  The observed abundance of high-redshift galaxies and clusters contains
precious information about the properties of the initial perturbations. We
present a method to compute analytically the number density of objects as a
function of mass and redshift for a range of physically motivated non-Gaussian
models. In these models the non-Gaussianity can be dialed from zero and is
assumed to be small. We compute the probability density function for the
smoothed dark matter density field and we extend the Press and Schechter
approach to mildly non-Gaussian density fields. The abundance of high-redshift
objects can be directly related to the non-Gaussianity parameter and thus to
the physical processes that generated deviations from the Gaussian behaviour.
Even a skewness parameter of order 0.1 implies a dramatic change in the
predicted abundance of $z\gap 1$ objects. Observations from NGST and X-ray
satellites (XMM) can be used to accurately measure the amount of
non-Gaussianity in the primordial density field.
