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