The cosmic distance duality relation with strong lensing and gravitational waves: an opacity-free test

The cosmic distance duality relation (CDDR) is a fundamental assumption in cosmological studies. Given the redshift $z$, it relates luminosity distance $D^L$ with angular diameter distance $D^A$ through $(1+z)^2D^A/D^L\equiv1$. Many efforts have been devoted to test the CDDR with various observational approaches. However, to the best of our knowledge, those methods are always affected by cosmic opacity which could make the CDDR violated due to the non-conservation of photon number. Such mechanism is more related with astropartical physics. In this work, to directly study the nature of space-time, i.e., to disentangle it from astropartical physics, we propose a new strategy to test the CDDR with strong lensing providing $D^A$ and gravitational wave providing $D^L$. It is known that the propagation of gravitational wave is unaffected by cosmic opacity. We demonstrate distances from optical lensing observation are also opacity-free. These two kinds of distance measurements make it possible to test any metric theories of gravity where photons travel along null geodesics. Our results show the constraints on deviations of the CDDR will be very competitive to current techniques.