Probing the formation of the seeds of supermassive black holes with gravitational waves

The existence of black holes in the intermediate-mass interval between 10²M_⊙ and 10⁵M_⊙, filling the gap between the stellar and the supermassive black holes is a key prediction to explain the origin of luminous QSOs at redshifts z as large as 7. There is a sheer difficulty in forming a 10⁹M_⊙ black hole in less than one billion years. This has led to the concept of seed black holes. They are high-z intermediate-mass black holes that formed during the dark ages of the universe. Seeds are a transient population, which later grew massive through episodes of accretion and mergers. In this chapter we explore the possibility of discovering seed black holes and track their growth across all cosmic epochs, by detecting the gravitational wave signal they emit at the time of their coalescence, if/when they pair to form close binaries. We show that the ESA's LISA mission for the detection of low-frequency gravitational waves will be paramount in granting this insight. Gravitational waves travel unimpeded through the cosmos and carry exquisite information on the masses and spins of the merging black holes. To this purpose we introduce key concepts on the gravitational wave emission from binaries, describing briefly their formation pathway during halo-halo mergers and galaxy collisions.