Optimal Trispectrum Estimators and WMAP Constraints

We present an implementation of an optimal CMB trispectrum estimator which accounts for anisotropic noise and incomplete sky coverage. We use a general separable mode expansion which can and has been applied to constrain both primordial and late-time models. We validate our methods on large angular scales using known analytic results in the Sachs-Wolfe limit. We present the first near-optimal trispectrum constraints from WMAP data on the cubic term of local model inflation $g_{\rm NL} = (1.6 \pm 7.0)\times 10^5$, for the equilateral model $t_{\rm NL}^{\rm{equil}}=(-3.11\pm 7.5)\times 10^6$ and for the constant model $t_{\rm NL}^{\rm{const}}=(-1.33\pm 3.62)$. These results, particularly the equilateral constraint, are relevant to a number of well-motivated models (such as DBI and K-inflation) with closely correlated trispectrum shapes. We also use the trispectrum signal predicted for cosmic strings to provide a conservative upper limit on the string tension $G\mu \le 1.1\times 10^{-6}$ (at 95% confidence), which is largely background and model independent. All these new trispectrum results are consistent with a Gaussian Universe. We discuss the importance of constraining general classes of trispectra using these methods and the prospects for higher precision with the Planck satellite.