Polynomial Interpretation of Multipole Vectors

Abstract

Copi, Huterer, Starkman and Schwarz introduced multipole vectors in a tensor context and used them to demonstrate that the first-year WMAP quadrupole and octopole planes align at roughly the 99.9% confidence level. In the present article the language of polynomials provides a new and independent derivation of the multipole vector concept. Bezout's Theorem supports an elementary proof that the multipole vectors exist and are unique (up to rescaling). The constructive nature of the proof leads to a fast, practical algorithm for computing multipole vectors. We illustrate the algorithm by finding exact solutions for some simple toy examples, and numerical solutions for the first-year WMAP quadrupole and octopole. We then apply our algorithm to Monte Carlo skies to independently re-confirm the estimate that the WMAP quadrupole and octopole planes align at the 99.9% level.

Version 1: 6 pages. Version 2: added uniqueness proof to Corollary 2; added proper citation (to Starkman et al.) for Open Question; other minor improvements