Comparing the redshift-space density field with the real-space velocity field

I derive a non-linear local relation between the redshift-space density field and the real-space velocity field. The relation accounts for the radial character of the redshift distortions and is not restricted to the limit of the distant observer. Direct comparisons between the observed redshift-space density fields and the real-space velocity fields possess all of the advantages of the conventional redshift-space analyses, while at the same time they are free of their disadvantages. In particular, neither the model-dependent reconstruction of the density field in real space, nor the reconstruction of the non-linear velocity field in redshift space is necessary, the latter being questionable because of its vorticity at second order. The non-linear redshift-space velocity field is irrotational only in the distant observer limit, and that limit is not a good approximation for the shallow catalogues of peculiar velocities currently available. Unlike the conventional redshift-space comparisons, the comparison proposed here does not have to be restricted to the linear regime. Accounting for non-linear effects removes one of the sources of bias in the estimation of beta. Moreover, the non-linear effects break the Omega -bias degeneracy plaguing all analyses based on linear theory.