Biased Estimates of Ω from Comparing Smoothed Predicted Velocity Fields to Unsmoothed Peculiar Velocity Measurements

We show that a regression of unsmoothed peculiar velocity measurements against peculiar velocities predicted from a smoothed galaxy density field leads to a biased estimate of the cosmological density parameter Ω, even when galaxies trace the underlying mass distribution and galaxy positions and velocities are known perfectly. The bias arises because the errors in the predicted velocities are correlated with the predicted velocities themselves. We investigate this bias using cosmological N-body simulations and analytic arguments. In linear perturbation theory, for cold dark matter power spectra and Gaussian or top hat smoothing filters, the bias in Ω is always positive, and its magnitude increases with increasing smoothing scale. This linear calculation reproduces the N-body results for Gaussian smoothing radii Rs ∼> 10h Mpc, while non-linear effects lower the bias on smaller smoothing scales, and for Rs ∼< 3h Mpc Ω is underestimated rather than overestimated. The net bias in Ω for a given smoothing filter depends on the underlying cosmological model. The effect on current estimates of Ω from velocity-velocity comparisons is probably small relative to other uncertainties, but taking full advantage of the statistical precision of future peculiar velocity data sets will require either equal smoothing of the predicted and measured velocity fields or careful accounting for the biases discussed here. Subject headings: cosmology: theory, galaxies: distances and redshifts, methods: numerical