Maxwell's Multipole Vectors and the CMB

Abstract

The recently re-discovered multipole vector approach to understanding the harmonic decomposition of the cosmic microwave background traces its roots to Maxwell's Treatise on Electricity and Magnetism. Taking Maxwell's directional derivative approach as a starting point, the present article develops a fast algorithm for computing multipole vectors, with an exposition that is both simpler and better motivated than in the author's previous work. Tests show the resulting algorithm, coded up as a Mathematica notebook, to be both fast and robust. This article serves to announce the free availability of the code. The algorithm is then applied to the much discussed anomalies in the low-ell CMB modes, with sobering results. Simulations show the quadrupole-octopole alignment to be unusual at the 98.7% level, corroborating earlier estimates of a 1-in-60 alignment while showing recent reports of 1-in-1000 (using non-unit-length normal vectors) to have unintentionally relied on the near orthogonality of the quadrupole vectors in one particular data set. The alignment of the quadrupole and octopole vectors with the ecliptic plane is confirmed at better than the 2sigma level.