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

Abstract: 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, $Ω{\rm k}$, in a non-flat $Λ$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 $Ω{\rm k}$ to a $1σ$ 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 $Ω_{\rm k}$, with $1σ$ 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.

• The paper leverages Fisher forecasting on bright siren data from IMBBHs and BNSs to achieve percent-level H0 and Omega_m constraints.
• It demonstrates that enhanced sky localization from detectors like LGWA significantly improves EM follow-up for host galaxy identification.
• Although GW-derived Omega_k constraints remain looser than those from CMB+BAO, they provide crucial, systematics-independent cross-checks for cosmology.

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

Motivation and Context

The Lambda-CDM model underpins contemporary cosmology, but persistent tensions in parameter estimates—particularly $H_0$ and $S_8$—have prompted interest in independent probes. Gravitational wave (GW) standard sirens, leveraging compact binary merger events, offer a calibration-independent means of measuring cosmic distances and expansion. Direct redshift measurements are accessible when electromagnetic (EM) counterparts are identified, yielding "bright sirens" as opposed to statistical ("dark") sirens. This work quantitatively investigates the constraining power of future GW observatories, focusing on the spatial curvature parameter $\Omega_{\rm k}$ within non-flat Lambda-CDM, utilizing both intermediate-mass binary black holes (IMBBHs) and binary neutron stars (BNSs) as bright sirens.

Detectors and GW Source Populations

The study considers advanced terrestrial GW detectors—two Cosmic Explorer (CE) sites and Einstein Telescope (ET)—with prospective sensitivity extending to $z\sim10$ due to improved low-frequency performance. Non-terrestrial detectors, LISA and the Lunar Gravitational Wave Antenna (LGWA), augment observational coverage in sub-Hz bands, facilitating multiband analysis for IMBBHs, whose inspiral phases persist at low frequencies.

IMBBHs, exemplified by GW231123 ($\sim236\,M_\odot$ at $z\sim0.4$), are chosen for their merger rate uncertainty and detectability. Three merger rate density models—uniform in comoving volume (pragmatic), Madau-Dickinson-Belczynski-Ng (optimistic), and Fragione+ (pessimistic)—parameterize population scenarios. BNSs are modeled with a double Gaussian mass distribution and the Madau-Dickinson redshift evolution, constrained to $z<5$ and inclination $<30^\circ$ to reflect EM counterpart visibility.

Figure 1: Merger rate densities for IMBBH populations across pragmatic, optimistic, and pessimistic scenarios.

EM counterparts for IMBBHs are plausible in AGN environments, through post-merger flares, jets, and disk interactions, enabling unique host identification as bright sirens. However, their realistic occurrence rate remains speculative; the association of GW190521 with ZTF19abanrhr demonstrates only marginal evidence. BNS mergers robustly produce EM signals (sGRB, kilonova, afterglows), enabling accurate redshift determination.

Methodology: Forecasting Cosmological Constraints

The primary analytic tool is Fisher information forecasting, estimating $1\sigma$ uncertainties in binary parameters (luminosity distance $D_L$, masses, spins, extrinsic properties) propagated to cosmological parameters ($S_8$0, $S_8$1, $S_8$2). SNR computation employs IMRPhenomXPHM and IMRPhenomPv2_NRtidalv2 approximants for IMBBHs and BNSs, respectively; events with ground-based SNR $S_8$3 are selected. Multiband detectability requires SNR $S_8$4 in LISA/LGWA.

Figure 2: Left: Cumulative SNR distribution of IMBBH events in LISA and LGWA; Right: Cumulative sky localization errors for 2CE+ET, 2CE+ET+LISA, and 2CE+ET+LGWA networks.

Figure 3: Relative luminosity distance errors versus redshift for IMBBH and BNS populations in a 2CE+ET network.

Cosmological parameter constraints are synthesized for IMBBH and BNS cohorts, utilizing realistic redshift distributions and propagation of $S_8$5 uncertainties. The analysis also explores multiband enhancements and the additional benefit of LGWA for sky localization in EM follow-up.

