The grand challenges of contemporary fundamental physics—dark matter, dark energy, vacuum energy, inflation and early universe cosmology, singularities and the hierarchy problem—all involve gravity as a key component. And of all gravitational phenomena, black holes stand out in their elegant simplicity, while harbouring some of the most remarkable predictions of General Relativity: event horizons, singularities and ergoregions. The hitherto invisible landscape of the gravitational Universe is being unveiled before our eyes: the historical direct detection of gravitational waves by the LIGO-Virgo collaboration marks the dawn of a new era of scientific exploration. Gravitational-wave astronomy will allow us to test models of black hole formation, growth and evolution, as well as models of gravitational-wave generation and propagation. It will provide evidence for event horizons and ergoregions, test the theory of General Relativity itself, and may reveal the existence of new fundamental fields. The synthesis of these results has the potential to radically reshape our understanding of the cosmos and of the laws of Nature. The purpose of this work is to present a concise, yet comprehensive overview of the state of the art in the relevant fields of research, summarize important open problems, and lay out a roadmap for future progress. This write-up is an initiative taken within the framework of the European Action on ‘Black holes, Gravitational waves and Fundamental Physics’.

The dynamics of two massive black holes in a rotationally supported nuclear disc of mass Mdisc= 108 M⊙ is explored using N‐body/smoothed particle hydrodynamics simulations. Gas and star particles are copresent in the disc. Described by a Mestel profile, the disc has a vertical support provided by turbulence of the gas, and by stellar velocity dispersion. A primary black hole of mass 4 × 106 M⊙ is placed at the centre of the disc, while a secondary black hole is set initially on an eccentric corotating orbit in the disc plane. Its mass is in a 1:1, 1:4, and 1:10 ratio, relative to the primary. With this choice, we mimic the dynamics of black hole pairs released in the nuclear region at the end of a gas‐rich galaxy merger. It is found that, under the action of dynamical friction, the two black holes form a close binary in ∼10 Myr. The inspiral process is insensitive to the mass fraction in stars and gas present in the disc and is accompanied by the circularization of the orbit. We detail the gaseous mass profile bound to each black hole that can lead to the formation of two small Keplerian discs, weighing ≈2 per cent of the black hole mass, and of size ∼0.01 pc. The mass of the tightly (loosely) bound particles increases (decreases) with time as the black holes spiral into closer and closer orbits. Double active galactic nucleus activity is expected to occur on an estimated time‐scale of ≲10 Myr, comparable to the inspiral time‐scale. The double nuclear point‐like sources that may appear during dynamical evolution will have typical separations of ≲10 pc.

We study supermassive black holes (BHs) in merging galaxies, using a suite of hydrodynamical simulations with very high spatial (∼10 pc) and temporal (∼1 Myr) resolution, where we vary the initial mass ratio, the orbital configuration, and the gas fraction. (i) We address the question of when and why, during a merger, increased BH accretion occurs, quantifying gas inflows and BH accretion rates. (ii) We also quantify the relative effectiveness in inducing AGN activity of merger-related versus secularrelated causes, by studying different stages of the encounter: the stochastic (or early) stage, the (proper) merger stage, and the remnant (or late) stage. (iii) We assess which galaxy mergers preferentially enhance BH accretion, finding that the initial mass ratio is the most important factor. (iv) We study the evolution of the BH masses, finding that the BH mass contrast tends to decrease in minor mergers and to increase in major mergers. This effect hints at the existence of a preferential range of mass ratios for BHs in the final pairing stages. (v) In both merging and dynamically quiescent galaxies, the gas accreted by the BH is not necessarily the gas with low angular momentum, but the gas that loses angular momentum.

We consider super-critical accretion with angular momentum onto stellar-mass black holes as a possible mechanism for growing billion-solar-mass holes from light seeds at early times. We use the radiatively-inefficient "slim disk" solution -advective, optically thick flows that generalize the standard geometrically thin disk model -to show how mildly super-Eddington intermittent accretion may significantly ease the problem of assembling the first massive black holes when the universe was less than 0.8 Gyr old. Because of the low radiative efficiencies of slim disks around non-rotating as well as rapidly rotating holes, the mass e-folding timescale in this regime is nearly independent of the spin parameter. The conditions that may lead to super-critical growth in the early universe are briefly discussed.