Optimal analysis of the CMB trispectrum

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: $\dot\sigma^4$, $\dot\sigma^2 (\partial\sigma^2)$, and $(\partial\sigma)^4$. 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 $g_{NL}^{\dot\sigma^4}$ and $g_{NL}^{(\partial\sigma)^4}$, in addition to the coefficient $g_{NL}^{\rm loc}$ of $\zeta^3$-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 $g_{NL}^{\rm loc} = (-3.80 \pm 2.19) \times 10^5$, $g_{NL}^{\dot\sigma^4} = (-3.20 \pm 3.09) \times 10^6$, and $g_{NL}^{(\partial\sigma)^4} = (-10.8 \pm 6.33) \times 10^5$ after correcting for the effects of CMB lensing. No evidence of a nonzero inflationary four-point function is seen.