*Astron. Astrophys. 318, 667-672 (1997)*
## 1. Introduction
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
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