Action Principle Solutions for Galaxy Motions within 3000 Kilometers per Second

If we could reconstruct the orbits that galaxies followed to arrive at their present observed positions and velocities, there is much we would learn about the distribution of mass and the initial conditions of fluctuations. A discussion of how to achieve this reconstruction is begun here. The analysis requires a redshift catalog of mass tracers. The numerical action variational principle is used to find fully nonlinear solutions for the orbits of the mass tracers given their present redshifts and angular positions and the cosmological boundary condition that the peculiar velocities are small at high redshift. A solution predicts the distances of the mass tracers and is tested by a comparison with measured distances. The current numerical results use 289 luminosity-line width distance measurements designed to be close to unbiased. A catalog of 1138 tracers approximates the luminosity distribution of galaxies in the vicinity of the Local Supercluster, at redshifts cz -1 These mass tracers include groups with crossing times less than the Hubble time and isolated galaxies. In this preliminary computation, we assign each mass tracer the same mass-to-light ratio M/L. The tracer masses are fixed by apparent magnitudes and model distances. The only two free parameters in this model are M/L and the expansion time t0. The measure of merit of a solution is the sum of the mean square differences between the predicted and observed distance moduli. In the 3000 km s-1 sample, this reduced χ2 statistic has a well-defined minimum value at M/L = 175 and t0 = 10.0 Gyr, and χ2 at the minimum is about 1.29 times the value expected from just the standard deviation of the distance measurements. We have tested for the effect of the mass at greater distance by using the positions of Abell clusters as a model for the largescale mass distribution. This external mass model reduces the minimum value of 2 by about 10% (approximately 1 σ). The value of the cosmological density parameter Ω0 is determined by the global mean mass-to-light ratio. Our preliminary analysis yields Ω0 = 0.17±0.10 at 1 σ. A tighter bound is expected to come out of a larger sample of measured distances now available.