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

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2. Prediction of the perturbation theory

In the second-order Eulerian perturbation theory, the bispectrum [FORMULA], which is the Fourier transform of the three-point correlation correlation function, is predicted to be (Fry 1984)

[EQUATION]

Strictly speaking, the above expression is derived only for the Einstein-de Sitter universe. However it has been shown that this expression is also a very accurate approximation for [FORMULA] (Bouchet et al. 1992, Catelan et al. 1995), so we will apply this equation to the low-density flat model with [FORMULA] as well (Sect. 3).

The three-point correlation function is then

[EQUATION]

After some tedious calculation, we can write [FORMULA] as

[EQUATION]

where [FORMULA], [FORMULA] is the two-point correlation function

[EQUATION]

and [FORMULA] and [FORMULA] (l is 0 or 2) are respectively

[EQUATION]

and

[EQUATION]

For a power-law power spectrum [FORMULA], from the above expressions, one can easily get the result of Fry (1984), i.e.

[EQUATION]

where [FORMULA] is the slope of [FORMULA] [ [FORMULA] ]. For a realistic power spectrum, to calculate the three-point correlation function, one has to integrate Eqs. (6-8) numerically as we do below.

The properties of [FORMULA] for scale-free power spectra are very instructive for understanding the results of [FORMULA] for realistic power spectra, therefore we first discuss [FORMULA] for scale-free power spectra. In this case, the normalized three-point correlation function Q only depends on the shape of the triangle and on the index n of the power spectrum. There are many ways to express the shape of a triangle. After some trials, we found that Peebles's variables r, u and v (Peebles 1980) provide a very good presentation of the three-point correlation function. For a triangle with three sides [FORMULA], r, u, and v are defined as:

[EQUATION]

Clearly, u and v characterize the shape and r the size for a triangle. With these variables, Q does not depend on r and depends very weakly on u for a scale-free [FORMULA]. The dependence of Q on the triangle shape is then mainly the dependence on v. Figure 1 shows Q as a function of v for three scale-free spectra with [FORMULA], 1.5 and 2 (or equivalently [FORMULA], -1.5 and -1). For each spectrum, u is fixed to be 1, 2 or 10. The plot confirms a weak dependence of Q on u. Depending on the index n, Q may depend on v strongly ([FORMULA]) or weakly ([FORMULA]), therefore the v -dependence provides a way to constrain the shape of the power spectrum.

[FIGURE] Fig. 1. The normalized three-point correlation function Q as a function of v, predicted by the second-order perturbation theory for scale-free power spectra with [FORMULA] (thin curves), 1.5 and 2 (thick curves). Solid lines are for [FORMULA], dashed lines for [FORMULA] and dot-dashed lines for [FORMULA].

In Fig. 2 we present our numerical results of Q for three realistic power spectra. The SCDM ([FORMULA]) and LCDM ([FORMULA]) spectra are taken from Bardeen et al. (1986), and the MDM spectrum is the power spectrum given by Klypin et al. (1993) for cold dark matter at [FORMULA] in the MDM model of [FORMULA]. As in the case of scale-free power spectra, the normalized function Q does not depend on the amplitude of the power spectrum. Here we are interested in the quasilinear regime (i.e. [FORMULA]) where the three-point correlation function can be estimated accurately both in observations and in N-body simulations and where the second-order approximation for [FORMULA] is likely to be valid according to previous simulation tests on [FORMULA]. In this regime, the free-streaming motion of neutrinos in the MDM model is not important and our treatment which considers only CDM is valid. The figure shows that the normalized three-point correlation function of SCDM is significantly different from those of LCDM and MDM. Even the LCDM and MDM models show sufficient difference in Q for some configurations (e.g. [FORMULA] and [FORMULA]), so that one can hope to distinguish between these two interesting models through analyzing the three-point correlation function for a large galaxy survey. The results can be easily understood with the results for scale-free power spectra. For the scales k we considered, the SCDM spectrum has the largest and the MDM the smallest effective index among the three spectra, therefore SCDM shows the strongest and MDM the weakest variation of Q with v.

[FIGURE] Fig. 2. The normalized three-point correlation function Q as a function of v, predicted by the second-order perturbation theory for realistic power spectra. Solid lines are for SCDM, dashed lines for LCDM and dot-dashed lines for MDM.
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© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998
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