Searching for extra dimensions with gravitational waves: Dark-siren constraints from GWTC-4

Higher-dimensional theories of gravity predict that gravitational waves (GWs) can propagate into extra spatial dimensions, leading to modified amplitude damping over cosmological distances. Measurements of GW luminosity distances therefore provide a unique probe of the dimensionality of spacetime. In this work, we constrain higher-dimensional GW propagation using the dark-siren method with the Gravitational-Wave Transient Catalog 4.0 (GWTC-4). We adopt a phenomenological parameterization motivated by braneworld scenarios, in which deviations from General Relativity are characterized by the spacetime dimension number D and a crossover scale Rc governing the transition between four- and higher-dimensional gravity. We perform a hierarchical Bayesian analysis combining 141 compact binary coalescences from GWTC-4 with line-of-sight galaxy information from the GLADE+ catalog. For a prior H₀ ∈ [65,77]\ km~s⁻¹Mpc⁻¹ and log(Rc/ Mpc) ∈ [2.7,4.0], we obtain D = 4.38+1.91-1.01 (68% credible interval). We also find that the inferred posterior distribution of Rc accumulates near the upper prior boundary, indicating that the crossover scale remains poorly constrained by current observations. We further show that the inferred constraint on D depends sensitively on the assumed prior range of Rc, which determines the characteristic distance scale at which deviations from General Relativity become significant. Our results provide the first GWTC-4 dark-siren constraints on higher-dimensional GW propagation and demonstrate that current observations remain consistent with four-dimensional General Relativity.