Shape of U: Measuring the Curvature of the Universe with Gravitational Waves

Gravitational waves (GWs) from compact binary mergers are standard sirens that can measure distances across the Universe without external calibrators. When an electromagnetic counterpart enables an independent redshift measurement, such "bright sirens" can be used to probe the expansion history of the Universe and constrain cosmological models. In this work, we investigate the ability of future GW observatories to measure the spatial curvature parameter, $\Omega_{\rm k}$, in a non-flat $\Lambda$CDM cosmology. We focus on intermediate-mass binary black hole mergers (with masses similar to GW231123) as bright siren sources, motivated by their detectability to high redshifts with next-generation ground-based detectors and by the possibility that mergers in active galactic nucleus disks may produce electromagnetic counterparts. Using Fisher matrix forecasts, we find that a network consisting of two Cosmic Explorer detectors and Einstein Telescope can constrain $\Omega_{\rm k}$ to a $1\sigma$ uncertainty of $0.029$ with these bright sirens. We further show that multiband observations with LISA or the Lunar Gravitational Wave Antenna do not significantly improve these cosmological constraints, because the additional signal-to-noise ratios accumulated in their bands are modest. Further, a population of binary neutron stars as bright sirens provides substantially broader constraints on $\Omega_{\rm k}$, with $1\sigma$ error of $0.055$. Our results show that bright intermediate-mass binary black hole and binary neutron star mergers observed with next-generation GW detectors together can provide an independent and informative probe of spatial curvature, with systematics distinct from those of other cosmological observations.