Trispectrum estimator in equilateral type non-Gaussian models

We investigate an estimator to measure the primordial trispectrum in equilateral type non-Gaussian models such as k-inflation, single field DBI inflation and multi-field DBI inflation models from Cosmic Microwave Background (CMB) anisotropies. The shape of the trispectrum whose amplitude is not constrained by the bispectrum in the context of effective theory of inflation and k-inflation is known to admit a separable form of the estimator for CMB anisotropies. We show that this shape is $87 \%$ correlated with the full quantum trispectrum in single field DBI inflation, while it is $33 \%$ correlated with the one in multi-field DBI inflation when curvature perturbation is originated from purely entropic contribution. This suggests that $g_{\rm NL} ^{equil}$, the amplitude of this particular shape, provides a reasonable measure of the non-Gaussianity from the trispectrum in equilateral non-Gaussian models. We relate model parameters such as the sound speed, $c_s$ and the transfer coefficient from entropy perturbations to the curvature perturbation, $T_{\mathcal{R} S}$ with $g_{\rm NL} ^{equil}$, which enables us to constrain model parameters in these models once $g_{\rm NL}^{equil}$ is measured in WMAP and Planck.