The formation, merging, and accretion history of massive black holes along the hierarchical build–up of cosmic structures leaves a unique imprint on the background of gravitational waves at mHz frequencies. We study here, by means of dedicated simulations of black hole build–up, the possibility of constraining different models of black hole cosmic evolution using future gravitational wave space–borne missions, such as LISA. We consider two main scenarios for black hole formation, namely, one where seeds are light ( ≃ 10 2 M⊙, remnant of Population III stars), and one where seeds are heavy ( ∼>10 4 M⊙, direct collapse). In all the models we have investigated, massive black hole binary coalescences do not produce a stochastic GW background, but rather, a set of individual resolved events. Detection of several hundreds merging events in a 3 year LISA mission will be the sign of a heavy seed scenario with efficient formation of black hole seeds in a large fraction of high redshift halos. At the other extreme, a low event rate, about a few tens in 3 years, is peculiar of scenarios where either the seeds are light, and many coalescences do not fall into the LISA band, or seeds ar
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