Testing for Non-Gaussianity in the Wilkinson Microwave Anisotropy Probe Data: Minkowski Functionals and the Length of the Skeleton

TL;DR

This study uses morphology-based statistics—three Minkowski functionals and a newly defined skeleton length—to test Gaussianity in the WMAP first-year CMB data. By calibrating against 5000 Gaussian simulations with a running-index power spectrum and WMAP-like beam/noise, the authors examine area, boundary length, genus, and skeleton length across multiple smoothing scales and Galactic hemispheres. They find strong evidence for non-Gaussian behavior and a power asymmetry between the northern and southern hemispheres, most notably a large northern-genus amplitude and a high spectral parameter $\gamma$ around $\sim 3\!\deg$ scales, while full-sky statistics largely agree with Gaussian expectations. The results are robust to foreground treatment and point sources on the relevant scales, implying that deviations are not readily explained by systematics, and highlighting the potential need for primordial non-Gaussianity or other new physics. The work emphasizes the importance of multi-faceted, topology- and geometry-based tests for CMB data and sets the stage for future, higher-precision measurements such as those from the Planck mission.

Abstract

The three Minkowski functionals and the recently defined length of the skeleton are estimated for the co-added first-year Wilkinson Microwave Anisotropy Probe (WMAP) data and compared with 5000 Monte Carlo simulations, based on Gaussian fluctuations with the a-priori best-fit running-index power spectrum and WMAP-like beam and noise properties. Several power spectrum-dependent quantities, such as the number of stationary points, the total length of the skeleton, and a spectral parameter, gamma, are also estimated. While the area and length Minkowski functionals and the length of the skeleton show no evidence for departures from the Gaussian hypothesis, the northern hemisphere genus has a chi^2 that is large at the 95% level for all scales. For the particular smoothing scale of 3.40 degrees FWHM it is larger than that found in 99.5% of the simulations. In addition, the WMAP genus for negative thresholds in the northern hemisphere has an amplitude that is larger than in the simulations with a significance of more than 3 sigma. On the smallest angular scales considered, the number of extrema in the WMAP data is high at the 3 sigma level. However, this can probably be attributed to the effect of point sources. Finally, the spectral parameter gamma is high at the 99% level in the northern Galactic hemisphere, while perfectly acceptable in the southern hemisphere. The results provide strong evidence for the presence of both non-Gaussian behavior and an unexpected power asymmetry between the northern and southern hemispheres in the WMAP data.

Figures (10)

• Figure 1: Tracing contour lines and localizing stationary points by means of linear interpolation. (a) Temperature values $T_{ij}$ are known only at the centers of the HEALPix pixels, and linear interpolation is therefore used to approximate the true contour line. The true contour line is shown as a dashed curve in this figure, and the linear approximation as solid line segments. In this process it is useful to construct a set of secondary pixels defined by letting the centers of the HEALPix pixels define their vertices. In order to trace the contour line, one then has to check all such secondary pixels, searching for edges for which the two vertices have different signs relative to the contour line. Once such an edge has been located, the point of intersection (marked as $p_i$ in the figure) is approximated by linear interpolation, according to Equation (\ref{['eq:crossing']}). (b) Localizing stationary points is similar to tracing the contour lines. In this case one focuses on the zero-contours of the first derivative maps and searches for secondary pixels in which both derivatives are zero. Once such a pixel is found, the position of intersection is approximated by linear interpolation, i.e., by solving Equation (\ref{['eq:localizing_extrema']}) for $s$ and $t$. If $0<s,t<1$, then the point of intersection lies within the secondary pixel under consideration.
• Figure 2: A $10^{\circ} \times 10^{\circ}$ patch from the CMB sky as measured by WMAP, with (a) $dT/d\theta = 0$ marked in blue and $dT/d\phi = 0$ marked in red, and (b) the skeleton of the field marked in yellow. In both figures red filled circles indicate maxima, blue filled circles indicate minima and green crosses indicate saddle points.
• Figure 3: Masks used in the computations of the Minkowski functionals and the skeleton for FWHMs between $0\fdg53$ and $4\fdg26$. The masked regions at high latitudes are extended "semi-point" sources excluded by the Kp0 mask.
• Figure 4: Minkowski functionals and differential skeleton measured from the Kp0 sky, smoothed with a $1\fdg28$ FWHM Gaussian beam. Gray bands indicate 1, 2, and $3\,\sigma$ bands, and the solid line indicates the ensemble average. Filled circles show the observed functions.
• Figure 5: Results from the Minkowski functional and skeleton measurements, with the median subtracted from each bin. Without such a subtraction it is not possible to distinguish between a function that follows the median and one that lies in the $2\,\sigma$ region, as a result of the extremely narrow confidence bands -- usually only a few percent wide. The gray bands indicate 1 and $2\,\sigma$ confidence regions as computed from the simulations.