Optimal Polyspectra Estimation

Abstract

We show that full inverse covariance weighting can be naturally incorporated into modal estimation methods making them optimal for the CMB power spectrum, bispectrum and trispectrum, as well as in other 3D applications. Modal estimation methods are highly efficient requiring inversion of only an nmax x nmax covariance matrix, where nmax is the limited number of modes needed to describe both the theoretical model and the noise which affects it (in contrast to the full lmax^2 x lmax^2 inverse covariance weighting). A WMAP resolution implementation for modal bispectrum estimation is discussed and shown to yield optimal constraints for local and other models at lmax =500 and nmax <= 50. We also discuss further improvements by characterising and preconditioning the covariance matrix using a single bispectrum variance shape induced by the noise and mask. We describe optimal modal estimation in general terms applicable to polyspectra of any order in a formalism that can be extended beyond the isotropic case. We use the example of anisotropic noise in bispectrum estimation to show the completeness of the isotropic modal basis and the necessity of subtracting a linear estimator term for optimality, removing cross-terms in the variance. We briefly discuss applications to the CMB power spectrum and trispectrum, as well as large-scale structure.