General CMB and Primordial Trispectrum Estimation

Abstract

We present trispectrum estimation methods which can be applied to general non-separable primordial and CMB trispectra. We present a general optimal estimator for the connected part of the trispectrum, for which we derive a quadratic term to incorporate the effects of inhomogeneous noise and masking. We describe a general algorithm for creating simulated maps with given arbitrary (and independent) power spectra, bispectra and trispectra. We propose a universal definition of the trispectrum parameter $T_{NL}$, so that the integrated bispectrum on the observational domain can be consistently compared between theoretical models. We define a shape function for the primordial trispectrum, together with a shape correlator and a useful parametrisation for visualizing the trispectrum. We derive separable analytic CMB solutions in the large-angle limit for constant and local models. We present separable mode decompositions which can be used to describe any primordial or CMB bispectra on their respective wavenumber or multipole domains. By extracting coefficients of these separable basis functions from an observational map, we are able to present an efficient estimator for any given theoretical model with a nonseparable trispectrum. The estimator has two manifestations, comparing the theoretical and observed coefficients at either primordial or late times. These mode decomposition methods are numerically tractable with order $l^5$ operations for the CMB estimator and approximately order $l^6$ for the general primordial estimator (reducing to order $l^3$ in both cases for a special class of models). We also demonstrate how the trispectrum can be reconstructed from observational maps using these methods.Comment: 38 pages, 9 figures. In v2 Figures 4-7 are altered slightly and some extra references are included in the bibliography. v3 matches version submitted to journal. Includes discussion of special case