The Analysis Of The Three-point Correlation Function From Simulations And Theory

In this thesis we focus on the properties of the bispectrum. We find that the two-dimensional density field has the same statistical properties as the three-dimensional field in Fourier space. With the 2D fast fourier transformation (FFT),therefore we can extend the analysis of the bispectrum into very small scales with little computer limitation. We measured the reduced bispectrum for four scale-free models of structure formations with spectral index of 1,0, -1, -2 and the cold dark matter models (ACDM and SCDM).From the scale-free N-body simulations, we have carefully examined possible effects on the power spectrum and bispectrum measurements of numerical artifacts, such as the finite box size, force softening, and particle discreteness with the properties of self-similar evolution. We found that the bispectrum dependes not only on spectral index but also on the shape and size of the k-triangle, even in the strong nonlinear regime. It increases with wavenumber and decreases with the spectral index. These results are in contrast with the hypothesis that the reduced bispectrum is a constant in the strongly non-linear regime. Comparing our measured non-linear bispectrum with the weakly non-linear bispectrum obtained from the second order PT (hereafter PT2), we have arrived at a new formula for the bispectrum that is significantly more accurate than the previous fitting formula of Scoccimarro et al..With the criteria about numerical artifacts found from the results of the scale-free simulations, we measured the bispectrum at different redshift for ACDM and SCDM from the simulations in a comoving frame. The results show that the reduced bispectrum dependes on the shape and size of the k -triangle, increases with wavenumber, decreases with redshift for a fixed wavenumber and shape of the triangle. With the bispectrum predicted by halo model, we can analyze the trend of the strong nonlinear bispectrum. With the bispectrum measured from simulations and predicted by halo model, we can improve the halo model.