Correlations between Maxwell’s multipoles for gaussian random functions on the sphere

Maxwell’s multipoles are a natural geometric characterisation of real functions on the sphere (with fixed l). The correlations between multipoles for gaussian random functions are calculated, by mapping the spherical functions to random polynomials. In the limit of high l, the 2-point function tends to a form previously derived by Hannay in the analogous problem for the Majorana sphere. The application to the cosmic microwave background (CMB) is discussed. PACS numbers: 02.30.Px, 05.45.Mt, 98.70.Vc A striking feature of randomness is the emergence of structure arising from independent processes. Here, I describe a simple, universal correlation structure associated with statistically isotropic random functions on the sphere: the correlations between Maxwell’s multipoles. In the multipole representation, any real spherical function Φ(θ, φ), with fixed l, may be represented by l unit vectors u1, . . . ,ul,