Omega from the skewness of the cosmic velocity divergence

Abstract

We propose a method for measuring the cosmological density parameter $\Omega$ from the statistics of the divergence field, $\theta \equiv H^{-1}÷v$, the divergence of peculiar velocity, expressed in units of the Hubble constant, $H\equiv 100hkm/s/Mpc$. The velocity field is spatially smoothed over $\sim 10h^{-1}Mpc$ 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 $26/7-\gamma$, where $\gamma =-d\log <{\theta }^2>/d\log R$. 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 $\Omega$ 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 $\Omega \sim 1$. 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 $\Omega$.

16 pages + 2 figures, uuencoded postscript file, also available by anonymous ftp from ftp.cita.utoronto.ca in directory /cita/francis/div_skewness, CITA 94-15

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