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 $\eta_1(z) = 1 + \eta_0 z$ and $\eta_2(z) = 1 + \eta_0 z/(1+z)$, and we incorporate the deviation parameter $\eta_0$ 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 $\eta_0$ by roughly a factor of two compared with Taiji-only observations. By combining 10 simulated events, the population-level constraints on $\eta_0$, quantified by the half width of the $95\%$ credible interval, reach approximately $2.61\times10^{-4}$ ($1.72\times10^{-4}$) for the $\eta_1(z)$ parameterization and $1.22\times10^{-3}$ ($6.86\times10^{-4}$) for $\eta_2(z)$ in the Taiji-only (Taiji+LISA) scenario, respectively. The inferred values of $\eta_0$ remain consistent with $\eta_0 = 0$ 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.