Beyond general relativity: probing gravity with gravitational redshifts

Despite the success of general relativity (GR), the unexplained nature of dark energy on cosmological scales leaves open the question of whether GR provides a complete description of gravity. This quest is further motivated by growing tensions among cosmological observations when interpreted within $\mathrm{\Lambda }$CDM. Gravitational redshifts of cluster member galaxies probe cluster potentials on megaparsec scales directly, complementing conventional large-scale structure tests. Here, we investigate how redshift precision and survey design propagate into constraints on modified gravity using an end-to-end pipeline run on mock catalogues, focusing on mis-centring and spectroscopic completeness. We find that competitive measurements require wide-field spectroscopic cluster surveys explicitly designed to maximise the number of spectroscopically confirmed members per cluster, to enable high-purity stacking, and to control systematic effects.