In this paper, we extend a previous work where we presented a method based on the N-point probability density function (pdf) to study the Gaussianity of the cosmic microwave background (CMB). We explore a local non-linear perturbative model up to third order as a general characterization of the CMB anisotropies. We focus our analysis in large-scale anisotropies (θ > 1°). At these angular scales (the Sachs–Wolfe regime), the non-Gaussian description proposed in this work defaults (under certain conditions) to an approximated local form of the weak non-linear coupling inflationary model. In particular, the quadratic and cubic terms are governed by the non-linear coupling parameters fNL and gNL, respectively. The extension proposed in this paper allows us to directly constrain these non-linear parameters. Applying the proposed methodology to Wilkinson Microwave Anisotropy Probe (WMAP) 5-year data, we obtain −5.6 × 105 < gNL < 6.4 × 105, at 95 per cent confidence level. This result is in agreement with previous findings obtained for equivalent non-Gaussian models and with different non-Gaussian estimators, although this is the first direct constraint on gNL from CMB data. A model selection test is performed, indicating that a Gaussian model (i.e. fNL≡ 0 and gNL≡ 0) is preferred to the non-Gaussian scenario. When comparing different non-Gaussian models, we observe that a pure fNL model (i.e. gNL≡ 0) is the most favoured case and that a pure gNL model (i.e. fNL≡ 0) is more likely than a general non-Gaussian scenario (i.e. fNL≠ 0 and gNL≠ 0). Finally, we have analysed the WMAP data in two independent hemispheres, in particular the ones defined by the dipolar pattern found by Hoftuft et al. We show that, whereas the gNL value is compatible between both hemispheres, it is not the case for fNL (with a p-value of ≈0.04). However, if, as suggested by Hoftuft et al., anisotropy of the data is assumed, the distance between the likelihood distributions for each hemisphere is larger than expected from Gaussian and anisotropic simulations, not only for fNL but also for gNL (with a p-value of ≈0.001 in the case of this latter parameter). This result is extra evidence for the CMB asymmetries previously reported in WMAP data.

We acknowledge partial financial support from the Spanish Ministerio de Ciencia e Innovación project AYA2007- 68058-C03-02. PV also thanks financial support from the Ramón y Cajal programme.

Peer reviewed