An Opacity-Free Test of the Cosmic Distance Duality Relation Using Strongly Lensed Gravitational Wave Signals with Space-Based Detector Networks

Abstract

The cosmic distance duality relation (CDDR), expressed as $d_L(z) = (1+z)^2 D_A(z)$, is a fundamental relation in modern cosmology. In this work, we apply a method to test the CDDR using simulated strongly lensed gravitational-wave (SLGW) signals from massive binary black holes (MBBH) as observed by proposed space-based detector networks. Our analysis is conducted under the point-mass lens model, considering the strong lensing scenario that produces two images. We generate 90 days of simulated SLGW data for 10 events based on the Population III stellar formation model, with source redshifts in the range $z_s \in [2,6]$ and lens redshifts in $z_L \in [0.2,1]$. The deviation of CDDR is parameterized by $η_1(z) = 1 + η_0 z$ and $η_2(z) = 1 + η_0 z/(1+z)$, and we incorporate the deviation parameter $η_0$ directly into the waveform model. Parameter estimation is performed within a Bayesian statistical framework, combining simulated data from both Taiji and LISA. For a single lensed event, the joint Taiji+LISA analysis improves the measurement precision of $η_0$ by roughly a factor of two compared with Taiji-only observations. By combining 10 simulated events, the population-level constraints on $η_0$, quantified by the half width of the $95\%$ credible interval, reach approximately $2.61\times10^{-4}$ ($1.72\times10^{-4}$) for the $η_1(z)$ parameterization and $1.22\times10^{-3}$ ($6.86\times10^{-4}$) for $η_2(z)$ in the Taiji-only (Taiji+LISA) scenario, respectively. The inferred values of $η_0$ remain consistent with $η_0 = 0$ within the estimated uncertainties, with no statistically significant evidence for deviations from the CDDR at the achieved precision. These results demonstrate the significant advantage of joint space-based observations for high-precision tests of the CDDR.

Figures (4)

• Figure 1: Schematic illustration of GW lensing. The vectors $\eta_L$ and $\xi$ represent the positions on the source plane and the lens plane, respectively. The angular diameter distances $D_A^{l}$, $D_A^{s}$, and $D_A^{ls}$ are measured in the observer's rest frame.
• Figure 2: Posterior distributions of the source, lensing, and cosmological parameters for a representative SLGW event, assuming a parameterized deviation from the CDDR of the form $\eta_1(z) = 1 + \eta_0 z$. Blue contours denote the constraints obtained with Taiji alone, while orange contours correspond to the joint Taiji--LISA analysis. The inner and outer contours enclose the 50% and 95% credible regions, respectively. The red vertical and horizontal lines indicate the injected values of the parameters. The inferred posterior of $\eta_0$ remains compatible with $\eta_0 = 0$ within the quoted uncertainties, indicating no statistically significant evidence for a deviation from the CDDR in this event.
• Figure 3: Same as Fig. \ref{['fig:case1']}, but assuming an alternative parameterization of the deviation from the CDDR, $\eta_2(z) = 1 + \eta_0 z/(1+z)$. The inferred posterior of $\eta_0$ is also consistent with $\eta_0 = 0$, indicating no statistically significant deviation from the CDDR.
• Figure 4: Population-level constraints on the deviation parameter $\eta_0$ obtained by progressively combining $10$ lensed events. Top panel: results assuming the parametrization $\eta_1(z) = 1 + \eta_0 z$. Bottom panel: results assuming $\eta_1(z) = 1 + \eta_0 z/(1+z)$. Blue and orange violins correspond to Taiji-only and Taiji+LISA observations, respectively. The horizontal dotted line indicates the fiducial value $\eta_0 = 0$. In both case, the population posteriors remain compatible with $\eta_0 = 0$, indicating no evidence for a violation of the CDDR.