An opacity-free method of testing the cosmic distance duality relation using strongly lensed gravitational wave signals

Abstract

The cosmic distance duality relation (CDDR), expressed as $D_L\left(z\right)={(1+z)}^2{D}_A\left(z\right)$, plays an important role in modern cosmology. In this paper, we propose a new method of testing CDDR using strongly lensed gravitational wave (SLGW) signals. Under the geometric optics approximation, we calculate the gravitational lens effects of two lens models, the point mass and singular isothermal sphere. We use functions of ${\eta }_1\left(z\right)=1+{\eta }_{0}z$ and ${\eta }_2\left(z\right)=1+{\eta }_{0}z/(1+z)$ to parameterize the deviation of CDDR. By reparameterizing the SLGW waveform with CDDR and the distance-redshift relation, we include the deviation parameters ${\eta }_0$ of CDDR as waveform parameters. We evaluate the ability of this method by calculating the parameter estimation of simulated SLGW signals from massive binary black holes. We apply the Fisher information matrix and Markov Chain Monte Carlo methods to calculate parameter estimation. We find that with only one SLGW signal, the measurement precision of ${\eta }_0$ can reach a considerable level of 0.5-1.3% for ${\eta }_1\left(z\right)$ and 1.1-2.6% for ${\eta }_2\left(z\right)$, depending on the lens model and parameters.