Omega from the skewness of the cosmic velocity divergence

We propose a method for measuring the cosmological density parameter Ω from the statistics of the expansion scalar, $$\theta\equiv H^{-1}\nabla\cdot\upsilon$$, which is the divergence of the peculiar velocity, expressed in units of the Hubble constant, $$H\equiv100 \enspace h$$ km s⁻¹ Mpc⁻¹. The velocity field is spatially smoothed over ∼ 10 h⁻¹ Mpc to remove strongly non-linear effects. We assume weakly non-linear gravitational evolution from Gaussian initial fluctuations, and using second-order perturbative analysis, we show that $$\left \langle \theta^{3} \right \rangle\propto -\Omega^{-0.6}\left \langle \theta^{2} \right \rangle^2$$. The constant of proportionality depends on the smoothing window. For a top-hat of radius R and volume-weighted smoothing, this constant is 26/7-γ, where γ = –d log $$\left \langle \theta^{2} \right \rangle$$/d log R. If the power spectrum is a power law, $$P(k)\propto k^n$$, then γ = 3 + n. A Gaussian window yields similar results. The resulting method for measuring Ω 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, favours Ω ∼ 1. Because of an uncertain sampling error, however, this result should be treated as an assessment of the feasibility of our method rather than a definitive measurement of Ω.