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

By J. Fergusson, M. Liguori, E. Shellard. 2009

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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.

Subfields: Astronomy and Astrophysics, Nuclear and High Energy Physics, Oceanography

View Paper: https://www.semanticscholar.org/paper/6ac957d4505a7cc79a784497b86d5137f3d4c92f

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Citations in the corpus

Large-scale bias and efficient generation of initial conditions for nonlocal primordial non-Gaussianity (2011, methodology): “For other approaches to generating non-Gaussian fields see [31, 35–39].”
Planck2018 results (2018, methodology): “Using modal techniques developed by Fergusson & Shellard (2009) and Fergusson et al. (2010, 2012), any shape can be factorized, with a close-to-optimal estimator.”
Primordial Non-Gaussianity and Bispectrum Measurements in the Cosmic Microwave Background and Large-Scale Structure (2010, methodology): “in [14] (see also [6]) and summarized in formula (60) that the separation can be performed directly in harmonic space on the reduced bispectrum b 1 2 3 , rather than on the primordial shape S(k1, k2, k3).”
Primordial Non-Gaussianities from Inflation Models (2010, methodology): “New methods that are applicable to non-factorizable bispectrum forms and are more model-independent are under active development [71].”
Planck 2018 results. IX. Constraints on primordial non-Gaussianity (2019, methodology): “As in our 2015 analysis, we examine even and odd `1 + `2 + `3 multipole domains, employing the original parity-even Modal estimator, f̂ evenNL (Fergusson et al. 2010, 2012; Shiraishi et al. 2019), and its parity-odd version, f̂ oddNL (Shiraishi et al. 2014, 2015, 2019), respectively.”
Planck 2015 results. XVII. Constraints on primordial non-Gaussianity (2015, unknown): “Modal estimators (Fergusson et al. 2010a, 2012) are based on decomposing the bispectrum (both from theory and from data) into a sum of uncorrelated separable templates, forming a complete basis in bispectrum space, and measuring the amplitude of each.”
Planck 2015 results: XIX. Constraints on primordial magnetic fields (2015, unknown): “…the non-factorizable magneticallyinduced bispectrum, we use an optimal estimator derived within the so-called separable modal methodology (see Fergusson et al. 2010, 2012; Shiraishi et al. 2014, 2015 for auto-bispectra; and Fergusson 2014, and Liguori et al., in prep. for cross-bispectra),…”