Optimal analysis of the CMB trispectrum

TL;DR

This work develops a comprehensive framework to analyze the CMB trispectrum, identifying a physically motivated three-parameter space of primordial shapes, and introduces practical, factorizable representations enabling fast estimators. It establishes two notions of trispectrum factorizability (contact and exchange), derives efficient algorithms (Monte Carlo and exact Fisher-matrix) to forecast and measure $g_{NL}$ parameters, and optimizes representations to manage computational cost. Applying these methods to WMAP data yields no significant detection of primordial trispectra, with robust, lensing-corrected constraints on $g_{NL}^{\rm loc}$, $g_{NL}^{\dot\sigma^4}$, and $g_{NL}^{(\partial\sigma)^4}$ that inform inflationary physics. The framework is poised for Planck-scale data and adaptable to future probes, including extensions to exchange trispectra and refined noise and lensing biases.

Abstract

We develop a general framework for data analysis and phenomenology of the CMB four-point function or trispectrum. To lowest order in the derivative expansion, the inflationary action admits three quartic operators consistent with symmetry: $\dotσ^4$, $\dotσ^2 (\partialσ^2)$, and $(\partialσ)^4$. In single field inflation, only the first of these operators can be the leading non-Gaussian signal. A Fisher matrix analysis shows that there is one near-degeneracy among the three CMB trispectra, so we parameterize the trispectrum with two coefficients $g_{NL}^{\dotσ^4}$ and $g_{NL}^{(\partialσ)^4}$, in addition to the coefficient $g_{NL}^{\rm loc}$ of $ζ^3$-type local non-Gaussianity. This three-parameter space is analogous to the parameter space $(f_{NL}^{\rm loc}, f_{NL}^{\rm equil}, f_{NL}^{\rm orth})$ commonly used to parameterize the CMB three-point function. We next turn to data analysis and show how to represent these trispectra in a factorizable form which leads to computationally fast operations such as evaluating a CMB estimator or simulating a non-Gaussian CMB. We discuss practical issues in CMB analysis pipelines, and perform an optimal analysis of WMAP data. Our minimum-variance estimates are $g_{NL}^{\rm loc} = (-3.80 \pm 2.19) \times 10^5$, $g_{NL}^{\dotσ^4} = (-3.20 \pm 3.09) \times 10^6$, and $g_{NL}^{(\partialσ)^4} = (-10.8 \pm 6.33) \times 10^5$ after correcting for the effects of CMB lensing. No evidence of a nonzero inflationary four-point function is seen.

Figures (2)

• Figure 1: Representation of a typical multifield potential. Modes of interest for observation cross the horizon about sixty $e$-foldings before the end of inflation. Therefore, effects coming from the evolution of the fields after horizon crossing can be treated locally in real space. The effective theory is more general than this example, as it does not assume that the inflaton is a scalar field. This example is however interesting in helping in visualizing the different scales in the problem.
• Figure 2: 68% and 95% confidence regions in the $(g_{NL}^{\dot\sigma^4}, g_{NL}^{(\partial\sigma)^4})$ plane, with the Lorentz invariant model in Eq. (\ref{['eq:S_lorentz']}) shown as the dashed line.