General CMB and Primordial Bispectrum Estimation I: Mode Expansion, Map-Making and Measures of f_NL

TL;DR

This paper develops a general, dual-mode framework for estimating and simulating non-Gaussianity in the CMB via bispectra that are not restricted to separable forms. By constructing a tetrapyd-domain-based, rapidly convergent set of separable basis functions and their orthonormal transforms, the authors derive primordial and late-time $f_{\rm NL}$ estimators that are efficient even for high-resolution data. They demonstrate two complementary pipelines—one starting from a primordial shape and one from a late-time CMB bispectrum—capable of generating non-Gaussian map simulations and extracting full bispectrum information, with a universal normalization $F_{\rm NL}$ for cross-model comparison. The methodology resolves computational bottlenecks associated with non-separable shapes and provides a scalable path toward Planck-era analyses and broader non-Gaussian models, including cosmic strings and other secondary effects. The results on simulated equilateral models validate the approach, showing fast convergence and consistent recovery of input non-Gaussian signals.

Abstract

We present a detailed implementation of two bispectrum estimation methods which can be applied to general non-separable primordial and CMB bispectra. The method exploits bispectrum mode decompositions on the domain of allowed wavenumber or multipole values. Concrete mode examples constructed from symmetrised tetrahedral polynomials are given, demonstrating rapid convergence for known bispectra. We use these modes to generate simulated CMB maps of high resolution (l > 2000) 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 and general f_NL estimator for a given theoretical model. The estimator has two versions comparing theoretical and observed coefficients at either primordial or late times, thus encompassing a wider range of models, including secondary anisotropies, lensing and cosmic strings. We provide examples and validation of both f_NL estimation methods by direct comparison with simulations in a WMAP-realistic context. In addition, we show how the full 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_NL for more consistent comparison between theoretical models. We obtain WMAP5 estimates of f_NL for the equilateral model from both our primordial and late-time estimators which are consistent with each other, as well as with results already published in the literature. These general bispectrum estimation methods should prove useful for the analysis of nonGaussianity in the Planck satellite data, as well as in other contexts.

Figures (17)

• Figure 1: Flow chart for the two general estimator methodologies described and implemented in this article using complete separable mode expansions. Note the overall redundancy which assists estimator validation and the independence of the extraction of expansion coefficients from theory $\alpha_n$ (cycle 1) and data $\beta_n$ (cycle 2). Explanations for the schematic equations can be found in the main text.
• Figure 2: Tetrahedral domain ('tetrapyd') for allowed multipole values $l$ for the CMB bispectrum $b_{l_1l_2l_3}$ or, with wavenumbers $k$ for the primordial bispectrum $B(k_1,k_2,k_3)$). The regular tetrahedral region defined up to the equilateral slice $l_1+l_2+l_3 \le 2l_\textrm{max}\equiv 2L$ (shaded brown) contains two thirds of the overall volume. The rest of the domain is given by the regular triangular pyramid on top which fills the volume to the corner of the encompassing cube defined by $l_1,l_2,l_3 \leq L$. An origami tetrapyd is also shown (right) with folding instructions.
• Figure 3: The reduced CMB bispectra for the equilateral model (left) and the local model (right) plotted on the tetrahedral region shown in figure \ref{['eq:tetrapydl']} (from Fergusson:2008ra). Several density contours are illustrated (light blue positive and magenta negative) and $b_{l_1l_2l_3}$ is normalised by scaling relative to the constant Sachs-Wolfe solution (\ref{['eq:constbispect']}) $b_{l_1l_2l_3}^{\rm model}/b_{l_1l_2l_3}^{\rm const.}$. Note the acoustic peaks induced by the transfer functions and the centre weighting for the equilateral model, contrasting with the corner-weighting for the local case Fergusson:2006pr.
• Figure 4: Scaling comparison of the multipole domain weight function $w(l_1,l_2,l_3)$ (or $w_{l_1 l_2 l_3}$) given in (\ref{['eq:lweight']}) and the modified weight function $w_s(l_1,l_2,l_3)$ given in (\ref{['eq:lweightsep']}), which is rescaled by a separable function. On the left, the equal-$l$ values are shown with the linear scaling of $w$ (dashed) contrasting with the constant $w_s$ (solid). On the right, a density plot of $w_s$ is shown on the $l_1+l_2+l_3 = 2L$ slice with $L=2000$. Note the uniformity $w_s\approx {\rm const.}$, except very close to the edges where there is about a factor of 4 rise to the maximum value on the perimeter.
• Figure 5: The orthonormal one-dimensional tetrahedral $q_n(x)$ plotted on the unit interval for $n=0$--$5$. The behaviour is smooth and bounded across the domain even for high $n$, except where the weight function $w(x)$ vanishes at $x=0$. Also plotted for comparison are the rescaled Legendre polynomials $P_n(2x-1)$ (dashed lines). Despite $q_n$ and $P_n$ sharing qualitative features such as $n$ nodal points, their properties and orthogonality on ${{\cal V}_{\cal T}}$ are very different.