R. Capasso, Chris A. Collins, Nicolas Clerc, D. P. Schneider, Nelson Padilla, K. Furnell, Alexis Finoguenov, Johan Comparat, S. Damsted, Andrea Merloni, John A. Peacock, A. Saro, C. T. Mpetha, Astrophysics Research Institute [Liverpool] (ARI), Liverpool John Moores University (LJMU), Institute for Astronomy [Edinburgh] (IfA), University of Edinburgh, Institut de recherche en astrophysique et planétologie (IRAP), Institut national des sciences de l'Univers (INSU - CNRS)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Observatoire Midi-Pyrénées (OMP), Météo France-Centre National d'Études Spatiales [Toulouse] (CNES)-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut de Recherche pour le Développement (IRD)-Météo France-Centre National d'Études Spatiales [Toulouse] (CNES)-Centre National de la Recherche Scientifique (CNRS)-Institut de Recherche pour le Développement (IRD)-Centre National de la Recherche Scientifique (CNRS), Department of Physics [Helsinki], Falculty of Science [Helsinki], University of Helsinki-University of Helsinki, Max Planck Institute for Extraterrestrial Physics (MPE), Max-Planck-Gesellschaft, Institute for Gravitation and the Cosmos [PennState], Pennsylvania State University (Penn State), Penn State System-Penn State System, Oskar Klein Centre [Stockholm], Stockholm University, Pontificia Universidad Católica de Chile (UC), INAF - Osservatorio Astronomico di Trieste (OAT), Istituto Nazionale di Astrofisica (INAF), Istituto Nazionale di Fisica Nucleare, Sezione di Trieste (INFN, Sezione di Trieste), Istituto Nazionale di Fisica Nucleare (INFN), Università degli studi di Trieste, Instituto de Astrofisica [Santiago], Department of Physics, Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut national des sciences de l'Univers (INSU - CNRS)-Observatoire Midi-Pyrénées (OMP), Institut de Recherche pour le Développement (IRD)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National d'Études Spatiales [Toulouse] (CNES)-Centre National de la Recherche Scientifique (CNRS)-Météo-France -Institut de Recherche pour le Développement (IRD)-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National d'Études Spatiales [Toulouse] (CNES)-Centre National de la Recherche Scientifique (CNRS)-Météo-France -Centre National de la Recherche Scientifique (CNRS), Helsingin yliopisto = Helsingfors universitet = University of Helsinki-Helsingin yliopisto = Helsingfors universitet = University of Helsinki, Università degli studi di Trieste = University of Trieste, Mpetha, C. T., Collins, C. A., Clerc, N., Finoguenov, A., Peacock, J. A., Comparat, J., Schneider, D., Capasso, R., Damsted, S., Furnell, K., Merloni, A., Padilla, N. D., and Saro, A.
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., Comment: Accepted for publication in MNRAS. 11 pages, 10 figures