Multipole Vectors--a new representation of the CMB sky and evidence for statistical anisotropy or non-Gaussianity at 2<=l<=8

TL;DR

The paper introduces multipole vectors as a novel, rotation-covariant representation of each CMB multipole $f_{\ell}(\Omega)$, encoding the $a_{\ell m}$ information via $\ell$ headless unit vectors $\hat{v}^{(\ell,i)}$ and a scalar $A^{(\ell)}$. It provides both a practical recursion-based peeling algorithm and an alternative tensor-based derivation to compute these vectors from observed maps, and then develops four vector-based statistics to test statistical isotropy and Gaussianity against Monte Carlo isotropic-Gaussian skies. Applying these methods to WMAP full-sky maps (ILC and Tegmark-cleaned), the authors find a strong, scale-dependent signal in oriented-area correlations, notably a near-maximal cross-cross rank for $(\ell_1,\ell_2)=(3,8)$ with $p$-values around $10^{-4}$ to $10^{-2}$ depending on map, and a $2.6\sigma$ level indication from the oriented-area $Q$-statistic, implying violations of isotropy and/or Gaussianity beyond foreground/systematics in some tests. The results persist under plausible dust contamination and beam/noise considerations and are not solely explained by the well-known quadrupole-octupole alignment, suggesting that multipole vectors may reveal new, physically informative signatures in the CMB that warrant further investigation, including cut-sky analyses and higher-$\ell$ extensions.

Abstract

We propose a novel representation of cosmic microwave anisotropy maps, where each multipole order l is represented by l 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). However, they are related to the familiar spherical harmonic coefficients, alm, in a nonlinear way, and are therefore sensitive to different aspects of the CMB anisotropy. Nevertheless, it is straightforward to determine the multipole vectors for a given CMB map and we present an algorithm to compute them. Using the WMAP full-sky maps, we perform several tests of the hypothesis that the CMB anisotropy is statistically isotropic and Gaussian random. We find that the result from comparing the oriented area of planes defined by these vectors between multipole pairs 2<=l1!=l2<=8 is inconsistent with the isotropic Gaussian hypothesis at the 99.4% level for the ILC map and at 98.9% level for the cleaned map of Tegmark et al. A particular correlation is suggested between the l=3 and l=8 multipoles, as well as several other pairs. This effect is entirely different from the now familiar planarity and alignment of the quadrupole and octupole: while the aforementioned is fairly unlikely, the multipole vectors indicate correlations not expected in Gaussian random skies that make them unusually likely. The result persists after accounting for pixel noise and after assuming a residual 10% dust contamination in the cleaned WMAP map. While the definitive analysis of these results will require more work, we hope that multipole vectors will become a valuable tool for various cosmological tests, in particular those of cosmic isotropy.

Figures (5)

• Figure 1: An image of the sky as decomposed into the $\ell=2$ through $8$ multipole moments based on the first year WMAP results wmap_results as cleaned by Tegmark et al.tegmark_cleaned. Shown are the $\sum_m a_{\ell m} Y_{\ell m}$ and the vectors calculated for these multipoles. The vectors are drawn as "sticks" since they only defined up to a sign (thus they are "headless" vectors). See http://www.phys.cwru.edu/projects/mpvectors/ for a full sized color picture.
• Figure 2: The accuracy in $\theta$ (filled circles) and $\phi$ (empty squares) of a chosen multipole vector as a function of noise added to the $a_{\ell m}$. Both the mean value and scatter in the shift of the angles are shown. The $x$-axis value of unity corresponds to the scatter in the $a_{\ell m}$ due to WMAP V-band pixel noise.
• Figure 3: A flowchart of the algorithm we apply in order to extract the likelihood of the statistic, $S$. The lower case bold faced letters, such as (a), refer to the itemized points in section V B where more information can be found about the step. Also, the (V C) and (VI) refer to the sections in the paper where the details of these boxes can be found.
• Figure 4: Ranks of the oriented area statistics for the Tegmark full-sky map. The mean values correspond to the actual extracted $a_{\ell m}$, while the error bars were obtained by adding the pixel noise (the error bars are not necessarily symmetric around the no-error values). Note that an unusually large fraction of the ranks are high. Also note the low value of $\ell_1=2$, $\ell_2=3$ rank which is due to the alignment of the quadrupole and octupole which has been noted earlier tegmark_cleaned.
• Figure 5: The statistic $Q$ for the oriented area statistic, computed from the Tegmark et al. cleaned WMAP map, is shown by vertical line. The shaded region around it corresponds to the uncertainty due to pixel noise, while the histogram shows the distribution of the statistic for MC-generated Gaussian maps. Only $1.07\%$ of MC values of $Q$ are smaller than the no-error value of the WMAP $Q$. The same fraction for the ILC map (not shown here) is $0.38\%$.