Strongly lensed gravitational waves as probes to test the cosmic distance duality relation *

Abstract The cosmic distance relation (DDR) associates the angular diameters ( ) and luminosity D_L by a simple formula, i.e., = (1+z)^2D_A . strongly lensed gravitational waves (GWs) provide unique way to measure simultaneously GW source, hence they can be used as probes test DDR. In this study, we investigated use of events from future Einstein Telescope We assumed possible deviation DDR (1+z)^2D_A/D_L \eta(z) , considered two different parametrizations namely, \eta_1(z) 1+\eta_0 z \eta_2(z) z/(1+z) Numerical simulations showed that, with about 100 observed ET, parameter \eta_0 was constrained at 1.3% 3% levels for first second parametrizations, respectively.