Multipole Vectors and the Area Statistic: How Often Random Skies Produce Octopole Planarity and Quadrupole-Octopole Vector Alignments

The octopole of the microwave sky is described as planar, its multipole vectors clustered so that their area vectors nearly coincide, and its orientation is tied to the quadrupole's in the alignment discussions. This paper computes how often isotropic skies produce area-vector statistics as extreme as the values quoted for the real sky. In two hundred thousand random skies the octopole area alignment statistic has mean 0.5 and 95th percentile 0.821; a value of 0.80, in the range reported for the real octopole, occurs in 6.18 per cent of random skies, and 0.90 in 1.56 per cent. The cross-alignment between a random quadrupole axis and the octopole's area directions reaches 0.80 in 3.58 per cent of skies. Planarity and vector alignment at the reported levels are therefore uncommon but well within the repertoire of chance, and their conjunction with the axis alignment should be counted with the dependence between the statistics in view, since all of them are functions of the same seven octopole numbers. The study is simulated, not measured.