Application Of DWT Method In The Large Scale Structure And Gravitational Lensing

This thesis mainly focuses on some of our studies on the statistical study of the large-scale structure in the universe and gravitational lensing, which are of the most active issues in the modern astrophysics. The studies on them are very important for understanding the matter distribution of the universe and the evolution of the Universe. This thesis consists of six chapters.First, we introduce some background knowledge of the large scale structure in the universe in Chapter 1. We briefly discuss the Standard Cosmology model, structure formation theories, the statistical analysis and the observations and numerical simulations of large scale structure.In Chapter 2, we introduce the basic theory of the gravitational lensing. We discuss the basis equations of gravitational lensing and its application in astrophysics. We also introduce the numerical methods in gravitational lensing, including how to compute the surface density, lensing potential, deflection angel, critical curves and caustics of lens.In Chapter 3, we introduce the application of wavelet methods in astrophysics, including the multi-resolution analysis of density fields and the wavelet denoising method based on DWT(discrete wavelet transform).In Chapter 4, we develop a new smoothing algorithm for computing surface densities from 3D numerical simulation samples. This algorithm is based on MRA (Multiresolution Analysis) wavelet method. We test the algorithm by applying it to two gravitational lensing simulation samples which are generated by monte-carlo method and have different mass resolution, the results show our algorithm can successfully reconstruct the surface density of lens. We also compute the critical curves and caustics of lens samples, the results show that they can fit the theoretic curves very well. We test three different wavelet bases and compare them, including Daub4, Daub6 and B-spline 3th. Our algorithm is very fast and is suitable for high resolution N-body simulations.The essential step in statistical analysis of a large data set in Fourier space using FFT is to make assignment onto mesh. It has been noted such a procedure brings up thealiasing effect in Fourier space. Furthermore, the resulting power spectrum estimationmay be biased from the true power spectrum. Actually, the power spectrum measuredwith FFT is a convolution of the true power spectrum and window function specified bymass assignment, In Chapter 5, based on the Beylkin's unequally spaced Fast FourierTransformation technique, we introduce a new precision method for extracting the truepower spectrum in a large data set. Numerically, we compare the traditional massassignment schemes with the new method using Daub6 and 3rd-order B-spline scalingfunctions. The results show that B-splines function could be an optimal choice formass assignment in sense that (1) it has a compact support in real space and thus yieldsan efficient algorithm (2) without any extra corrections, it leads to accurate recovery ofthe true power spectrum with errors less than 5% at k≤k_N. It is hopeful that sucha method could be applied to higher order statistics in Fourier space and enable us tohave a precision modeling of the non-Gaussian features in the large scale structure ofthe universe.Finally, the summary and future discussions are presented in Chapter 6.