Measuring gravitational redshifts in galaxy clusters

Wojtak et al. have stacked 7800 clusters from the Sloan Digital Sky Survey (SDSS) in redshift space. They find a small net blueshift for the cluster galaxies relative to the brightest cluster galaxies, which agrees quite well with the gravitational redshift predicted from general relativity. Zhao et al. have pointed out that, in addition to the gravitational redshift, one would expect to see transverse Doppler (TD) redshifts, so = - + /2 with beta the 3D source velocity in units of c, 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 = - + /2 + , with beta(x) 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 = - + /2, so the kinematical effect is then opposite to the usual transverse Doppler effect. In the Wojtak, Hansen & Hjorth 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 spectral energy distributions of the galaxies. Including these effects substantially modifies the blueshift profile. We show that in-fall and out-flow have very small effect over the relevant range of impact parameters but out-flow becomes significant and needs to be taken into account for measurements on larger scales.