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

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We generalize Doroshkevich's celebrated formulae for the eigenvalues of the initial shear field associated with Gaussian statistics to the local non-Gaussian fnl model. This is possible because, to at least second order in fnl, distributions at fixed overdensity are unchanged from the case fnl=0. We use this generalization to estimate the effect of fnl≠ 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 fnl. While a positive fnl 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 σ₈. Therefore, void abundances furnish complementary information to cluster abundances, and a joint comparison of both might provide interesting constraints on primordial non-Gaussianity.