation prediction of the perturbation theory through the skewness (see Sect. 1), there are two publications which tested the bispectrum (Eq. 3) and the function (Eq. 9) for scale-free power spectra. Fry et al. (1993) analysed the bispectrum for an ensemble of staggered-mesh simulations. Their results in the quasilinear regime agree with the SEPT prediction (Eq. 3), although the statistical errors in their N-body results are rather large (because of the small amount of independent triangles in the Fourier space in their analysis). More closely related to our work, Matsubara & Suto (1994) analyzed for a set of Tree N-body simulations with to particles. They concluded that Eq.(9) disagrees with their N-body results, consistent with our results here. It is worth pointing out that the discrepancy they found looks more severe than the one we found here. The reason is probably that their comparison between theory and simulations is not optimal, since they did not separate out the non-linear regime where the SEPT prediction is expected to be invalid and they ignored the finite-volume effect which is serious for separations greater than 0.1 box size. Furthermore, their method for quantifying a triangle mixed configurations of different v for a finite triangle-bin. For example, their triangles of shape-bin (1,10,10) could have all possible values 0 to 1 for v. Our discussions in Sect. 2 have shown that Q depends very sensitively on v in SEPT.
We have checked the skewness for our N-body simulations and found that the N-body skewness is in good agreement with the SEPT prediction, consistent with many earlier investigations (Sect. 1). Indeed, the skewness shows much better agreement between the perturbation theory and the N-body simulations than the three-point correlation function. The reason can be seen in our Fig. 3 where the N-body results of cross the SEPT prediction curves at for the two worse-fit models (SCDM and MDM). The skewness is an average of over r, u and v (Eq. 1) and the averaging improves the agreement for the skewness between the perturbation theory and the N-body simulations.
© European Southern Observatory (ESO) 1997
Online publication: July 3, 1998