  Primordial non-Gaussianity has emerged as one of the most promising probes of
the inflationary epoch. While the cosmic microwave background and large-scale
halo bias currently provide the most stringent constraints on the non-Gaussian
parameter fNL, the abundance of dark matter halos is a complementary probe
which may allow tests of Gaussianity which are independent of the precise form
of non-Gaussian initial conditions. We study the halo mass function in N-body
simulations with a range of non-Gaussian initial conditions. In addition to the
usual fNL model, we consider gNL Phi^3-type non-Gaussianity and models where
the 4-point amplitude tauNL is an independent parameter. We introduce a new
analytic form for the halo mass function in the presence of primordial
non-Gaussianity, the "log-Edgeworth" mass function, and find good agreement
with the N-body simulations. The log-Edgeworth mass function introduces no free
parameters and can be constructed from first principles for any model of
primordial non-Gaussianity.
