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Weakly Nonlinear Evolution of Topology of Large-scale Structure

The gravitational evolution of the genus and other statistics of isodensity contours of the density field is derived analytically in a weakly nonlinear regime using second-order perturbation theory. The effect of final smoothing in perturbation theory on the statistics of isodensity contours is also evaluated. The resulting analytic expression for the genus is compared with $N$-body numerical simulations and exhibits a good agreement.