Primordial non-Gaussianity and Bispectrum Measurements in the Cosmic Microwave Background and Large-Scale Structure

TL;DR

This work provides a comprehensive framework for probing primordial non-Gaussianity through the bispectrum in the CMB and large-scale structure. It develops shape-based formalisms and separable eigenmode expansions to efficiently compute and compare a wide class of inflationary models, including local, equilateral, and feature-type shapes, as well as late-time sources like cosmic strings. The authors derive optimal cubic estimators for the non-Gaussian amplitude $f_\textrm{NL}$, discuss their optimality, and address practical issues such as rotational invariance, instrumental noise, and masking, with detailed discussion of numerical implementations and Planck-era forecasts. They also extend the analysis to LSS, highlighting how primordial non-Gaussianity interacts with gravitational evolution and galaxy bias, and emphasize the need for joint CMB-LSS analyses to robustly constrain broad classes of models. Overall, the work outlines the path toward tighter, more general constraints on inflationary physics using both CMB observations and large-scale structure surveys, while acknowledging substantial challenges from foregrounds, secondary anisotropies, and nonlinear bias.

Abstract

The most direct probe of non-Gaussian initial conditions has come from bispectrum measurements of temperature fluctuations in the Cosmic Microwave Background and of the matter and galaxy distribution at large scales. Such bispectrum estimators are expected to continue to provide the best constraints on the non-Gaussian parameters in future observations. We review and compare the theoretical and observational problems, current results and future prospects for the detection of a non-vanishing primordial component in the bispectrum of the Cosmic Microwave Background and large-scale structure, and the relation to specific predictions from different inflationary models.

Figures (9)

• Figure 1: Triangle types contributing to the bispectrum corresponding to 'squeezed' or local configurations with $k_3\ll k_1,\,k_2$ (left), equilateral configurations with $k_3\approx k_1\approx k_2$ (centre) and flattened configurations with $k_3\approx k_1+ k_2$ (right).
• Figure 2: Tetrahedral domain for allowed wavenumber configurations $k_1,k_2,k_3$ contributing to the primordial bispectrum $B(k_1,k_2,k_3)$). A regular tetrahedron is shown satisfying $k_1+k_2+k_3 \le 2k_\textrm{max}\equiv 2K$.
• Figure 3: Shape functions for the scale-invariant equilateral (left) and local (right) models, $S(k_1,k_2,k_3) = S(\tilde{\alpha},\,\tilde{\beta})$ on transverse slices with $2\tilde{k} = k_1+k_2+k_3 = \hbox{const.}$
• Figure 4: The one-dimensional tetrahedral polynomials $q_n(k)$ on the domain (\ref{['eq:tetrapydk']}), rescaled to the unit interval for $n=0$--$5$. Also plotted are the shifted Legendre polynomials $P_n(2x-1)$ (dashed lines) which share qualitative features such as $n$ nodal points.
• Figure 5: Orthonormal eigenmode decomposition coefficients (\ref{['eq:orthobasis']}) for the equilateral and DBI models ( upper panel) and shape correlations (\ref{['eq:shapecorrelator']}) of the original bispectrum against the partial sum up to a given mode $n$ ( lower panel). The correlation plot includes both primordial and late-time CMB bispectra for the equilateral and DBI models, as well as the late-time CMB bispectrum from cosmic strings (refer to section \ref{['sec:CMB']}). In all cases, we find that we need at most 15 three-dimensional modes to obtain a correlation greater than 98% (primordial convergence without the acoustic peaks requires only 6 modes).