General CMB and primordial bispectrum estimation: Mode expansion, map making, and measures of F NL

We present a detailed implementation of two parallel bispectrum estimation methods which can be applied to general nonseparable primordial and cosmic microwave background (CMB) bispectra. The method exploits bispectrum mode decompositions on the tetrahedral domain of allowed wave number or multipole values, using both separable basis functions and related orthonormal modes. We provide concrete examples of such modes constructed from symmetrized tetrahedral polynomials, demonstrating the rapid convergence of expansions describing nonseparable bispectra. We use these modes to create rapid and robust pipelines for generating simulated CMB maps of high resolution ($lg2000$) given an arbitrary primordial power spectrum and bispectrum or an arbitrary late-time CMB angular power spectrum and bispectrum. By extracting coefficients for the same separable basis functions from an observational map, we are able to present an efficient ${f}_{\mathrm{NL}}$ estimator for a given theoretical model with a nonseparable bispectrum. The estimator has two manifestations, comparing the theoretical and observed coefficients at either primordial or late times, thus encompassing a wider range of models, including secondary anisotropies and lensing as well as active models, such as cosmic strings. We provide examples and validation of both ${f}_{\mathrm{NL}}$ estimation methods by direct comparison with simulations in a WMAP-realistic context. In addition, we demonstrate how the full primordial and CMB bispectrum can be extracted from observational maps using these mode expansions, irrespective of the theoretical model under study. We also propose a universal definition of the bispectrum parameter ${F}_{\mathrm{NL}}$, so that the integrated bispectrum on the observational domain can be more consistently compared between theoretical models. We obtain WMAP5 estimates of ${f}_{\mathrm{NL}}$ for the equilateral model from both our primordial and late-time estimators which are consistent with each other, as well as results already published in the literature. These general bispectrum estimation methods should prove useful for non-Gaussianity analysis with the Planck satellite data, as well as in other contexts.