The Redshift-Space Cluster-Galaxy Cross-Correlation as a Probe of Gravity on Cosmic Scales

Testing General Relativity (GR) on cosmological scales requires probes that are directly sensitive to gravitational dynamics. We present a Fisher matrix forecast of how well the Dark Energy Spectroscopic Instrument (DESI) Bright Galaxy Survey (BGS) can constrain departures from GR through the cluster infall velocity field, using the redshift-space cluster-galaxy cross-correlation function $ξ^{\rm s}_{\rm cg}(r_p, r_π)$ as the primary observable. We construct mock catalogs from the AbacusSummit N-body simulations, using a Halo Occupation Distribution (HOD) model for galaxies and a log-normal mass-observable relation for cluster selection. We parameterize departures from GR with a single multiplicative factor $F_{\rm G}$ that rescales halo infall velocities relative to the GR prediction. Marginalizing over the HOD, mass-observable, and cosmological parameters, we find that DESI BGS can constrain $F_{\rm G}$ to $σ(F_{\rm G}) = 0.033$ at $z \sim 0.3$. This precision is sufficient to distinguish representative $f(R)$ and Galileon gravity models from GR at $\gtrsim 4σ$. The strongest limitation is a degeneracy between $F_{\rm G}$ and the scatter in the cluster mass-observable relation, while the constraint shows no strong degeneracy with any individual HOD parameter. Perfect knowledge of the galaxy-halo connection would improve the constraint to $σ(F_{\rm G}) \approx 0.015$, and including the galaxy autocorrelation function in the data vector could recover part of this improvement. The same data vector yields tight constraints on the cosmological parameters, $σ(Ω_m) = 0.010$ and $σ(σ_8) = 0.010$, only moderately correlated with $F_{\rm G}$. These results demonstrate the potential of DESI BGS cluster-galaxy correlations to test gravity on quasi-linear scales, complementary to existing modified gravity probes and to weak-lensing cluster mass profiles.