Discrete Wavelet Transform Power Spectrum Research Articles

DWT Power Spectrum of the Two-Degree Field Galaxy Redshift Survey

The power spectrum of the two-degree Field Galaxy Redshift Survey (2dFGRS) sample is estimated with the discrete wavelet transform (DWT) method. The DWT power spectra within 0.035 < k < 2.2 h Mpc(-1) are measured for three volume-limited samples defined in consecutive absolute magnitude bins -19 similar to -18, -20 similar to -19 and -21 similar to -20. We show that the DWT power spectrum can effectively distinguish Lambda CDM models of sigma(8) = 0.84 and sigma(8) = 0.74. We adopt maximum likelihood method to perform three-parameter fitting of the bias parameter b, pairwise velocity dispersion sigma(pv) and redshift distortion parameter beta = Omega(0.6)(m)/b to the measured DWT power spectrum. The fitting results state that in a sigma(8) = 0.84 universe the best-fit values of Omega(m) given by the three samples are mutually consistent within the range 0.28 similar to 0.36, and the best fitted values of sigma(pv) are 398(-27)(+35), 475(-29)(+37) and 550 +/- 20km s(-1) for the three samples, respectively. In the model of sigma(8) = 0.74, our three samples give very different values of Omega(m). We repeated the fitting using the empirical formula of redshift distortion. The result of the model of low sigma(8) is still poor, especially, one of the best-fit values of sigma(pv) is as large as 10(3) km s(-1). We also repeated our fitting by incorporating a scale-dependent galaxy bias. This gave a slightly lower value of Omega(m). Differences between the models of sigma(8) = 0.84 and sigma(8) = 0.74 still exist in the fitting results. The power spectrum of 2dFGRS seems to disfavor models with low amplitude of density fluctuations if the bias parameter is assumed to be scale independent. For the fitting value of Omega(m) to be consistent with that given by WMAP3, strong scale dependence of the bias parameters is needed.

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

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.

Measuring the Galaxy Power Spectrum with Multiresolution Decomposition. II. Diagonal and Off‐Diagonal Power Spectra of the Las Campanas Redshift Survey Galaxies

We study the power spectrum estimator based on the discrete wavelet transform (DWT) for three-dimensional samples. The DWT estimator for higher than one-dimensional samples provides two types of spectra with respect to diagonal and off-diagonal modes, respectively. The two types of modes have different spatial invariance, and therefore the diagonal and off-diagonal DWT power spectra are very flexible for dealing with configuration-related problems in power spectrum detection. With simulation samples and the mock catalogs of the Las Campanas Redshift Survey (LCRS), we show that (1) the geometry of the LCRS does not affect the off-diagonal power spectrum with a slicelike mode; (2) the Poisson sampling with the LCRS selection function does not cause more than 1 sigma error in the DWT power spectrum; and (3) the powers of the mass or galaxy random velocity fluctuations, which cause the redshift distortion, are approximately scale-independent. These results ensure that the uncertainties of the power spectrum measurement are under control. The scatter of the DWT power spectra of the six strips of the LCRS survey is found to be rather small, less than 1 sigma of the cosmic variance of the mock sample in the wavenumber range 0.1< k<2 h Mpc(-1). To fit the detected LCRS diagonal DWT power spectrum with CDM models, we find that the best-fitting redshift distortion parameter beta is about the same as that obtained from the Fourier power spectrum. The velocity dispersions sigma (v) for SCDM and Lambda CDM models are also consistent with other sigma (v) detections with the LCRS. A systematic difference between the best-fitting parameters of diagonal and off-diagonal power spectra have been significantly measured. This indicates that the off-diagonal power spectra are capable of providing information about the power spectrum of the galaxy velocity field.

The Local Power Spectrum and Correlation Hierarchy of the Cosmic Mass Field

We analyze the power spectrum of a QSO's Lyα-transmitted flux in the discrete wavelet transform (DWT) representation. Although the mean DWT power spectrum is consistent with its counterpart in Fourier representation, the spatial distribution of the local power varies greatly; i.e., the local DWT power spectra show remarkably spiky structures on small scales. To measure these spiky features, we introduce the quantities roughness of the local power spectrum and correlation between spikes on different scales. We then test the predictions made by the correlation hierarchy model on the roughness and the scale-scale correlations of the local power spectrum. Using the Lyα-transmitted flux of the QSO HS 1700, we find that the underlying cosmic mass field of the transmitted flux at redshift around z 2.2 can be described by the hierarchical clustering model on physical scales from 2.5 h-1 Mpc to a few tens h-1 kpc in an Einstein-de Sitter universe. However, the nonlinear features of the clustering show differences on different scale ranges: (1) On physical scales larger than ~1.3 h-1 Mpc, the field is almost Gaussian. (2) On scales 1.3-0.3 h-1 Mpc, the field is consistent with the correlation hierarchy with a constant value for the coefficient Q4. (3) On scales less than 300 h-1 kpc, the field is no longer Gaussian but essentially intermittent. In this case, the field can still be fitted by the correlation hierarchy, but the coefficient, Q4, should be scale dependent. These three points are strongly supported by the following result: the scale dependencies of Q4 given by two statistically independent measures, i.e., Q by roughness and Q by scale-scale correlation, are the same in the entire scale range considered.

Non‐Gaussianity and the Recovery of the Mass Power Spectrum from the Lyα Forest

We investigate the eUect of non-Gaussianity on the reconstruction of the initial mass —eld from the Lya forest. We show that the transmitted —ux of QSO absorption spectra are highly non-Gaussian in terms of the statistics, the kurtosis spectrum, and the scale-scale correlation. These non-Gaussianities cannot be completely removed by the conventional algorithm of Gaussianization, and the scale-scale correlations are largely retained in the mass —eld recovered by the Gaussian mapping. Therefore, the mass power spectrum recovered by the conventional algorithm is systematically lower than the initial mass spectrum on scales at which the local scale-scale correlation is substantial. To reduce the non- Gaussian contamination, we present two methods. The —rst is to perform the Gaussianization scale-by- scale using the discrete wavelet transform (DWT) decomposition. We show that the non-Gaussian features of the Lya forest basically will no longer exist in the scale-by-scale Gaussianized mass —eld. The second method is to choose a proper orthonormal basis (representation) to suppress the eUect of the non-Gaussian correlations. In the quasi-linear regime of cosmic structure formation, the DWT power spectrum is efficient for suppressing the non-Gaussian contamination. These two methods signi—cantly improve the recovery of the mass power spectrum from the Lya forest. Subject headings: cosmology: theorydark matterlarge-scale structure of universe ¨ quasars: absorption lines