Under the assumption of GI and an assumed biasing relation between galaxies and mass, a redshift survey enables an independent reconstruction of the density and velocity fields. The comparison of the smoothed fields recovered from redshifts and from velocities is a very important tool for testing the basic hypotheses and for determining the cosmological parameters. The velocity is proportional to the gravitational acceleration, which ideally requires full knowledge of the distribution of mass in space. The integration is limited to r R max where the signal dominates over shot-noise. The convergence can be improved by increasing gradually during the iterations.
Relevant issues follow.
An accurate knowledge of the probability that a galaxy at a given distance be included in the sample is essential, especially at large distances where -1 can introduce large errors. For a given flux limit, can be evaluated together with the luminosity function using a maximum-likelihood technique independent of density inhomogeneities. Regions in the sky not covered by the survey have to be filled with mock galaxies by some method of extrapolation from nearby regions. Galaxies in three different positions along a line of sight through a contracting region may have the same redshift. Given a redshift in a collapsing region where the problem is not resolved by the smoothing used, one can either take some average of the three solutions, or make an intelligent choice between them, e.g. by using the velocity field derived from observed velocities.