The initial shear field in models with primordial local non-Gaussianity and implications for halo and void abundances

We generalize the 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) not equal 0 on the abundance of virialized haloes. 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 haloes, it makes fewer extremely large voids. Its effect thus is qualitatively different from a simple rescaling of the normalization 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.