Using a non-perturbative method developed in a previous work ( [ paper II ] ) , we derive the probability distribution { \cal P } ( \delta _ { R } ) of the density contrast within spherical cells in the quasi-linear regime for some non-Gaussian initial conditions .
We describe three such models .
The first one is a straightforward generalization of the Gaussian scenario .
It can be seen as a phenomenological description of a density field where the tails of the linear density contrast distribution would be of the form { \cal P } _ { L } ( \delta _ { L } ) \sim e ^ { - | \delta _ { L } | ^ { - \alpha } } , where \alpha is no longer restricted to 2 ( as in the Gaussian case ) .
We derive exact results for { \cal P } ( \delta _ { R } ) in the quasi-linear limit .
The second model is a physically motivated isocurvature CDM scenario .
Our approach needs to be adapted to this specific case and in order to get convenient analytical results we introduce a simple approximation ( which is not related to the gravitational dynamics but to the initial conditions ) .
Then , we find a good agreement with the available results from numerical simulations for the pdf of the linear density contrast for \delta _ { L,R } \ga 0 .
We can expect a similar accuracy for the non-linear pdf { \cal P } ( \delta _ { R } ) .
Finally , the third model corresponds to the small deviations from Gaussianity which arise in standard slow-roll inflation .
We obtain exact results for the pdf of the density field in the quasi-linear limit , to first-order over the primordial deviations from Gaussianity .