Abstract: In this talk I will discuss how we can use the cross-correlation of gravitational wave detections with the large-scale structure of the Universe to test cosmological models in a novel way that is complementary to standard measurements.
I will introduce the formalism and present forecasts obtained for a series of future experiments, focusing in particular on primordial black holes, testing early Universe physics and deviations from general relativity.
Finally, I will introduce some recent developments on how to include unequal time correlations.