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

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3. N-body simulation test

We use a large set of N-body simulations to test the SEPT results presented in the last section. The simulations were generated with a [FORMULA] N-body code. A description of the code can be found in Jing & Fang (1994). Each simulation has [FORMULA] to [FORMULA] particles. To remove possible effects of a finite simulation volume, we used box sizes [FORMULA] or 400 [FORMULA]. For each of the three power spectra, we have run 5 to 6 simulations in order to reduce the cosmic variances. Table 1 lists model and simulation parameters for these simulations.


Table 1. Parameters used in the simulations

The three-point correlation functions for simulation particles are estimated by


where [FORMULA] is the count of triplets with shape [FORMULA] in the simulation, [FORMULA] is the count of triplets expected if these particles were randomly distributed and [FORMULA] is the expected count of triplets formed by two simulation particles and one random point (Peebles 1980). In this paper we will use the procedure of Jing et al. (1995) to estimate [FORMULA]. Briefly, we estimate the two-point correlation function for particles and fit it by an analytical formula. Then we calculate [FORMULA] and [FORMULA] analytically. With counting triplets, we get [FORMULA] through Eq. (11).

It is non-trivial to count triplets for more than [FORMULA] particles with modern computers. Here we use the linked-list technique (Hockney & Eastwood 1981) to search for triplets. We limit our analysis to triangles with the longest side [FORMULA] less than 0.1 box side length (i.e. [FORMULA]) for several reasons. First and most importantly, the main aim of this paper is to test the SEPT prediction for [FORMULA] in the quasilinear regime, i.e. [FORMULA]. Second, on scales larger than [FORMULA], the three-point correlation function is very weak and it is not easy to measure it. Furthermore, the finite-volume effect becomes more and more important (see discussion at the end of this section). Finally the computational time grows rapidly with [FORMULA] ([FORMULA]). However, even with the linked-list technique and with this choice of [FORMULA], it is still not feasible to count triplets for more than [FORMULA] particles. Therefore we randomly select 100,000 particles (still a very big number) from each simulation for our analysis.

Our N-body results of the three-point correlation function are presented in Fig. 3. For all three models, the normalized function Q increases with the parameter v, qualitatively in agreement with the SEPT prediction. Quantitatively, the degree of the agreement between the N-body result and the SEPT prediction seems to depend on the power spectrum. For the LCDM model the agreement looks best, and for the SCDM model the agreement is worst. The N-body results of the LCDM model agree quite well with the SEPT prediction except at [FORMULA] where the N-body result is significantly higher. The disagreement between the N-body result and the SEPT prediction is statistically significant ([FORMULA]) for all three models. In the MDM model, the N-body Q shows a stronger v -dependence than the SEPT prediction. On the contrary, the SCDM simulations show a weaker v -dependence than SEPT predicts. As a result, the difference in the v -dependence among the models is smaller in the N-body simulations than the prediction by SEPT. This point can be clearly seen in Fig. 3c where we compare the N-body results of the three models.

[FIGURE] Fig. 3a-c. The normalized three-point correlation function [FORMULA] estimated from the N-body simulations (symbols), compared with the predictions by the second-order perturbation theory (the solid lines). The error bars are estimated from the fluctuation among different realizations. a for LCDM; b for MDM; and c for SCDM. For comparison, we draw again the N-body results of LCDM (dotted lines) and MDM (dashed lines) on the SCDM plot.

Numerical artifacts of simulations and non-linear effects of evolution can both weaken the agreement between the simulation results and the perturbative predictions. Among many possible numerical artifacts, only the finite volume effect could have significantly influenced our simulation results since the scales we are interested in are above [FORMULA]. The finite volume effect is caused by the fact that the simulation cannot include density fluctuations on scales above the simulation box size. We can quantitatively estimate this effect on our results by calculating two SEPT values of Q. One is calculated as in Sect. 2, and the other is calculated with the lower integral limits of Eqs.(6-8) set to [FORMULA] the fundamental wavenumber of the simulation. For the triangle configurations in Fig. 3 and for the three power spectra studied in this paper, the difference between these two SEPT values is small (either the absolute difference is less than 0.1 or/and the relative difference is less 10%). Therefore, the finite volume effect on our results is negligible, and the non-linear effects must have influenced the three-point correlation function in the quasilinear regime ([FORMULA]).

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© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998