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}}=(-3.11pm 7.5)times 10^6 $ and for the constant model $t_{rm NL}^{rm{const}}=(-1.33pm 3.62)$. 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 $Gmu le 1.1times 10^{-6}$ (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.
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