The shape of primordial non-Gaussianity and the CMB bispectrum

TL;DR

This work addresses constraining primordial non-Gaussianity via the CMB bispectrum, including general, non-separable primordial shapes. It develops a comprehensive framework combining a shape function $S(k_1,k_2,k_3)$, a fast shape correlator, and a two-dimensional eigenmode decomposition, augmented by numerical innovations such as the flat sky approximation and cubic interpolation to efficiently compute the CMB reduced bispectrum $b_{l_1 l_2 l_3}$ from any $B_\Phi(k_1,k_2,k_3)$ through transfer functions $\Delta_l(k)$. The authors identify five independent shape classes (equilateral, local, warm, flat, feature), demonstrate strong agreement between shape- and CMB-based correlators, and propose a standardized normalization $\bar{f}_{NL}$ based on the shape autocorrelator to enable robust, model-agnostic comparisons of non-Gaussianity. These methods enable Planck-era constraints to discriminate between distinct primordial shapes and guide theoretical model-building by clarifying which shapes are observationally separable. The approach provides a practical, scalable pipeline for interpreting future CMB bispectrum measurements in the quest to understand the physics of the early universe.

Abstract

We present a set of formalisms for comparing, evolving and constraining primordial non-Gaussian models through the CMB bispectrum. We describe improved methods for efficient computation of the full CMB bispectrum for any general (non-separable) primordial bispectrum, incorporating a flat sky approximation and a new cubic interpolation. We review all the primordial non-Gaussian models in the present literature and calculate the CMB bispectrum up to l <2000 for each different model. This allows us to determine the observational independence of these models by calculating the cross-correlation of their CMB bispectra. We are able to identify several distinct classes of primordial shapes - including equilateral, local, warm, flat and feature (non-scale invariant) - which should be distinguishable given a significant detection of CMB non-Gaussianity. We demonstrate that a simple shape correlator provides a fast and reliable method for determining whether or not CMB shapes are well correlated. We use an eigenmode decomposition of the primordial shape to characterise and understand model independence. Finally, we advocate a standardised normalisation method for $f_{NL}$ based on the shape autocorrelator, so that observational limits and errors can be consistently compared for different models.

Figures (20)

• Figure 1: (a) The region of $k$-space allowed by the triangle inequality, i.e., for which the primordial bispectrum is valid. The red lines are $k_1=k_2,\, k_3=0;\; k_2=k_3,\, k_1=0;\; k_3=k_1,\, k_2=0$ and the allowed region is in yellow. (b) This area can be parametrised into slices represented by the green triangle and the distance ${2}|\vec{k}|/{\sqrt{3}}$ of the centre of the triangle from the origin.
• Figure 2: Coordinate transformation from a shaded subtriangle on the equilateral $(\alpha,\,\beta)$ slice to the uniform square $(x,\,y)$ domain suitable for an eigenmode expansion, i.e.$\alpha = 1 - x\,,\;\beta = yx/3\,$. Here, we illustrate the transformation with contour plots for the warm inflation shape function. Note the non-trivial behaviour in the corner region where the function diverges and the sign changes.
• Figure 3: Eigenvalues for the two-dimensional eigenmode expansion of the warm inflation shape function. Here, we denote the conventions with $m$ (for $R_m(x)$ incrementing in the horizontal $x$-direction and the $n$ (for $\bar{P}_n(y)$) in the vertical $y$-direction. Note the dominance of $R_m(x)P_1(y)$ modes. The colour coding (used also in figure \ref{['fig:eigen_all']}) is such that only blue and red colours can contribute at above the 10% level to the autocorrelator $C_k(S, S')$, with yellow and pale green below 1%.
• Figure 4: Eigenmodes for different shape functions using the conventions and colour scale defined in figure \ref{['fig:eigen_warm']}. Strong similarities are apparent between the equilateral family of models which are all highly correlated and would prove very difficult to distinguish observationally. The independence of the local and warm models is also apparent from the orthogonality of the dominant eigenmodes.
• Figure 5: The shape function of models in the equilateral class. Clockwise from top left we have the equilateral, DBI, single and ghost models. All four of these models have the majority of their signal concentrated in the equilateral limit corresponding to the centre of the triangle. Despite significant variations in the flattened limit, particularly around the edges of the triangle, all are strongly correlated by 96% or greater to the equilateral model