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Using third-order perturbation theory, we derive a relation between the meandivergence of the peculiar velocity given density and the density itself. Ourcalculations assume Gaussian initial conditions and are valid for Gaussianfiltering of the evolved density and velocity fields. The mean velocitydivergence turns out to be a third-order polynomial in the density contrast. Wetest the power spectrum dependence of the coefficients of the polynomial forscale-free and standard CDM spectra and find it rather weak. Over scales largerthan about 5 megaparsecs, the scatter in the relation is small compared to thatintroduced by random errors in the observed density and velocity fields. Therelation can be useful for recovering the peculiar velocity from the associateddensity field, and also for non-linear analyses of the anisotropies ofstructure in redshift surveys.Comment: 8 pages, 1 figure, uses mn.sty and epsf.tex, slightly amended  abstract and summary, accepted for publication in MNRA

Recovery of the cosmological peculiar velocity from the density field in the weakly non-linear regime