2. Prediction of the perturbation theory
In the second-order Eulerian perturbation theory, the bispectrum , which is the Fourier transform of the three-point correlation correlation function, is predicted to be (Fry 1984)
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 (Bouchet et al. 1992, Catelan et al. 1995), so we will apply this equation to the low-density flat model with as well (Sect. 3).
The three-point correlation function is then
After some tedious calculation, we can write as
where , is the two-point correlation function
and and (l is 0 or 2) are respectively
For a power-law power spectrum , from the above expressions, one can easily get the result of Fry (1984), i.e.
where is the slope of [ ]. 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 for scale-free power spectra are very instructive for understanding the results of for realistic power spectra, therefore we first discuss 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 , r, u, and v are defined as:
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 . 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 , 1.5 and 2 (or equivalently , -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 () or weakly (), therefore the v -dependence provides a way to constrain the shape of the power spectrum.
In Fig. 2 we present our numerical results of Q for three realistic power spectra. The SCDM () and LCDM () 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 in the MDM model of . 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. ) 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 is likely to be valid according to previous simulation tests on . 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. and ), 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.
© European Southern Observatory (ESO) 1997
Online publication: July 3, 1998