The Angular Trispectra of CMB Temperature and Polarization

TL;DR

This work develops a complete angular trispectrum formalism for the CMB temperature and polarization, enabling rotationally invariant four-point statistics and their optimal estimators. It applies the framework to inflationary-origin non-Gaussianity, modeling curvature fluctuations with parameters $f_1$ and $f_2$ and deriving the corresponding curvature and angular trispectra and their detectability. The results show that trispectra can achieve sensitivity comparable to the temperature bispectrum at high multipoles, with polarization trispectra providing additional information, while secondary effects like gravitational lensing offer further opportunities. Overall, the methodology provides a robust toolkit for probing primordial physics and secondary anisotropies in CMB data analysis.

Abstract

We develop the formalism necessary to study four-point functions of the cosmic microwave background (CMB) temperature and polarization fields. We determine the general form of CMB trispectra, with the constraints imposed by the assumption of statistical isotropy of the CMB fields, and derive expressions for their estimators, as well as their Gaussian noise properties. We apply these techniques to initial non-Gaussianity of a form motivated by inflationary models. Due to the large number of four-point configurations, the sensitivity of the trispectra to initial non-Gaussianity approaches that of the temperature bispectrum at high multipole moment. These trispectra techniques will also be useful in the study of secondary anisotropies induced for example by the gravitational lensing of the CMB by the large scale structure of the universe.

Figures (4)

• Figure 1: Geometrical interpretation of the configuration of a trispectrum. The four-point quadrilateral in harmonic space is specified using the pairs $(l_1,l_2)$ along with the diagonal $L$ to define a triangle.
• Figure 2: Signal-to-noise ratio in the temperature trispectrum as a function of the maximum multipole $l_{\text{max}}$ in the Sachs-Wolfe approximation (Eq. \ref{['ReducedTrispecSW']}). The dotted line corresponds to $f_1=1,f_2=0$, the dashed line to $f_1=0,f_2=1$, and the solid line to $f_1=1,f_2=1$.
• Figure 3: $(S/N)^2/f_1^4$ vs. $l_{\text{max}}$ in the temperature trispectrum, using the approximation (\ref{['BetterEstimate']}).
• Figure 4: Relation between Euler angles $(\alpha \beta \gamma)$ and the original rotation angles $(\theta \phi)$ and $(\theta' \phi')$ for the weighted sky maps (\ref{['WeightedSkyMaps']}), with the identification $\hat{\bm{n}}\rightarrow (\theta' \phi')$, $\hat{\bm{q}}\rightarrow (\theta \phi)$, so that $\phi_{\hat{\bm{n}}}=\alpha$, and $\hat{\bm{n}}\cdot\hat{\bm{q}}=\beta$.