Testing the cosmological principle of isotropy: local power spectrum estimates of the WMAP data

We apply the Gabor transform methodology proposed in (Hansen et al. 2002, 2003) to the WMAP data in order to test the statistical properties of the CMB fluctuation field and specifically to evaluate the fundamental assumption of cosmological isotropy. In particular, we apply the transform with several apodisation scales, thus allowing the determination of the positional dependence of the angular power spectrum with either high spatial localisation or high angular resolution (ie. narrow bins in multipole space). Practically, this implies that we estimate the angular power spectrum locally in discs of various sizes positioned in different directions: small discs allow the greatest sensitivity to positional dependence, whereas larger discs allow greater sensitivity to variations over different angular scales. In addition, we determine whether the spatial position of a few outliers in the angular power spectrum could suggest the presence of residual foregrounds or systematic effects. For multipoles close to the first peak, the most deviant local estimates from the best fit WMAP model are associated with a few particular areas close to the Galactic plane. Such deviations also include the “dent” in the spectrum just shortward of the first peak which was remarked upon by the WMAP team. Estimating the angular power spectrum excluding these areas gives a slightly higher first Doppler peak amplitude. Finally, we probe the isotropy of the largest angular scales by estimating the power spectrum on hemispheres and reconfirm strong indications of a north-south asymmetry previously reported by other authors. Indeed, there is a remarkable lack of power in a region associated with the north ecliptic pole. With the greater fidelity in l-space allowed by this larger sky coverage, we find tentative evidence for residual foregrounds in the range l = 2 − 4, which could be associated with the low measured quadrupole amplitudes and other anomalies on these angular scales (eg. planarity and alignment). However, over the range l = 5−40 the observed asymmetry is much harder to explain in terms of residual foregrounds and known systematic effects. By reorienting the coordinate axes, we partition the sky into different hemispheres and search for the reference frame which maximises the asymmetric distribution of power. The north pole for this coordinate frame is found to intersect the sphere at (80◦, 57◦) in Galactic co-latitude and longitude over almost the entire multipole range l = 5− 40. Furthermore, the strong negative outlier at l = 21 and the strong positive outlier at l = 39 as determined from the global power spectrum by the WMAP team, are found to be associated with the northern and southern hemispheres respectively (in this frame of maximum asymmetry). Thus, these two outliers follow the general tendency of the multipoles l = 5 − 40 to be of systematically lower amplitude in the north and higher in the south. Such asymmetric distributions of power on the sky provide a serious test for the cosmological principle of isotropy.