Probing modified gravitational wave propagation with strongly lensed coalescing binaries

It has been recently shown that quadruply lensed gravitational-wave (GW) events due to coalescing binaries can be localized to one or just a few galaxies, even in the absence of an electromagnetic counterpart. We discuss how this can be used to extract information on modified GW propagation, which is a crucial signature of modifications of gravity at cosmological scales. We show that, using quadruply lensed systems, it is possible to constrain the parameter $\Xi_0$ that characterizes modified GW propagation, without the need of imposing a prior on $H_0$. A LIGO/Virgo/Kagra network at target sensitivity might already get a significant measurement of $\Xi_0$, while a third generation GW detector such as the Einstein Telescope could reach a very interesting accuracy.