X iv : a st ro - p h / 02 11 15 9 v 1 8 N ov 2 00 2 One - point Statistics of the Cosmic Density Field in Real and Redshift Spaces with A Multiresolutional Decomposition

In this paper, we develop a method of performing the one-point statistics of a perturbed density field with a multiresolutional decomposition based on the discrete wavelet transform (DWT). We establish the algorithm of the one-point variable and its moments in considering the effects of Poisson sampling and selection function. We also establish the mapping between the DWT one-point statistics in redshift space and real space, i.e. the algorithm for recovering the DWT one-point statistics from the redshift distortion of bulk velocity, velocity dispersion, and selection function. Numerical tests on N-body simulation samples show that this algorithm works well on scales from a few hundreds to a few h −1 Mpc for four popular cold dark matter models. Taking the advantage that the DWT one-point variable is dependent on both the scale and the shape (configuration) of decomposition modes, one can design estimators of the redshift distortion parameter (β) from combinations of DWT modes. Comparing with conventional β estimators, such as quadrupole-to-monopole ratio, the DWT β estimators are scale-decomposed. It is useful to consider scale-dependent effects. When the non-linear redshift distortion is not negligible, the quadrupole-to-monopole ratio is a function of scale. This esti-mator would not work without adding information about the scale-dependence, such as the power-spectrum index or the real-space correlation function of the random field. The DWT β estimators, however, do not need such extra information. Moreover, the scale-decomposed β estimators would also be able to reveal the scale-dependence of the bias parameter of galaxies. Numerical tests show that the proposed DWT estimators are able to determine β robustly with less than 15% uncertainty in the redshift range 0 ≤ z ≤ 3. Subject headings: cosmology: theory-large-scale structure of the universe – 2 –