Extreme Mass Ratio Inspirals in Galactic Nuclei: Formation, Dynamics, and Predictions for LISA

Extreme mass ratio inspirals (EMRIs) are among the most promising gravitational wave sources for the upcoming Laser Interferometer Space Antenna (LISA). These binary systems consist of a stellar-mass compact object orbiting a massive black hole. Their long-lived, highly structured signals will enable precision tests of general relativity, measurements of black hole masses and spins, and investigations of galactic nuclear environments. Despite their scientific importance, accurate theoretical modelling of EMRI formation mechanisms and predictions for their typical orbital parameters remain an open challenge, largely due to the complexity of the relativistic, multi-body dynamics governing their formation. This thesis presents a series of advances in the modelling of EMRI formation and evolution, focusing primarily on the standard two-body relaxation channel. In this picture, a stellar-mass black hole orbiting a massive black hole in a dense nuclear star cluster undergoes frequent, weak gravitational encounters with the surrounding stars; the cumulative effect of these encounters is a diffusive random walk in the orbital energy and angular momentum. Because the former relaxes on a much longer timescale than the latter, it is primarily the angular momentum that evolves randomly with time, occasionally driving the compact object onto a highly eccentric orbit that brings it close to the massive black hole. If the pericentre is small enough, gravitational wave emission eventually takes over forming an EMRI, and drives the system to merger after thousands of orbital cycles. If instead a single encounter drives the pericentre within the innermost stable orbit radius, the compact object is swallowed by the massive black hole directly, forming a so-called direct plunge. We develop a novel post-Newtonian Monte Carlo code that evolves the orbital dynamics of a stellar-mass black hole around a non-spinning central massive black hole. We integrate the system at 2.5 post-Newtonian order, the lowest order at which gravitational wave radiation reaction enters the equations of motion, and we incorporate the effects of two-body relaxation locally along the orbit rather than through the standard orbit-averaging approximation. Applying this framework to nuclear star clusters across a range of massive black hole masses, we study the ratio of EMRIs to direct plunges as a function of the initial orbital semi-major axis, which has direct implications for estimating EMRI rates. We furthermore confirm the existence of cliffhanger EMRIs, a population of inspirals forming from initially wide orbits through a single powerful gravitational wave burst near the separatrix which splits bound and plunging orbits, and show that they are common for low massive black hole masses. We further show that both the post-Newtonian treatment of the orbital dynamics and the local description of two-body relaxation substantially affect the predicted EMRI populations, highlighting the shortcomings of approximations widely adopted in the literature. Building on these results, we derive realistic eccentricity distributions for EMRIs at different stages of their inspiral, up to the final plunge, by integrating the endpoint configurations of our Monte Carlo simulations using the FastEMRIWaveforms (FEW) package, and weighting each trajectory by its astrophysical formation rate. We find that EMRIs retain significant eccentricities at plunge, with initially wide, cliffhanger EMRIs being systematically more eccentric than those formed from tighter orbits through the standard two-body relaxation channel. We further demonstrate that the numerical flux grids currently implemented in the FEW package do not adequately cover the full orbital parameter space occupied by the astrophysical EMRI population, particularly for lower massive black hole masses. This represents a significant limitation for LISA data analysis and motivates the development of extended waveform models and detection algorithms in the so-called global-fit effort. We further extend our modelling of EMRI formation beyond the two-body relaxation channel by studying EMRIs produced through the Hills mechanism, in which the tidal separation of a stellar-mass black hole binary by the central massive black hole deposits one component onto a tight orbit around it. We develop a four-step framework that follows the deposition of black hole binaries within their tidal radius via two-body relaxation, their tidal disruption during a close encounter with the massive black hole, the subsequent relaxation and gravitational wave driven evolution of the deposited black hole, and its final relativistic inspiral to plunge. We find that Hills EMRIs reach substantially lower eccentricities than those formed through the relaxation channel, both at plunge and at entry into the LISA band, because they largely retain the pericentre distance set by the original binary, rather than diffusing to progressively smaller pericentres as their relaxation-channel counterparts do. This distinction favours more monochromatic, longer-lived gravitational wave signals, and offers a possible explanation for the low eccentricities inferred for the EMRIs thought to source part of the recurring soft X-ray flares observed from some galactic nuclei, dubbed quasi-periodic eruptions. At the same time, we show the Hills channel to be intrinsically less efficient at producing EMRIs than two-body relaxation, so that only the most circular observed EMRIs might be attributable to it, unless a large number of black hole binaries is present in nuclear star clusters. Finally, we outline early progress towards extending our formation models to spinning massive black holes. Using an approximate separatrix-shift prescription, we extend the eccentricity distributions derived for non-spinning massive black holes to the Kerr case, finding that prograde orbits, despite starting their inspiral at higher eccentricity, circularise more efficiently and end up more circular at plunge than retrograde orbits. We further present the development of a numerical framework that integrates the exact geodesic dynamics of a compact object around a Kerr massive black hole, self-consistently coupled to a spin and inclination dependent treatment of the loss cone, laying the groundwork for fully self-consistent EMRI formation simulations in Kerr spacetime. Taken together, the results presented in this thesis substantially revise the theoretical picture of EMRI formation and observable properties, with direct implications for LISA science exploitation, waveform development, and our understanding of the dynamics of galactic nuclei.