Gravitational redshifting of galaxies in the SPIDERS cluster catalogue
  Data from the SPectroscopic IDentification of ERosita Sources (SPIDERS) are
searched for a detection of the gravitational redshifting of light from
$\sim\!20\,000$ galaxies in $\sim\!2500$ galaxy clusters using three
definitions of the cluster centre: its Brightest Cluster Galaxy (BCG), the
redMaPPer identified Central Galaxy (CG), or the peak of X-ray emission.
Distributions of velocity offsets between galaxies and their host cluster's
centre, found using observed redshifts, are created. The quantity
$\hat{\Delta}$, the average of the radial velocity difference between the
cluster members and the cluster systemic velocity, reveals information on the
size of a combination of effects on the observed redshift, dominated by
gravitational redshifting. The change of $\hat{\Delta}$ with radial distance is
predicted for SPIDERS galaxies in General Relativity (GR), and $f(R)$ gravity,
and compared to the observations. The values of $\hat{\Delta}=-13.5\pm4.7$ km
s$^{-1}$, $\hat{\Delta}=-12.5\pm5.1$ km s$^{-1}$, and
$\hat{\Delta}=-18.6\pm4.8$ km s$^{-1}$ for the BCG, X-ray and CG cases
respectively broadly agree with the literature. There is no significant
preference of one gravity theory over another, but all cases give a clear
detection ($>2.5\sigma$) of $\hat{\Delta}$. The BCG centroid is deemed to be
the most robust method in this analysis, due to no well defined central
redshift when using an X-ray centroid, and CGs identified by redMaPPer with no
associated spectroscopic redshift. For future gravitational redshift studies,
an order of magnitude more galaxies, $\sim\!500\,000$, will be required-a
possible feat with the forthcoming Vera C. Rubin Observatory, Euclid and
eROSITA.
