We have discovered a tantalizing connection between the statistical behavior of a stellar system and liquid crystals, and used this connection to construct state-of-the-art numerical models to understand their behavior. This connection is related to the way the orbits of stars and black holes interact gravitationally around a supermassive black hole and relax into a "thermodynamic" equilibrium. Over long timescales, these orbits represent rings, which efficiently reorient due to a process called vector resonant relaxation. The basic ingredient of this process is its Hamiltonian. We have obtained the Hamiltonian of vector resonant relaxation, which is the sum of the pairwise gravitational potential energy between the stellar orbits “smeared out over their precessing orbits”. We have shown that this Hamiltonian is algebraically very similar to the Hamiltonian of liquid crystals. This tantalizing analogy may provide great insight for understanding the behavior of stellar orbits in galactic nuclei. Using the Hamiltonian we have constructed a numerical algorithm, N-ring, to simulate the evolution as a function of time. We constructed a stochastic model to analytically understand the time-evolution during vector resonant relaxation. We derived the statistical equilibrium of vector resonant relaxation for a one-component system using mean field theory including the effect of angular momentum conservation. We showed that the system exhibits phase transitions. The system forms an ordered disk at low temperatures and a nearly isotropic disordered configuration at high temperatures. We also found stable negative absolute temperature states which are nearly isotropic. The effect of angular momentum is similar to the effect of a background magnetic field for liquid crystals. The overall implication is that stellar mass black holes are distributed in a disk in the centers of galaxies.
We have also examined the composition and distribution of objects in galactic nuclei. The gamma ray emission from the Galactic Center reveals a population of objects, magnetized rapidly spinning neutron stars, which were formed in dense stellar environments like globular clusters orbiting in the halo of the Galaxy. These clusters sank to the Galactic Center due to "dynamical friction" and facilitate the formation of the Galactic Center.
We have examined the distribution of binaries in the Galactic center as they interact with the star cluster and the central supermassive black hole. We found that a significant fraction of the binaries get destroyed within their lifetime. In active galaxies where gas falls onto the supermassive black hole to produce spectacularly luminous radiation in a disk, stars and black hole binaries get captured by the gaseous disk. These sources may be important sources of gravitational waves. We found that black holes get efficiently captured in the disk and merge due to hydodynamical interaction with gas. This produces hierarchical black hole mergers in active galactic nuclei detectable with gravitational wave instruments such as LIGO, VIRGO, and KAGRA, and may lead to the formation of intermediate mass black holes.
We constructed methods to examine the astrophysical origin of black hole mergers discovered by LIGO. In galactic nuclei we showed that the Kozai-Lidov process driven by the supermassive black hole may produce mergers observable by LIGO. We also showed that triple and quadruple systems also facilitate the mergers of black holes and lead to tidal dispruption events.
We showed that the distribution of astrophysical parameters is different for different astrophysical processes leading to a merger. The event rate distribution of gravitational wave merger events may have implications on the source environment of the observed black hole mergers with Earth-based detectors. In particular we determined the mass distribution of gravitational wave capture binaries in galactic nuclei, primoridial black hole binaries formed in the early universe, and the binaries formed dynamically in dense stellar systems without a supermassive black hole such as globular clusters.
