General CMB and Primordial Trispectrum Estimation

TL;DR

This work develops a comprehensive framework for estimating and reconstructing general primordial and CMB trispectra, extending beyond separable templates by using separable mode expansions. It introduces optimal and generalized estimators that account for anisotropic noise and masking, defines a universal $T_{NL}$ parameter for cross-model comparisons, and builds a full pipeline from shape definitions and domain deformations to practical map-making and trispectrum recovery. A central innovation is the mode-based decomposition into five-dimensional separable bases, enabling tractable $\mathcal{O}(\ell_{ m max}^5)$ or $\mathcal{O}(\ell_{ m max}^3)$ operations, depending on degeneracies, and the construction of both primordial and late-time estimators. The framework also provides visualization tools via shape decomposition and a principled approach to reconstructing the trispectrum from data, with explicit treatment of the Local and Equilateral models and a clear path toward non-Gaussian map simulations. Together, these methods significantly broaden the reach of trispectrum analyses for Planck-scale data and future surveys, offering robust diagnostics and interpretable measures of non-Gaussianity across cosmological probes.

Abstract

We present trispectrum estimation methods which can be applied to general non-separable primordial and CMB trispectra. We present a general optimal estimator for the connected part of the trispectrum, for which we derive a quadratic term to incorporate the effects of inhomogeneous noise and masking. We describe a general algorithm for creating simulated maps with given arbitrary (and independent) power spectra, bispectra and trispectra. We propose a universal definition of the trispectrum parameter $T_{NL}$, so that the integrated bispectrum on the observational domain can be consistently compared between theoretical models. We define a shape function for the primordial trispectrum, together with a shape correlator and a useful parametrisation for visualizing the trispectrum. We derive separable analytic CMB solutions in the large-angle limit for constant and local models. We present separable mode decompositions which can be used to describe any primordial or CMB bispectra on their respective wavenumber or multipole domains. By extracting coefficients of these separable basis functions from an observational map, we are able to present an efficient estimator for any given theoretical model with a nonseparable trispectrum. The estimator has two manifestations, comparing the theoretical and observed coefficients at either primordial or late times. These mode decomposition methods are numerically tractable with order $l^5$ operations for the CMB estimator and approximately order $l^6$ for the general primordial estimator (reducing to order $l^3$ in both cases for a special class of models). We also demonstrate how the trispectrum can be reconstructed from observational maps using these methods.

Figures (9)

• Figure 1: Quadrilateral defined by the four wave-vectors ${\bf k}_i$. The diagonal is represented for ${\bf K}$.
• Figure 2: Quadrilateral defined by the four wavenumbers $k_i$, the diagonal $K$, and the angle $\theta_4$ out of the plane of the first triangle.
• Figure 3: Three-dimensional shape function domain for a fixed value of $\epsilon$, i.e. for a particular ratio of the perimeters of the triangles created by the wavenumbers $(k_1,k_2,K)$ and $(k_3,k_4,K)$ respectively. Note that the triangle conditions on these two wavenumber sets restrict them to the two tetrahedral domains illustrated (right), slices through which are mapped as shown into the full domain, a rectangular pyramid (left).
• Figure 4: Local $A$ model \ref{['LocA']}. The peak as $\beta\rightarrow 1$ corresponds to $K\rightarrow 0$, i.e. the 'doubly-squeezed' limit. The other peak corresponds to $k_1\rightarrow 0, k_3\rightarrow (\epsilon-1)k$. As $\epsilon$ rises above unity (i.e. for triangle $(k_3,k_4,K)$ bigger than triangle $(k_1,k_2,K)$) we expect this peak to become suppressed as observed.
• Figure 5: Local $B$ model \ref{['LocB']}. The peaks at $\alpha=1$ correspond to $k_2\rightarrow 0$, while the peaks at $\alpha=-1$ correspond to $k_1\rightarrow 0$. For $\epsilon=1$ (i.e. equal triangle sizes $(k_1,k_2,K)$ and $(k_3,k_4,K)$) we see a more confined peaking at $\gamma=-1,\gamma=1$, i.e. $k_3\rightarrow 0, k_4\rightarrow 0$ respectively. We observe the peaking of the local B models to be somewhat orthogonal to the local A model.