The inverse redshift-space operator: reconstructing cosmological density and velocity fields

We present the linear inverse redshift-space operator that maps the galaxy density field derived from redshift surveys from redshift space to real space. Expressions are presented for observers in both the cosmic microwave background radiation and Local Group rest frames. We show how these results can be generalized to flux-limited galaxy redshift surveys. These results allow the straightforward reconstruction of real-space density and velocity fields without resort to iterative or numerically intensive inverse methods. As a corollary to the inversion of the density in the Local Group rest frame we present an expression for estimating the real-space velocity dipole from redshift space, allowing one to estimate the Local Group dipole without full reconstruction of the redshift survey. We test these results on some simple models and find that the reconstruction is very accurate. A new spherical harmonic representation of the redshift distortion and its inverse is developed, which simplifies the reconstruction and allows analytic calculation of the properties of the reconstructed redshift survey. We use this representation to analyse the uncertainties in the reconstruction of the density and velocity fields from redshift space, caused by only a finite volume being available. Both sampling and shot-noise variance terms are derived and we discuss the limits of reconstruction analysis. We compare the reconstructed velocity field with the true velocity field and show that reconstruction in the Local Group rest frame is preferable, since this eliminates the major source of uncertainty from the dipole mode. These results can be used to transform redshift surveys to real space and may be used as part of a full likelihood analysis to extract cosmological parameters.