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

Abstract

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 = 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 8 . Therefore, void abundances furnish complementary information to cluster abundances, and a joint comparison of both might provide interesting constraints on primordial non-Gaussianity.