There is strong evidence for a fundamental symbiosis between the growth of black holes in the centres of galaxies and the growth of the galaxies themselves. This evidence includes known close relationship between the central black hole's mass, the surrounding mass in stars and the velocities of the stars.

A number of possible mechanisms for how the close relationship arose have already been investigated. But there is no clear consensus on the driving process on when the relationship was first established, nor on whether the relationship has evolved over cosmic time.

LISA will be sensitive to the crucial mass range that witnesses the growth from black hole seeds to supermassive black holes, corresponding to the first spurts of mass growth. The best diagnostics for tracing this process are black hole spin measurements. They will offer crucial insights on whether accretion occurs continuously from large-scale gas reservoirs in the host galaxies, or episodically by eating small lumps of material chaotically falling onto the black holes. LISA will measure the masses and spins of the black holes prior to coalescence, offering unprecedented details on how black holes are fed by their host galaxy.

LISA observations, when coupled with near-infrared and radio/sub-mm observations of the evolution of the galactic masses, star formation rate and gas content, will help to understand the relationship between the growth processes. A variety of observations will give information on faint or quiescent black holes in the relatively nearby universe, and others on luminous QSOs tracing the most massive black holes undergoing exceptional growth phases, in the farther distant universe. Black hole coalescences, in the new window of LISA, trace the growth history at moderate to large redshifts for up to 10⁷ M⊙ for black holes that may be inactive, thus electromagnetically invisible, and for which we are still blind with current electromagnetic techniques. By measuring black hole masses and spins with extremely high precision in coalescing binaries as a function of redshift, LISA will provide a strong constraint on how, and how fast, black hole masses evolved over cosmic time along with their host galaxies.