The dynamics of black hole seeds in high redshift galaxies is key to understand their ability to grow via accretion and to pair in close binaries during galactic mergers. To properly follow the dynamics of black holes we develop a physically motivated model to capture unresolved dynamical friction from stars, dark matter and gas. We first validate the model and then we use it to investigate the dynamics of seed black holes born at z ∼ 9 in dwarf proto-galaxies. We perform a suite of zoom cosmological simulations with spatial resolution as high as 10 pc and with a stellar and dark matter mass resolution of 2×10 3 M and 2×10 5 M respectively. We first explore the dynamics of a seed black hole in the galaxy where it is born and show that it is highly erratic if the seed mass is less than 10 5 M . The dynamics is dominated by the stellar component, whose distribution is irregular and patchy, thus inducing stochasticity in the orbits: the black hole may be anywhere in the proto-galaxy. When this dwarf merges into a larger galaxy, it is paramount to simulate the process with very high spatial and mass resolution in order to correctly account for the stripping of the stellar envelope of the satellite black hole. The outcome of the encounter could be either a tight binary or, at least temporary, a wandering black hole, leading to multiple black holes in a galaxy, each inherited from a different merger.

Massive black hole (MBH) coalescences are powerful sources of low-frequency gravitational waves. To study these events in the cosmological context we need to trace the large-scale structure and cosmic evolution of a statistical population of galaxies, from dim dwarfs to bright galaxies. To cover such a large range of galaxy masses, we analyse two complementary simulations: Horizon-AGN with a large volume and low resolution which tracks the high-mass (>107 M⊙) MBH population, and NewHorizon with a smaller volume but higher resolution that traces the low-mass (<107 M⊙) MBH population. While Horizon-AGN can be used to estimate the rate of inspirals for Pulsar Timing Arrays, NewHorizon can investigate MBH mergers in a statistical sample of dwarf galaxies for LISA, which is sensitive to low-mass MBHs. We use the same method to analyse the two simulations, post-processing MBH dynamics to account for time delays mostly determined by dynamical friction and stellar hardening. In both simulations, MBHs typically merge long after galaxies do, so that the galaxy morphology at the time of the MBH merger is no longer determined by the structural disturbances engendered by the galaxy merger from which the MBH coalescence has originated. These time delays cause a loss of high-z MBH coalescences, shifting the peak of the MBH merger rate to z ∼ 1 − 2. This study shows how tracking MBH mergers in low-mass galaxies is crucial to probing the MBH merger rate for LISA and investigate the properties of the host galaxies.