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Register a new userMeasuring the galaxy power spectrum with multiresolution decomposition -- II. diagonal and off-diagonal power spectra of the LCRS galaxies
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The power spectrum estimator based on the discrete wavelet transform (DWT) for 3-dimensional samples has been studied. The DWT estimator for multi-dimensional samples provides two types of spectra with respect to diagonal and off-diagonal modes, which are very flexible to deal with configuration-related problems in the power spectrum detection. With simulation samples and mock catalogues of the Las Campanas redshift survey (LCRS), we show (1) the slice-like geometry of the LCRS doesnt affect the off-diagonal power spectrum with ``slice-like mode; (2) the Poisson sampling with the LCRS selection function doesnt cause more than 1-$sigma$ error in the DWT power spectrum; and (3) the powers of peculiar velocity fluctuations, which cause the redshift distortion, are approximately scale-independent. These results insure 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. It is less than 1-$sigma$ of the cosmic variance of mock samples 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 has been significantly measured. This indicates that the off-diagonal power spectra are capable of providing information about the power spectrum of galaxy velocity field.
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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 of both the scale and shape (configuration) of the decomposition modes, it is sensitive to distortion of 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 to detect the effect of redshift distortion. We first established the mapping between the DWT power spectra in redshift and real space. The mapping depends on the redshift distortion effects of (1) bulk velocity, (2) selection function and (3) pairwise peculiar velocity. We then proposed $beta$-estimators using the DWT off-diagonal power spectra. These $beta$-estimators are model-free even when the non-linear redshift distortion effect is not negligible. Moreover, these estimators do not rely on the assumption of whether the pairwise velocity dispersion being scale-dependent. The tests with N-body simulation samples show that the proposed $beta$-estimators can yield reliable measurements of $beta$ 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. The numerical test also shows that the real power spectrum can be well recovered from the redshift distorted power spectrum.
In this paper, we develop the method of analyzing the velocity field of cosmic matter with a multiresolution decomposition. This is necessary in calculating the redshift distortion of power spectrum in the discrete wavelet transform (DWT) representation. We show that, in the DWT analysis, the velocity field can be described by discrete variables, which are given by assignment of the number density and velocity into the DWT modes. These DWT variables are complete and not redundant. In this scheme, the peculiar velocity and pairwise velocity of galaxies or particles are given by field variables. As a consequence, the velocity dispersion (VD) and pairwise velocity dispersion (PVD) are no longer measured by number-counting or pair-counting statistic, but with the ensemble of the field variables, and therefore, they are free from the bias due to the number-counting and pair-counting. We analyzed the VD and PVD of the velocity fields given by the N-body simulation for models of the SCDM, $tau$CDM and $Lambda$CDM. The spectrum (scale-dependence) of the VD and PVD show that the length scale of the two-point correlation of the velocity field is as large as few tens h$^{-1}$ Mpc. Although the VD and PVD show similar behavior in some aspects, they are substantially different from each other. The VD-to-PVD ratio shows the difference between the scale-dependencies of the VD and PVD. More prominent difference between the VD and PVD is shown by probability distribution function. The one-point distribution of peculiar velocity is approximately exponential, while the pairwise velocitys is lognormal, i.e. of long tail. This difference indicates that the cosmic velocity field is typically intermittent.
We describe non-conventional localization of the midband E=0 state in square and cubic finite bipartite lattices with off-diagonal disorder by solving numerically the linear equations for the corresponding amplitudes. This state is shown to display multifractal fluctuations, having many sparse peaks, and by scaling the participation ratio we obtain its disorder-dependent fractal dimension $D_{2}$. A logarithmic average correlation function grows as $g(r) sim eta ln r$ at distance $r$ from the maximum amplitude and is consistent with a typical overall power-law decay $|psi(r)| sim r^{-eta}$ where $eta $ is proportional to the strength of off-diagonal disorder.
The large-scale structure of the Universe should soon be measured at high redshift during the Epoch of Reionization (EoR) through line-intensity mapping. A number of ongoing and planned surveys are using the 21 cm line to trace neutral hydrogen fluctuations in the intergalactic medium (IGM) during the EoR. These may be fruitfully combined with separate efforts to measure large-scale emission fluctuations from galactic lines such as [CII], CO, H-$alpha$, and Ly-$alpha$ during the same epoch. The large scale power spectrum of each line encodes important information about reionization, with the 21 cm power spectrum providing a relatively direct tracer of the ionization history. Here we show that the large scale 21 cm power spectrum can be extracted using only cross-power spectra between the 21 cm fluctuations and each of two separate line-intensity mapping data cubes. This technique is more robust to residual foregrounds than the usual 21 cm auto-power spectrum measurements and so can help in verifying auto-spectrum detections. We characterize the accuracy of this method using numerical simulations and find that the large-scale 21 cm power spectrum can be inferred to an accuracy of within 5% for most of the EoR, reaching 0.6% accuracy on a scale of $ksim0.1,text{Mpc}^{-1}$ at $left< x_i right> = 0.36$ ($z = 8.34$ in our model). An extension from two to $N$ additional lines would provide $N(N-1)/2$ cross-checks on the large-scale 21 cm power spectrum. This work strongly motivates redundant line-intensity mapping surveys probing the same cosmological volumes.
Our peculiar velocity with respect to the CMB rest frame is known to induce a large dipole in the CMB. However, the motion of an observer has also the effect of distorting the anisotropies at all scales, as shown by Challinor and Van Leeuwen (2002), due to aberration and Doppler effects. We propose to measure independently our local motion by using off-diagonal two-point correlation functions for high multipoles. We study the observability of the signal for temperature and polarization anisotropies. We point out that Planck can measure the velocity $beta$ with an error of about 30% and the direction with an error of about 20 degrees. This method constitutes a cross-check, which can be useful to verify that our CMB dipole is due mainly to our velocity or to disentangle the velocity from other possible intrinsic sources. Although in this paper we focus on our peculiar velocity, a similar effect would result also from other intrinsic vectorial distortion of the CMB which would induce a dipolar lensing. Measuring the off-diagonal correlation terms is therefore a test for a preferred direction on the CMB sky.
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