It is well known that the high-order correlation functions are very important cosmological measures which contain information in addition to the most widely used measure - the two-point correlation function (Peebles 1980). Determination of the high-order correlation functions, however, requires much better observational data than those needed for the two-point correlation function. Thanks to large angular and redshift surveys of galaxies which have been available recently or will be available, the high-order correlation functions have attracted much more attention.
According to the Second-order Eulerian Perturbation Theory (hereafter SEPT, Peebles 1980), the skewness , which is related to the three-point correlation function through
depends only on the shape of the linear power spectrum if the primordial density fluctuation is Gaussian (Fry 1984; Bouchet et al. 1992; Juszkiewicz et al. 1993; Catelan et al. 1995). The SEPT prediction has been shown to be in very good agreement with the results of N-body simulations in the quasilinear regime (Juszkiewicz et al. 1993; Luchin et al. 1994; Bernardeau 1994; Baugh et al. 1995; Colombi et al. 1996). It is also expected that the skewness is a statistic sensitive to a possible bias of the galaxy distribution (Fry & Gaztañaga 1993; Mo et al. 1996) and to a possible non-Gaussianity of the initial density fluctuation (Fry & Scherrer 1994). The statistical analysis of the APM galaxy catalogue (Gaztañaga 1994, 1995) has yielded a skewness for galaxies which is compatible with the theoretical prediction for the mass skewness, provided that the primordial fluctuation is Gaussian and the power spectrum is the same as that measured for the APM galaxies (Gaztañaga & Freiman 1994). An important implication is then that there exists little bias between the distributions of the galaxies and of the underlying mass. Mo et al. (1996) have recently studied the skewness for two plausible bias models that identify either primordial density peaks or dark matter halos as 'galaxies'. They found that for a power spectrum with a similar shape to that observed in the APM survey, the skewness of these 'galaxies' agrees with the APM galaxy skewness only when the spatial bias between the 'galaxies' and the underlying mass is small.
The three-point correlation function contains much richer information than the skewness, since the latter is an integral of the former (Eq.1). Based on SEPT, Fry (1984) calculated the three-point correlation function for scale-free power spectra . He pointed out that the normalized three-point correlation function Q, which is defined as @
depends on the shape of the triangle constructed from the three points , and . The shape dependence in turn depends on the index n of the power spectrum (see also Fry 1994 and Sect. 2). If the SEPT prediction holds for in the quasilinear regime as for and if galaxies trace mass as the previous studies on suggest, we would expect such a shape dependence in the three-point correlation function of galaxies. This shape dependence would certainly be important for determining the primordial power spectrum and for understanding the bias processes. Therefore, it is very important to study and to test with N-body simulations the SEPT predictions for the three-point correlation function.
In this paper, we will calculate the three-point correlation functions for realistic power spectra in SEPT. We consider the power spectra of two Cold Dark Matter (CDM) models and one Mixed Dark Matter (MDM) model. The two CDM spectra, which are specified by the parameter (where is the current density parameter and h is the Hubble constant in unit of , Bardeen et al. 1986), have and respectively. The first CDM spectrum is well known as the Standard CDM (SCDM) power spectrum, and the second is usually regarded as a power spectrum for a low-density CDM universe (LCDM). The MDM model assumes an Einstein-de Sitter universe with one of species neutrino with density , CDM density , baryon density and the Hubble constant . Both LCDM and MDM spectra are known to be compatible with the large scale structures observed in the local Universe, therefore it is very important to find statistics to distinguish between them (Bahcall 1995, Efstathiou et al. 1992, Jing et al. 1994, Klypin et al. 1993). A very encouraging result from this calculation is that the dependence of on the triangle shape is so sensitive to the shape of the power spectrum that the LCDM and MDM spectra can hopefully be discriminated by the three-point correlation function in the quasilinear regime.
Although the SEPT prediction for the skewness has been shown to be valid in the quasilinear regime by a number of authors, there is no a priori reason to believe that the SEPT prediction for the three-point correlation function is equally valid since the skewness is related to the three-point correlation function through an integral (Eq. 1). As shown in recent work by Jing et al. (1995), two quite different forms of the three-point correlation function can result in an indistinguishable skewness. Therefore, it is valuable to use N-body simulations to test the SEPT prediction for . In this paper, we will use a large set of N-body simulations to test the SEPT prediction for . The N-body test will not only tell us whether the SEPT results of can be applied to real observations but will also show us, as a result of general interest, whether the SEPT prediction for agrees with N-body simulations as accurately as the prediction for the skewness.
© European Southern Observatory (ESO) 1997
Online publication: July 3, 1998