Multipole vectors of completely random microwave skies for l≤50

Probability Density of the Multipole Vectors for a Gaussian Cosmic Microwave Background

We review Maxwell’s multipole vectors, and elucidate some of their mathematical properties, with emphasis on the application of this tool to the cosmic microwave background (CMB). In particular, for a completely random function on the sphere (corresponding to the statistically isotropic Gaussian model of the CMB), we derive the full probability density function of the multipole vectors. This fu...

Autoregressive Gaussian Random Vectors of First Order

Harmonic projection and multipole Vectors

We show that the multipole vector decomposition, recently introduced by Copi et al., is a consequence of Sylvester’s theorem, and corresponds to the Maxwell representation. Analyzing it in terms of harmonic polynomials, we show that this decomposition results in fact from the application of the harmonic projection operator and its inverse. We derive the coefficients of the usual harmonic decomp...

Estimation of Spectral Measure of Stable Random Vectors

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Multipole vectors: A new representation of the CMB sky and evidence for statistical anisotropy or non-Gaussianity at 2ÏøÏ8

We propose a novel representation of cosmic microwave anisotropy maps, where each multipole order , is represented by , unit vectors pointing in directions on the sky and an overall magnitude. These ‘‘multipole vectors and scalars’’ transform as vectors under rotations. Like the usual spherical harmonics, multipole vectors form an irreducible representation of the proper rotation group SO(3). H...