Numerical Results and Comparative Analysis

For one year of GW observation with 2CE+ET, $S_8$6 IMBBH bright sirens (pragmatic case) yield $S_8$7 uncertainties of $S_8$8 in $S_8$9, $\Omega_{\rm k}$0 in $\Omega_{\rm k}$1, and $\Omega_{\rm k}$2 in $\Omega_{\rm k}$3; BNS bright sirens provide $\Omega_{\rm k}$4, $\Omega_{\rm k}$5, and $\Omega_{\rm k}$6, respectively. Even in the optimistic scenario ($\Omega_{\rm k}$7 annual IMBBHs), the forecasted $\Omega_{\rm k}$8 uncertainty is an order of magnitude larger than CMB+BAO constraints (Planck 2018: $\Omega_{\rm k}$9).

Figure 4: $z\sim10$0 corner plots for cosmological parameters comparing IMBBH and BNS populations across detector networks.

Figure 5: Median fractional errors in $z\sim10$1 and $z\sim10$2; median absolute error in $z\sim10$3 for three detector networks, under different merger rates.

Multiband inclusion of LISA or LGWA confers only modest gains in SNRs and cosmological parameter uncertainties; improvement is limited due to low SNRs in non-terrestrial bands for GW231123-like IMBBHs. However, LGWA yields up to two orders of magnitude enhancement in sky localization (median $z\sim10$4 area $z\sim10$5 deg$z\sim10$6), critically useful for EM counterpart identification.

Combined IMBBH+BNS data marginally tightens cosmological constraints, with IMBBH dominance due to superior SNRs and distance accuracy. High-redshift events contribute less to parameter constraints due to increased $z\sim10$7 uncertainty; most constraining power derives from low-$z\sim10$8, high-SNR mergers.

Figure 6: $z\sim10$9 corner plots for cosmological parameters with GW231123-like and GW190521-like IMBBH populations sampled from a uniform in comoving volume merger rate.

Theoretical and Practical Implications

The forecasted constraints show that GW bright siren cosmology will not surpass current EM-based bounds (CMB+BAO) on $\sim236\,M_\odot$0 in precision, even with optimistic IMBBH merger rates and network coverage. Nevertheless, these sirens provide systematics-distinct measurements, valuable for cross-validating flatness and probing beyond-Lambda-CDM scenarios. Strong parameter degeneracy, especially between $\sim236\,M_\odot$1 and $\sim236\,M_\odot$2, accentuates sensitivity to population composition and source parameter estimation.

Improvement in sky localization with LGWA or similar lunar/space detectors will substantially increase the fraction of uniquely identified bright sirens, facilitating host galaxy assignment and redshift measurement—crucial for expanding the cosmological utility of GW observations. Population uncertainties for IMBBHs and realistic EM counterpart rates remain a limiting factor; future work should characterize population-averaged IMBBH properties and merger environments more robustly.

Future Directions

To further constrain cosmological parameters, particularly in extended cosmological models (e.g., $\sim236\,M_\odot$3CDM, $\sim236\,M_\odot$4CDM), incorporation of next-generation global networks—LIGO-India, TianQin, Taiji, B-DECIGO—and improved source modeling (including lensing and calibration systematics) will be essential. Enhanced multiband strategies with higher-mass IMBBHs or improved low-frequency detector capabilities could potentially yield tighter bounds. Integration of GW cosmology with EM observations will remain critical for resolving parameter tensions and investigating new physics.

Conclusion

This study demonstrates that future GW detector networks, especially through IMBBH bright sirens, can provide percent-level constraints on $\sim236\,M_\odot$5 and $\sim236\,M_\odot$6, with spatial curvature limits several times weaker than current CMB+BAO measurements. LGWA enhances practical utility via improved sky localization. While GW-based cosmological parameter estimation will not supersede EM probes in precision, it will supply valuable, independent constraints and cross-checks, particularly relevant in the context of persistent cosmological tensions and prospective deviations from the standard model.