Cluster gravitational redshifts: uncertainties and survey requirements

We investigate the impact of observational and modelling uncertainties in cluster gravitational redshifts as a probe of modified gravity using an end-to-end forecasting pipeline. We use a generative model to build a halo catalogue with M500 ≥ 3 × 1013 M⊙, populate haloes with member galaxies via a five-parameter halo occupation distribution (HOD), assign projected positions from radial density profiles, apply survey-like selections, and infer a linear rescaling of the gravitational potential, αMG, to parameterise modifications to general relativity (GR). We vary redshift uncertainties, radial and mass–redshift completeness, member abundance, minimum mass and maximum redshift, as well as mis-specify the clusters density and velocity profiles, centres, and mass function. We find that the intracluster velocity dispersion sets an effective floor: improving redshift precision beyond σz ~ 10−4(1 + z) brings no improvement in the precision of αMG. Realistic redshift and mass cuts primarily remove low-mass haloes and have minimal impact on the αMG precision. In this setting, we find that shallow, narrower spectroscopic surveys are preferable to deep, wide photometric ones for precise constraints on αMG. We further find that mis-centring can mimic significant departures from GR. Baryonic deviations from a Navarro-Frenk-White profile and velocity anisotropies do not introduce appreciable biases. In the high-S/N regime of upcoming surveys, accurate determination of cluster centres will be essential to avoid interpreting systematic effects as new physics. The Spectroscopic Stage-5 Experiment and the Wide Field Spectroscopic Telescope provide a promising route toward establishing gravitational redshifts as a valuable and complementary probe of modified gravity.