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Large-scale density from velocity expansion and shear

I derive up to second order in Eulerian perturbation theory a new relation between the weakly nonlinear density and velocity fields. In the case of unsmoothed fields, density at a given point turns out to be a purely local function of the expansion (divergence) and shear of the velocity field. The relation depends on the cosmological parameter Omega, strongly by the factor f(Omega) = Omega^{0.6} and weakly by the factors K(Omega) and C(Omega) proportional to Omega^{-2/63} and Omega^{-1/21} respectively. The Gramann solution is found to be equivalent to the derived relation with the weak Omega-dependence neglected. To make the relation applicable to the real world, I extend it for the case of smoothed fields. The resulting formula, when averaged over shear given divergence, reproduces up to second order the density-velocity divergence relation of Chodorowski & Lokas; however, it has smaller spread. It makes the formula a new attractive local estimator of large-scale density from velocity.