Omega from the skewness of the cosmic velocity divergence

Abstract

We propose a method for measuring the cosmological density parameter $\hat{I}\text{c◯}$ from the statistics of the divergence field, $\hat{I}¸\equiv H^{-1}\tilde{A}\cdot v$, 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 $<\hat{I}{¸}^3>\propto -\hat{I}{\text{c◯}}^{-0.6}<\hat{I}{¸}^2{>}^2$. The constant of proportionality depends on the smoothing window. For a top-hat of radius R and volume-weighted smoothing, this constant is $26/7-{\hat{I}}^3$, where ${\hat{I}}^3=-d\log <\hat{I}{¸}^2>/d\log R$. If the power spectrum is a power law, $P(k)\propto k^n$, then ${\hat{I}}^3=3+n$. A Gaussian window yields similar results. The resulting method for measuring $\hat{I}\text{c◯}$ 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 $\hat{I}\text{c◯}\sim 1$. 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 $\hat{I}\text{c◯}$.

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