Published on October 15, 2007
Slide1: Abstract Cosmological data analysis  Wavelets on the sphere  Conclusions Development of scale-space signal processing tools on the sphere and application to cosmology Wavelets on the sphere and filter steerability Fast directional correlation algorithm Analysis of the Cosmic Microwave Background (CMB) radiation Scale-Space Signal Processing Scale-space signal processing: The high resolution analysis of signals on the sphere is rendered affordable in scales, positions, and local orientations Fast Algorithm  Correspondence Principle Definition: any rotation of around itself by an angle may be expressed as a linear combination of a finite number M of basis filters m. By linearity of the transform, this is valid for wavelet coefficients as well The wavelet transform is defined as the directional correlation between a signal F(), with = (,), and a wavelet dilated by a>0, rotated on itself by [0, 2 ], and translated on the sphere at the point 0 S2: The wavelet coefficients characterize the signal for each analysis scale a, local orientation and position 0 Remark: the wavelet must satisfy an admissibility condition which ensures the reconstruction of the signal from its coefficients The inverse stereographic projection -1 of a wavelet on the plane automatically leads to a wavelet on the sphere This Correspondence Principle allows to transfer properties of wavelets on the plane, such as the notion of steerability Steerable Wavelets The priori asymptotic complexity of directional correlation is of order O(L5) for signals of band-limit L (maps of O(L2) pixels) Fast Directional correlation with steerable filters The filter steerability and the separation of variables technique for spherical harmonics allows to achieve an O(L3) algorithm independently of the pixelization For each analysis scale a, the computation of the wavelet coefficients of mega-pixels maps (L 103) is typically reduced from years to tens of seconds on a single standard computer Remark: A sampling theorem allows to achieve the exactness of the algorithm on equi-angular pixelizations Questioning the Cosmological Principle CMB Data and Simulations Analysis with Steerable Wavelets Results: Total Weights and Anomalies The second derivative of a Gaussian on the sphere is steerable (M=3) x y xy = /4 We question the global universe isotropy by probing the alignment of local structures in the CMB radiation on the celestial sphere The experimental mega-pixels temperature map used for the analysis is obtained from the data of the ongoing WMAP experiment For each pixel 0 outside the exclusion mask, the local orientation 0(0) for which the wavelet coefficient is maximum is selected 12 scales are selected between 5° and 30° for the analysis wavelet. At each scale, an exclusion mask is used to avoid considering pixels contaminated by foreground emissions 10000 simulations of an isotropic universe are produced to define confidence levels for the analysis results The directions of the sky lying on the great circle defined by the selected local orientation are considered to be highlighted by the local structure identified, and are weighted by the wavelet coefficient Summing the great circles corresponding to all the pixels of the original signal gives a map of total weights, which depicts the anisotropy distribution on the sky Mollweide projection maps of the total weights (left) and anomalies (right) at a scale around 8°, normalized by the corresponding standard deviation of the simulations. A rare detection is observed at a scale around 8°. It identifies 20 directions qualified as anomalous at 99.99%, with an associated positive total weight higher than in any of the 10000 simulations. The analysis of the WMAP data identifies a mean preferred plane with a normal direction close to the CMB dipole axis, and a mean preferred direction in this plane, very close to the ecliptic poles axis Application to cosmology: A synthesis of CMB statistical anisotropy studies is achieved. Further analyses are required to establish the systematics, foreground, or cosmological origin of the anisotropy References:  Y. Wiaux, L. Jacques, and P. Vandergheynst, Astrophys. J. 632, 15 (2005).  Y. Wiaux , L. Jacques, P. Vielva, and P. Vandergheynst, preprint astro-ph/0508516.  Y. Wiaux, P. Vielva, E. Martinez-Gonzalez, and P. Vandergheynst, Phys. Rev. Lett. 96 151303 (2006).