In this paper we extend a previous work ( 45 ) where we presented a method based on the N-point probability distribution ( 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 ( \theta > 1 ^ { \circ } ) .
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 f _ { \mathrm { NL } } and g _ { \mathrm { NL } } , respectively .
The extension proposed in this paper allows us to directly constrain these non-linear parameters .
Applying the proposed methodology to WMAP 5-yr data , we obtain -5.6 \times 10 ^ { 5 } < { g } _ { \mathrm { NL } } < 6.4 \times 10 ^ { 5 } , at 95 % CL .
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 constrain on g _ { \mathrm { NL } } from CMB data .
A model selection test is performed , indicating that a Gaussian model ( i.e .
f _ { \mathrm { NL } } \equiv 0 and g _ { \mathrm { NL } } \equiv 0 ) is preferred to the non-Gaussian scenario .
When comparing different non-Gaussian models , we observe that a pure f _ { \mathrm { NL } } model ( i.e .
g _ { \mathrm { NL } } \equiv 0 ) is the most favoured case , and that a pure g _ { \mathrm { NL } } model ( i.e .
f _ { \mathrm { NL } } \equiv 0 ) is more likely than a general non-Gaussian scenario ( i.e .
f _ { \mathrm { NL } } \neq 0 and g _ { \mathrm { NL } } \neq 0 ) .
Finally , we have analyzed the WMAP data in two independent hemispheres , in particular the ones defined by the dipolar pattern found by ( 23 ) .
We show that , whereas the g _ { \mathrm { NL } } value is compatible between both hemispheres , it is not the case for f _ { \mathrm { NL } } ( with a p-value \approx 0.04 ) .
However , if , as suggested by ( 23 ) , 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 f _ { \mathrm { NL } } , but also for g _ { n } l ( with a p-value of \approx 0.001 in the case of this latter parameter ) .
This result is an extra evidence for the CMB asymmetries previously reported in WMAP data .