OMEGA FROM THE SKEWNESS OF THE COSMIC VELOCITY DIVERGENCE

We propose a method for measuring the cosmological density parameter Omega from the statistics of the expansion scalar, theta = H(-1)del.upsilon, which is the divergence of the peculiar velocity, expressed in units of the Hubble constant, H = 100 h km s(-1) Mpc(-1). The velocity field is spatially smoothed over similar to 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 [theta(3)]proportional to -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 26/7-gamma, where gamma = - d log[theta>(2)]/d log R. If the power spectrum is a power law, P(k) proportional to k(n), then gamma = 3+n. A Gaussian window yields similar results. The resulting method for measuring 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, favours Omega similar to 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 Omega.