The multipole vectors of the Wilkinson Microwave Anisotropy Probe, and their frames and invariants

We investigate the statistical isotropy (SI) and Gaussianity of the cosmic microwave background fluctuations, using a set of multipole vector functions capable of separating these two issues. In general a multipole is broken into a frame and 2ℓ− 3 ordered invariants. The multipole frame is found to be suitably sensitive to Galactic cuts. We then apply our method to real Wilkinson Microwave Anisotropy Probe (WMAP) data sets; a co-added masked map, the interlinear combinations map, and Wiener filtered and cleaned maps. Taken as a whole, multipoles in the range ℓ= 2–10 or ℓ= 2–20 show consistency with SI, as proved by the Kolmogorov test applied to the Euler angles of the frame. This result is not inconsistent with previous claims for a preferred direction in the sky for ℓ= 2, …, 5. The multipole invariants also show overall consistency with Gaussianity apart from a few anomalies of limited significance (98 per cent), listed at the end of this paper. However, we stress that this method in practice is very limited by noise.