The Angular Trispectrum of the CMB

TL;DR

This paper builds a complete all-sky framework for the CMB temperature trispectrum, deriving how rotational, permutation, and parity symmetries constrain its form and Gaussian-noise properties. It develops estimators for the trispectrum and shows how to compress its information into various quadratic (and cubic) statistics with optimal filters, focusing on the divergence of the temperature–weighted gradient as the best statistic for weak lensing. The authors quantify the signal-to-noise for the lensing trispectrum, revealing that Planck could achieve a total $S/N$ of order $\sqrt{3100}\approx 60$ and that the divergence statistic can reduce the convergence-power-spectrum variance by more than an order of magnitude compared with previous gradient-based methods. This configuration-aware approach provides a powerful, general toolkit for extracting non-Gaussian information from the CMB and has immediate implications for precision lensing measurements and cosmological parameter inference.

Abstract

We study the general properties of the CMB temperature four-point function, specifically its harmonic analogue the angular trispectrum, and illustrate its utility in finding optimal quadratic statistics through the weak gravitational lensing effect. We determine the general form of the trispectrum, under the assumptions of rotational, permutation, and parity invariance, its estimators on the sky, and their Gaussian noise properties. The signal-to-noise in the trispectrum can be highly configuration dependent and any quadratic statistic used to compress the information to a manageable two-point level must be carefully chosen. Through a systematic study, we determine that for the case of lensing a specific statistic, the divergence of a filtered temperature-weighted temperature-gradient map, contains the maximal signal-to-noise and reduces the variance of estimates of the large-scale convergence power spectrum by over an order of magnitude over previous gradient-gradient techniques. The total signal-to-noise for lensing with the Planck satellite is of order 60 for a LCDM cosmology.

Figures (5)

• Figure 1: The power spectrum of the deflection angle in the fiducial $\Lambda$CDM model. Errors boxes represent the $1 \sigma$ errors from Gaussian noise on the divergence statistic binned in the bands shown. The divergence estimator of Eqn. (\ref{['eqn:divweight']})-(\ref{['eqn:divfilter']}) is optimal for the low multipoles and reduces the variance in the power spectrum estimation by more than an order of magnitude as compared with the gradient-gradient statistics of ZalSel99.
• Figure 2: Contributions to the $(S/N)^2$ from trispectra configurations with a fixed diagonal $L$ and maximum side length $l_1$, summed over the remaining three sides. Solid lines represent the full calculation of the trispectrum terms; dashed lines represent the pairwise approximation of Eqn. (\ref{['eqn:trispectapprox']}). The signal-to-noise in the low $L$ trispectrum is highly dependent on the configuration.
• Figure 3: Cumulative signal to noise in the trispectra configurations with the diagonal $L$ summed over $l_1\ldots l_4$. Dashed lines represent an ideal experiment where $C_l=C_l^{\rm tot}$ out to a maximum $l=l_{\rm max}$ ; solid lines represent the Planck experiment. Lines represent the approximation of Eqn. (\ref{['eqn:trispectapprox']}); points represent the calculation using the full trispectrum for an ideal experiment.
• Figure 4: Approximate total $(S/N)^2$ in the trispectrum for an ideal experiment out to $l=l_{\rm max}$ and the Planck experiment. The Planck experiment approximates an ideal experiment of $l \approx 1600$ with a $(S/N)^2 \sim 3100$.
• Figure 5: Degradation in the total $(S/N)^2$ in the power spectrum due to covariance from gravitational lensing. The degradation is minimal for the Planck experiment or any that is cosmic variance limited only out to $l \sim 2000$.