Measuring the Galaxy Power Spectrum and Scale‐Scale Correlations with Multiresolution‐decomposed Covariance. I. Method

Abstract

We present a method for measuring the galaxy power spectrum based on multiresolution analysis of the discrete wavelet transformation (DWT). Apart from the technical advantages of the computational feasibility for data sets with a large volume and complex geometry, the DWT scale-by-scale decomposition provides a physical insight into the covariance matrix of the cosmic mass field. Since the DWT representation has a strong capability for suppressing the off-diagonal components of the covariance for self-similar clustering, the DWT covariance for all popular models of the cold dark matter cosmogony is generally diagonal, or j (scale) diagonal in the scale range in which the second or higher order scale-scale correlations are weak. In this range, the DWT covariance gives a lossless estimation of the power spectrum, which is equal to the corresponding Fourier power spectrum banded with a logarithmical scaling. This DWT estimator is optimized in the sense that the spatial resolution is automatically adaptive to the perturbation wavelength to be studied. In the scale range in which the scale-scale correlation is significant, the accuracy of a power spectrum detection depends on the scale-scale or band-band correlations. In this case, for a precision measurements of the power spectrum, or a precision comparison of the observed power spectrum with models, a measurement of the scale-scale or band-band correlations is needed. We show that the DWT covariance can be employed to measure both the band-power spectrum and second-order scale-scale correlation. We also present the DWT algorithm of the binning and Poisson sampling with real observational data. We show that the so-called alias effect appeared in usual binning schemes can exactly be eliminated by the DWT binning. Since the Poisson process possesses diagonal covariance in the DWT representation, the Poisson sampling and selection effects on the power spectrum and second order scale-scale correlation detection are suppressed into a minimum. Moreover, the effect of the non-Gaussian features of the Poisson sampling can also be calculated in this frame. The DWT method is open, i.e., one can add further DWT algorithms on the basic decomposition in order to estimate other effects on the power spectrum detection, such as non-Gaussian correlations and bias models.