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

The cosmic distance relation(DDR) associates the angular diameters distance(D) and luminosity distance(D) by a simple formula,i.e.,D=(1+z)~2 D.The strongly lensed gravitational waves(GWs) provide a unique way to measure Dand Dsimultaneously to the GW source,hence they can be used as probes to test DDR.In this study,we investigated the use of strongly lensed GW events from the future Einstein Telescope to test DDR.We assumed the possible deviation of DDR as(1+z)~2 DA/D=η(z),and considered two different parametrizations of η(z),namely,η1(z)=1+ηz and η2(z)=1+ηz/(1+z).Numerical simulations showed that,with about 100 strongly lensed GW events observed by ET,the parameter ηwas constrained at 1.3% and 3% levels for the first and second parametrizations,respectively.