The non-linear probability distribution function in models with local primordial non-Gaussianity

We use the spherical evolution approximation to investigate non-linear evolution from the non-Gaussian initial conditions characteristic of the local f _nl model. We provide an analytic formula for the non-linearly evolved probability distribution function (PDF) of the dark matter which shows that the underdense tail of the non-linear PDF in the f _nl model should differ significantly from that for Gaussian initial conditions. Measurements of the underdense tail in numerical simulations may be affected by discreteness effects, and we use a Poisson counting model to describe this effect. Once this has been accounted, our model is in good quantitative agreement with the simulations. In principle, our calculation is an important first step in programs which seek to reconstruct the shape of the initial PDF from observations of large-scale structures in the Lyα forest and the galaxy distribution at later times.