Measuring Gravitational Redshifts in Galaxy Clusters

Wojtak {\it et al} have stacked 7,800 clusters from the SDSS survey in redshift space. They find a small net blue-shift for the cluster galaxies relative to the brightest cluster galaxies, which agrees quite well with the gravitational redshift from GR. Zhao {\it et al.} have pointed out that, in addition to the gravitational redshift, one would expect to see transverse Doppler (TD) redshifts, and that these two effects are generally of the same order. Here we show that there are other corrections that are also of the same order of magnitude. The fact that we observe galaxies on our past light cone results in a bias such that more of the galaxies observed are moving away from us in the frame of the cluster than are moving towards us. This causes the observed average redshift to be $\langle \delta z \rangle = -\langle \Phi \rangle + \langle \beta^2 \rangle / 2 + \langle \beta_x^2 \rangle$, with $\beta_x$ is the line of sight velocity. That is if we average over galaxies with equal weight. If the galaxies in each cluster are weighted by their fluence, or equivalently if we do not resolve the moving sources, and make an average of the mean redshift giving equal weight per photon, the observed redshift is then opposite to the usual transverse Doppler effect. In the WHH experiment, the weighting is a step-function because of the flux-limit for inclusion in the spectroscopic sample and the result is different again, and depends on the details of the luminosity function and the SEDs of the galaxies. Including these effects substantially modifies the blue-shift profile. We show that in-fall and out-flow have very small effect over the relevant range of impact parameters but become important on larger scales.Comment: accepted for publication in MNRA

Non-Linear Cluster Lens Reconstruction

We develop a method for general non-linear cluster lens reconstruction using the observable distortion of background galaxies. The distortion measures the combination $\gamma/(1-\kappa)$ of shear $\gamma$ and surface density $\kappa$. From this we obtain an expression for the gradient of $\log (1 - \kappa)$ in terms of directly measurable quantities. This allows one to reconstruct $1 - \kappa$ up to an arbitrary constant multiplier. Recent work has emphasised an ambiguity in the relation between the distortion and $\gamma/(1-\kappa)$. Here we show that the functional relation depends only on the parity of the images, so if one has data extending to large radii, and if the critical lines can be visually identified (as lines along which the distortion diverges), this ambiguity is resolved. Moreover, we show that for a generic 2-dimensional lens it is possible to locally determine the parity from the distortion. The arbitrary multiplier, which may in fact take a different value in each region bounded by the contour $\kappa = 1$, can be determined by requiring that the mean surface excess vanish at large radii and that the gradient of $\kappa$ should be continuous across $\kappa = 1$. We show how these ideas might be implemented to reconstruct the surface density, if necessary without use of the data in regions where determination of the parity is insecure, and we show how one can measure the mass contained within an aperture, again, if necessary, without using data within the aperture.Comment: 6 pages, uuencoded compressed postscript, CITA-94-3