Primordial non-Gaussianity and the CMB bispectrum

TL;DR

This work develops a general, numerically efficient framework to compute the CMB bispectrum today from an arbitrary primordial bispectrum by propagating it through full linear radiation transfer functions. By reformulating the four-dimensional integral into a two-dimensional integral on a triangular k-space domain and introducing a triangular, scale-invariant parametrisation, the method accommodates both local and equilateral shape limits and non-separable forms. It combines analytic checks in the large-angle regime with a robust adaptive 2D integration and a basis-based estimator, achieving sub-percent accuracy up to multipoles $l \,\lesssim\,1800$ for realistic transfer functions and enabling detailed discrimination among inflationary scenarios, including DBI-like higher-derivative models. The approach offers a pathway to accurate, model-agnostic predictions and practical estimators for primordial non-Gaussianity in current and upcoming CMB data.

Abstract

We present a new formalism, together with efficient numerical methods, to directly calculate the CMB bispectrum today from a given primordial bispectrum using the full linear radiation transfer functions. Unlike previous analyses which have assumed simple separable ansatze for the bispectrum, this work applies to a primordial bispectrum of almost arbitrary functional form, for which there may have been both horizon-crossing and superhorizon contributions. We employ adaptive methods on a hierarchical triangular grid and we establish their accuracy by direct comparison with an exact analytic solution, valid on large angular scales. We demonstrate that we can calculate the full CMB bispectrum to greater than 1% precision out to multipoles l<1800 on reasonable computational timescales. We plot the bispectrum for both the superhorizon ('local') and horizon-crossing ('equilateral') asymptotic limits, illustrating its oscillatory nature which is analogous to the CMB power spectrum.

Figures (19)

• Figure 1: Region of integration for the bispectrum calculation. 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 region of integration is in yellow. This area can be parametrised into slices represented by the green triangle and the distance $\frac{2}{\sqrt{3}}k$ of the centre of the triangle from the origin.
• Figure 2: $F^{SI}$ plotted on the $\alpha\beta$-triangle for the 'local' superhorizon shape (left) and the 'equilateral' horizon-crossing shape (right). The equilateral case has been scaled to that the centres of both plots are at the same height.
• Figure 3: A plot of the spherical Bessel function $l=200$. Note how the function essentially vanishes up to $l\approx 180$.
• Figure 4: The equilateral triangle we need to integrate over to obtain the bispectrum for $l_1=l_2=l_3=20$. Note the small scale structure to the right of the peak. The first few oscillations make a significant contribution and must be sampled intensively for an accurate result. The rest of the oscillations, however, do not because of cancellations. Hence any adaptive algorithm must be able to determine the importance of any structure it finds.
• Figure 5: Refinement method. We start with the black cell defined by the three black points at the corners. The cell is then divided into four by calculating the three red points and the change in area is then $\frac{1}{4}\left|\sum black - \sum red\right|$.