Cosmological data sets are usually analysed in terms of their harmonics, i.e. Fourier plane waves for three-dimensional galaxy data or spherical harmonics for data projected on the sky (especially the cosmic microwave background). In this talk One of the reasons such an approach is adopted is that the "Gaussian" initial conditions for structure formation are generally supposed to involve these fluctuations having random-phase harmonic modes.
In this talk I begin with a brief discussion of phase evolution in the Fourier modes of density fluctuations in order to describe the relationship between phases and more familiar descriptors such as the bispectrum. I then describe various methods of studing the statistical properties of spherical harmonic modes of temperature maps of the cosmic microwave background.
Unlike other studies, which focus mainly on properties of the amplitudes of these modes, I look instead at their phases. In particular, I present a simple measure of phase correlation that can be diagnostic of departures from the standard assumption of Gaussian initial conditions. The method I discuss checks for the uniformity of the distribution of phase angles using a non-parametric descriptor known as Kuiper's statistic. The particular advantage of this method is that, when coupled to the judicious use of Monte Carlo simulations, it can deliver very interesting results from small data samples. In particular, it is useful for studying the properties of spherical harmonics at low l for which there are only small numbers of independent values of m and which therefore furnish only a small number of phases for analysis.I describe an application of the method to the COBE-DMR and WMAP sky maps, and find departures from uniformity in both. In the case of WMAP, our results probably reflect Galactic contamination or the known variation of signal-to-noise across the sky rather than primordial non-Gaussianity.
I also describe how correlations in the phases of the spherical harmonic coefficients of the CMB temperature pattern are associated with matched pairs of circles seen in the sky in universes with non-trivial topology. Unlike other statistics developed to hunt for these matched circles, the method is computationally fast. I describe the result of applying our phase method to a range of simulations with topologies of various forms and on different scales. A characteristic form of phase correlation is found in the simulations, but no evidence of similar effects is found in the WMAP data.