Phase Information and the Evolution of Cosmological Density Perturbations

The Fourier transform of cosmological density perturbations can be represented in terms of amplitudes and phases for each Fourier mode. We investigate the phase evolution of these modes using a mixture of analytical and numerical techniques. Using a toy model of one-dimensional perturbations evolving under the Zel'dovich approximation as an initial motivation, we develop a statistic that quantifies the information content of the distribution of phases. Using numerical simulations beginning with more realistic Gaussian random-phase initial conditions, we show that the information content of the phases grows from zero in the initial conditions, first slowly and then rapidly when structures become non-linear. This growth of phase information can be expressed in terms of an effective entropy: Gaussian initial conditions are a maximum entropy realisation of the initial power spectrum, gravitational evolution decreases the phase entropy. We show that our definition of phase entropy results in a statistic that explicitly quantifies the information stored in the phases of density perturbations (rather than their amplitudes) and that this statistic displays interesting scaling behaviour for self-similar initial conditions.Comment: Accepted for publication in MNRAS with added comments on future work. For high-resolution Figure 1, or postscript file, please see http://www-star.qmw.ac.uk/~lyc

Non-linearity and Non-Gaussianity through Phase Information

In the standard picture of structure formation, initially random-phase fluctuations are amplified by non-linear gravitational instability to produce a final distribution of mass which is highly non-Gaussian and has highly coupled Fourier phases. Second-order statistics, such as the power spectrum, are blind to this kind of phase association. We discuss the information contained in the phases of cosmological density fluctuations and their possible use in statistical analysis tools. In particular, we show how the bispectrum measures a particular form of phase association called quadratic phase coupling, show how to visualise phase association using colour models. These techniques offer the prospect of more complete tests of initial non-Gaussianity than those available at present.Comment: 7 pages, 1 figure (two parts). To appear in the proceedings of The MPA/ESO/MPE Joint Astronomy Conference "Mining the Sky" held in Garching, Germany, July 31 - August 4 2000. To be published in the Springer-Verlag series "ESO Astrophysics Symposia

The Importance of Fourier Phases for the Morphology of Gravitational Clustering

The phases of the Fourier modes appearing in a plane-wave expansion of cosmological density fields play a vital role in determining the morphology of gravitationally-developed clustering. We demonstrate this qualitatively and quantitatively using simulations. In particular, we use cross-correlation and rank-correlation techniques to quantify the agreement between a simulated distribution and phase-only reconstructions. The phase-only reconstructions exhibit a high degree of correlation with the original distributions, showing how meaningful spatial reconstruction of cosmological density fields depends more on phase accuracy than on amplitudes.Comment: 5 pages, 5 figures (with 2 added figures), accepted for publication in MNRA

Cross-Power Spectrum and Its Application on Window Functions in the WMAP data

Cross-power spectrum is a quadratic estimator between two maps that can provide unbiased estimate of the underlying power spectrum of the correlated signals, which is therefore used for extracting the power spectrum in the WMAP data. In this paper we discuss the limit of cross-power spectrum and derive the residual from uncorrelated signal, which is the source of error in power spectrum extraction. We employ the estimator to extract window functions by crossing pairs of extragalactic point sources. We desmonstrate its usefulness in WMAP Difference Assembly maps where the window functions are measured via Jupiter and then extract the window functions of the 5 WMAP frequency band maps.Comment: added the part of applying cross power spectrum on WMAP DA maps and frequency band maps and submitted to Ap