Random-Walk Statistics and the Spherical Harmonic Representation of CMB Maps

Abstract

We investigate the properties of the (complex) coefficients obtained in a spherical harmonic representation of temperature maps of the cosmic microwave background (CMB). We study the effect of the coefficient phase only, as well as the combined effects of phase and amplitude. The method used to check for anomalies is to construct a ``random walk'' trajectory in the complex plane where the step length and direction are given by the amplitude and phase (respectively) of the harmonic coefficient. If the fluctuations comprise a homogeneous and isotropic Gaussian random field on the sky, the path so obtained should be a classical ``Rayleigh flight'' with very well known statistical properties. We illustrate the use of this random-walk representation by using the net walk length as a test statistic, and apply the method to the coefficients obtained from a Wilkinson Microwave Anisotropy Probe (WMAP) preliminary sky temperature map.

5 pages, 1 figure, uses mn.sty. Version accepted for publication in MNRAS. Includes toy models of non-random walks

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• Testing Global Isotropy of Three-Year Wilkinson Microwave Anisotropy Probe (WMAP) Data: Temperature Analysis
• Testing Gaussian random hypothesis with the cosmic microwave background temperature anisotropies in the three-year WMAP data
• Measuring Statistical Isotropy of CMB Anisotropy
• Unveiling Hidden Patterns in CMB Anisotropy Maps
• A new family of solvable Pearson-Dirichlet random walks
• Statistical Characterization of Temperature Patterns in Anisotropic Cosmologies
• Phase analysis of the cosmic microwave background from an incomplete sky coverage
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• Local properties of the large-scale peaks of the CMB temperature
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