The initial shear field in models with primordial local non-Gaussianity
  and implications for halo and void abundances
  We generalize Doroshkevich's celebrated formulae for the eigenvalues of the
initial shear field associated with Gaussian statistics to the local
non-Gaussian f_{nl} model. This is possible because, to at least second order
in f_{nl}, distributions at fixed overdensity are unchanged from the case
f_{nl}=0. We use this generalization to estimate the effect of f_{nl}\ne 0 on
the abundance of virialized halos. Halo abundances are expected to be related
to the probability that a certain quantity in the initial fluctuation field
exceeds a threshold value, and we study two choices for this variable: it can
either be the sum of the eigenvalues of the initial deformation tensor (the
initial overdensity), or its smallest eigenvalue. The approach based on a
critical overdensity yields results which are in excellent agreement with
numerical measurements. We then use these same methods to develop
approximations describing the sensitivity of void abundances on f_{nl}. While a
positive f_{nl} produces more extremely massive halos, it makes fewer extremely
large voids. Its effect thus is qualitatively different from a simple rescaling
of the normalisation of the density fluctuation field \sigma_8. Therefore, void
abundances furnish complementary information to cluster abundances, and a joint
comparison of both might provide interesting constraints on primordial
non-Gaussianity.
