level. 4. Higher order correlation functions Higher order correlations, such as bispectrum B ζ and trispectrum T ζ (defined in Sec. II C 4) can also constrain the inflationary dynamics and the interactions, (see [23] for a review). The amount of non-gaussianities has been recently constrained by the WMAP data [13]. As pointed out in Eq. (59), even a single field inflation model in a slow-roll regime generates small non-Gaussianities at the level, f$_{NL}$ ∼ ϵ, η ∼ 10 − 2 [21, 156], though the current limits are around four orders of magnitude above this level. The first constraint on full bispectrum was computed for the COBE data [269] but due to the computational cost the WMAP employ the “KSW estimator” [270] that combine squeezed triangular configurations in the harmonic space to construct an optimal estimator for f$_{NL}$ . For example inflation, curvaton, and preheating induced non-Gaussianity belong to this category, where ζ and ζ 2 are evaluated at the same location in space [13]. In addition, the equilateral non-linear coupling parameter f$_{NL eq}$ provides a complementary description of the bispectrum, combining triangular configurations in the harmonic space that are equilateral $^{19}$. The constraints from the most recent observations are currently given by WMAP 5-years data [13] $^{20}$: meaning that there is a hint in favor of a non-vanishing positive f$_{NL}$ . The lower bound on f$_{NL}$ is even raised above zero when the bispectrum maps are less restrictive. This is the origin of the discrepancy between these results and the prior claimed detection of non- Gaussianities from [272] using the WMAP 3-years data. Note that using another statistics − 3500 ≤ f$_{NL}$ ≤ 2000 , at 2 σ , (73) − 9 < f$_{NL}$ < 115 , at 2 σ , − 151 < f$_{NL eq}$ < 253 , at 2 σ , (74) 19 Indeed, models can generate a large amount of non-Gaussianities only in one configuration. Note that several other configurations and statistics to estimate f$_{NL}$ have been proposed and searched for, such as the point-source bispectrum B src measured in the WMAP 5-years data [13], the flattened triangle configurations from models with departure from the Bunch-Davies initial condition [271]. 20 Note that there is no known constraint on the trispectrum parameter, τ$_{NL}$ . It is expected to be of order τ$_{NL}$ ∼ ( f$_{NL}$ ) $^{2}$. 32