Measuring the Galaxy Power Spectrum with Multiresolution Decomposition. IV. Redshift Distortion

Abstract

In this paper, we develop a theory of redshift distortion of the galaxy power spectrum in the discrete wavelet transform (DWT) representation. Because the DWT power spectrum is dependent on both the scale and shape (configuration) of the decomposition modes, it is sensitive to distortion of the shape of the field. On the other hand, the redshift distortion causes a shape distortion of distributions in real space with respect to redshift space. Therefore, the shape-dependent DWT power spectrum is useful for detecting the effect of redshift distortion. We first established the mapping between the DWT power spectra in redshift and real space. The mapping depended on the redshift-distortion effects of (1) bulk velocity, (2) selection function, and (3) pairwise peculiar velocity. We then proposed β estimators using the DWT off-diagonal power spectra. These β estimators are model-free, even when the nonlinear redshift-distortion effect is not negligible. Moreover, these estimators do not rely on the assumption of whether the pairwise velocity dispersion is scale dependent. Tests with N-body simulation samples show that the proposed β estimators can yield reliable measurements of β with about 20% uncertainty for all popular dark matter models. We also develop an algorithm for reconstruction of the power spectrum in real space from the redshift-distorted power spectrum. Numerical tests also show that the real power spectrum can be well recovered from the redshift-distorted power spectrum.