Multipole invariants and non-Gaussianity

We propose a framework for separating the information contained in the cosmic microwave background multipoles, alm, into algebraically independent components that cleanly separate information pertaining to the power spectrum, non-Gaussianity and preferred axis effects. The formalism builds upon the recently proposed multipole vectors and we elucidate a few features regarding these vectors, namely their lack of statistical independence for a Gaussian random process. In a few cases we explicitly relate our proposed invariants to components of the n-point correlation function (power spectrum, bispectrum). We find the distributions of the invariants using a mixture of analytical and numerical methods. We also evaluate them for the co-added WMAP first-year map.