Algorithms for bispectra: forecasting, optimal analysis, and simulation

TL;DR

This work addresses the challenge of connecting predicted CMB bispectra to data with feasible computational methods. It introduces a factorizability ansatz for $b_{}$ and develops a toolkit of fast algorithms for Fisher forecasting, optimal estimation (with sky cuts and inhomogeneous noise), and non-Gaussian map simulation, all applicable to any factorizable bispectrum. Key contributions include an optimization procedure to reduce the number of factorizable terms, a general Monte Carlo framework for bispectrum estimation, and a fast T/$ abla T$ evaluation method; Planck-era forecasts show independent constraints on local, equilateral, and secondary bispectra with important cross-effects (e.g., ISW-lensing bias). The approach enables practical analysis of multiple bispectrum shapes and robust simulations, making Planck-like datasets amenable to comprehensive non-Gaussian studies and enabling broader use of bispectrum physics in CMB cosmology.

Abstract

We propose a factorizability ansatz for angular bispectra which permits fast algorithms for forecasting, analysis, and simulation, yet is general enough to encompass many interesting CMB bispectra. We describe a suite of general algorithms which apply to any bispectrum which can be represented in factorizable form. First, we present algorithms for Fisher matrix forecasts and the related problem of optimizing the factorizable representation, giving a Fisher forecast for Planck as an example. We show that the CMB can give independent constraints on the amplitude of primordial bispectra of both local and equilateral shape as well as those created by secondary anisotropies. We also show that the ISW-lensing bispectrum should be detected by Planck and could bias estimates of the local type of non-Gaussianity if not properly accounted for. Second, we implement a bispectrum estimator which is fully optimal in the presence of sky cuts and inhomogeneous noise, extends the generality of fast estimators which have been limited to a few specific forms of the bispectrum, and improves the running time of existing implementations by several orders of magnitude. Third, we give an algorithm for simulating random, weakly non-Gaussian maps with prescribed power spectrum and factorizable bispectrum.

Figures (9)

• Figure 1: Contribution to $B^{\rm loc}_{\ell_1\ell_2\ell_3}$ as a function of conformal distance $r$, with dominant contribution from recombination ($r \sim 14000$ Mpc), for $(\ell_1,\ell_2\ell_3) = (2,300,300)$, a typical squeezed triple with high signal-to-noise.
• Figure 2: Distribution of factorizable terms in the $(r,t)$ plane for the "optimized" higher-derivative bispectrum $b^{\rm hd}_{\ell_1\ell_2\ell_3}$.
• Figure 3: Fisher matrix errors vs. $\ell_{\rm max}$ for the ISW-lensing, local, equilateral, and gravitational forms of the bispectrum, assuming Planck noise levels throughout.
• Figure 4: Contour plots of $dF/d(\log\ell_{\rm max})d(\ell_{\rm min}/\ell_{\rm max})$, defined in Eq. (\ref{['eq:Fll']}), showing the contribution to the Fisher matrix error as a function of $(\ell_{\rm min},\ell_{\rm max})$, for the ISW-lensing, local, equilateral, and gravitational forms of the bispectrum.
• Figure 5: Values of $b_{\ell_1,\ell_2\ell_3}/(C_{\ell_1}C_{\ell_2}C_{\ell_3})^{1/2}$, showing oscillations in the ISW-lensing bispectrum but not the local bispectrum, plotted for $\ell_1=10$ and $\ell_3=\ell_2+6$. This is a typical "squeezed" triangle which contributes high signal-to-noise in both cases.