Omega from the skewness of the cosmic velocity divergence

We propose a method for measuring the cosmological density parameter $\mathrm{\Omega }$ from the statistics of the divergence field, $\theta \equiv H^{-1}÷v$, the divergence of peculiar velocity, expressed in units of the Hubble constant, $H \equiv 100 h km/s/Mpc$\mathrm{.}Thevelocityfieldisspatiallysmoothedover$\sim 10 h^{-1} Mpc$ to remove strongly nonlinear effects. Assuming weakly-nonlinear gravitational evolution from Gaussian initial fluctuations, and using second-order perturbative analysis, we show that $ <\theta^3> \propto -\Omega^{-0.6} <\theta^2>^2$. The constant of proportionality depends on the smoothing window. For a top-hat of radius R and volume-weighted smoothing, this constant is $26\mathrm{/}7-\gamma$, where $\gamma =-d\log <{\theta }^2>\mathrm{/}d\log R$. If the power spectrum is a power law, $P(k)\propto k^n$, then $\gamma =3+n$. A Gaussian window yields similar results. The resulting method for measuring $\mathrm{\Omega }$ is independent of any assumed biasing relation between galaxies and mass. The method has been successfully tested with numerical simulations. A preliminary application to real data, provided by the POTENT recovery procedure from observed velocities favors $\mathrm{\Omega }\sim 1$. However, because of an uncertain sampling error, this result should be treated as an assessment of the feasibility of our method rather than a definitive measurement of $\mathrm{\Omega }$.