Measuring the galaxy power spectrum and scale-scale correlations with multiresolution-decomposed covariance. I. Method

We present a method for measuring the galaxy power spectrum based on multir esolution analysis of the discrete wavelet transformation (DWT). Apart from the technical advantages of the computational feasibility for data sets wi th a large volume and complex geometry, the DWT scale-by-scale decompositio n provides a physical insight into the covariance matrix of the cosmic mass held. 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 cosmogon y 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, whi ch is equal to the corresponding Fourier power spectrum banded with a logar ithmical scaling. This DWT estimator is optimized in the sense that the spa tial resolution is automatically adaptive to the perturbation wavelength to be studied. In the scale range in which the scale-scale correlation is sig nificant, the accuracy of a power spectrum detection depends on the scale-s cale or band-band correlations. In this case, for a precision measurements of the power spectrum, or a precision comparison of the observed power spec trum with models, a measurement of the scale-scale or band-band correlation s is needed. We show that the DWT covariance can be employed to measure bot h the band-power spectrum and second-order scale-scale correlation. We also present the DWT algorithm of the binning and Poisson sampling with real ob servational data. We show that the so-called alias effect appeared in usual binning schemes can exactly be eliminated by the DWT binning. Since the Po isson process possesses diagonal covariance in the DWT representation, the Poisson sampling and selection effects on the power spectrum and second ord er scale-scale correlation detection are suppressed into a minimum. Moreove r, 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 fur ther DWT algorithms on the basic decomposition in order to estimate other e ffects on the power spectrum detection, such as non-Gaussian correlations a nd bias models.