Publication : t94/063

Abstract:
We propose a method for measuring the cosmological density parameter $ \Omega $ from the statistics of the expansion scalar, $ \theta \equiv H^{-1} {\bf \nabla} . {\bf v} , $ - the divergence of peculiar velocity, expressed in units of the Hubble constant, $ H\equiv 100 {\rm h\ km\ s}^{ {\rm -1}} {\rm Mpc}^{ {\rm -1}}. $ The velocity field is spatially smoothed over $ \sim 10 {\rm h}^{ {\rm -1}} {\rm Mpc} $ to remove strongly nonlinear effects. Assuming weakly-nonlinear 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+\gamma , $ where $ \gamma = {\rm d\ log} \left\langle \theta^ 2 \right\rangle / {\rm d\ log} \ R. $ If the power spectrum is a power law, $ P(k)\propto k^ {\bf n}, $ then $ \gamma =-(3+n). $ A Gaussian window yields similar results. The resulting method for measuring $ \Omega $ is independent of any assumed biaising 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 $ \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 $ \Omega . $