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LISA, the Laser Interferometer Space Antenna, will usher in a new era in gravitational-wave astronomy. As the first anticipated space-based gravitational-wave detector, it will expand our view to the millihertz gravitational-wave sky, where a spectacular variety of interesting new sources abound: from millions of ultra-compact binaries in our Galaxy, to mergers of massive black holes at cosmological distances; from the beginnings of inspirals that will venture into the ground-based detectors' view to the death spiral of compact objects into massive black holes, and many sources in between. Central to realising LISA's discovery potential are waveform models, the theoretical and phenomenological predictions of the pattern of gravitational waves that these sources emit. This white paper is presented on behalf of the Waveform Working Group for the LISA Consortium. It provides a review of the current state of waveform models for LISA sources, and describes the significant challenges that must yet be overcome., Comment: 239 pages, 11 figures, white paper from the LISA Consortium Waveform Working Group, invited for submission to Living Reviews in Relativity, updated with comments from community

In this paper, we investigate the dynamics of a binary system that orbits a rotating supermassive black hole. Our approach employs Fermi-Walker transport to construct a local inertial reference frame, and to set up a Newtonian binary system. We consider a scenario in which a circular geodesic observer is positioned around a Kerr black hole, and thereby derive the equations of motion governing the binary system. To eliminate the interaction terms between the center of mass (CM) of the binary and its relative coordinates, we introduce a small acceleration for the observer. This adjustment leads to the CM closely following the observer's orbit, deviating from a circular geodesic. Here, we first focus on elucidating the stability conditions in a hierarchical triple system. Subsequently, we discuss the phenomenon of von Zeipel-Lidov-Kozai (vZLK) oscillations, which manifest when the binary system is compact and the initial inclination exceeds a critical angle. In hard binary systems, these oscillations exhibit regular behavior, while in soft binary systems, they exhibit a chaotic character, characterized by irregular periods and amplitudes, albeit remaining stable. Additionally, we observe an orbital flip under circumstances of large initial inclination. As for the motion of the CM, we observe deviations from a purely circular orbit that transform into stable yet chaotic oscillations characterized by minute amplitude variations., Comment: 32 pages, 18 figures. arXiv admin note: substantial text overlap with arXiv:2303.16553

We discuss motion of a binary system around a supermassive black hole. Using Fermi-Walker transport, we construct a local inertial reference frame and set up a Newtonian binary system. Assuming a circular geodesic observer around a Schwarzschild black hole, we write down the equations of motion of a binary. Introducing a small acceleration of the observer, we remove the interaction terms between the center of mass (CM) of a binary and its relative coordinates. The CM follows the observer's orbit, but its motion deviates from an exact circular geodesic. We first solve the relative motion of a binary system, and then find the motion of the CM by the perturbation equations with the small acceleration. We show that there appears the Kozai-Lidov (KL) oscillations when a binary is compact and the initial inclination is larger than a critical angle. In a hard binary system, KL oscillations are regular, whereas in a soft binary system, oscillations are irregular both in period and in amplitude, although stable. We find an orbital flip when the initial inclination is large. As for the motion of the CM, the radial deviations from a circular orbit become stable oscillations with very small amplitude., Comment: 34 pages,12 figures, 2 tables

Extreme-mass-ratio inspirals (EMRIs) are promising target sources for space-based interferometers such as LISA, Taiji, and Tianqin. Depending on the astrophysical environment, such as close perturbers or an accretion disk, EMRI orbital evolution may deviate from the predictions of general relativity in vacuum. In particular, we focus on the resonance jumps, i.e., the changes of the conserved quantities induced by a stationary perturbation to the background Kerr geometry. Using Hamiltonian formulation, we provide a closed relation between the jump in Carter constant and that in the axial component of angular momentum. It is also shown that the obtained relation is consistent with the fitting formulae computed for the tidal resonance in previous works., Comment: 9 pages, 1 figure, submitted to Class. Quantum Grav

Extreme-mass-ratio inspirals are one of the most exciting and promising target sources for space-based interferometers (such as LISA, TianQin). The observation of their emitted gravitational waves will offer stringent tests on general theory of relativity, and provide a wealth of information about the dense environment in galactic centers. To unlock such potential, it is necessary to correctly characterize EMRI signals. However, resonances are a phenomena that occurs in EMRI systems and can impact parameter inference, and therefore the science outcome, if not properly modeled. Here, we explore how to model resonances and develop an efficient implementation. Our previous work has demonstrated that tidal resonances induced by the tidal field of a nearby astrophysical object alters the orbital evolution, leading to a significant dephasing across observable parameter space. Here, we extensively explore a more generic model for the tidal perturber with additional resonance combinations, to study the dependence of resonance strength on the intrinsic orbital and tidal parameters. To analyze the resonant signals, accurate templates that correctly incorporate the effects of the tidal field are required. The evolution through resonances is obtained using a step function, whose amplitude is calculated using an analytic interpolation of the resonance jumps. We benchmark this procedure by comparing our approximate method to a numerical evolution. We find that there is no significant error caused by this simplified prescription, as far as the astronomically reasonable range in the parameter space is concerned. Further, we use Fisher matrices to study both the measurement precision of parameters and the systematic bias due to inaccurate modeling. Modeling of self-force resonances can also be carried out using the implementation presented in this study, which will be crucial for EMRI waveform modeling., Comment: 18 pages, 14 figures