  The abundance of collapsed objects in the universe, or halo mass function, is
an important theoretical tool in studying the effects of primordially generated
non-Gaussianities on the large scale structure. The non-Gaussian mass function
has been calculated by several authors in different ways, typically by
exploiting the smallness of certain parameters which naturally appear in the
calculation, to set up a perturbative expansion. We improve upon the existing
results for the mass function by combining path integral methods and saddle
point techniques (which have been separately applied in previous approaches).
Additionally, we carefully account for the various scale dependent combinations
of small parameters which appear. Some of these combinations in fact become of
order unity for large mass scales and at high redshifts, and must therefore be
treated non-perturbatively. Our approach allows us to do this, and to also
account for multi-scale density correlations which appear in the calculation.
We thus derive an accurate expression for the mass function which is based on
approximations that are valid over a larger range of mass scales and redshifts
than those of other authors. By tracking the terms ignored in the analysis, we
estimate theoretical errors for our result and also for the results of others.
We also discuss the complications introduced by the choice of smoothing filter
function, which we take to be a top-hat in real space, and which leads to the
dominant errors in our expression. Finally, we present a detailed comparison
between the various expressions for the mass functions, exploring the accuracy
and range of validity of each.
