The analytical effective-one-body formalism for extreme-mass-ratio limits: general orbits

Extreme-mass-ratio-inspiral (EMRI) is one of the most important sources for future space-borne gravitational wave detectors. In such kind of systems, the compact objects orbit around the central supermassive black holes with complicated trajectories. Usually, the trajectory is approximated as a geodesic of a test-particle in Kerr space-time in a short duration, and the orbital evolution is simulated with the help of adiabatic approximation. However, this omits the influence of the compact object on the background. In the present paper, employing effective-one-body formalism, we analytically calculate out the trajectories of a compact object around a massive Kerr black hole with generally three-dimension orbits and express the fundamental orbital frequencies in explicit forms. In addition, by constructing an approximate similar with the constant, we transfer the dynamical quantities such as energy, angular momentum and Carter constant to semilatus rectum, eccentricity and orbital inclination with mass-ratio corrections. Our formalism includes the first-order corrections of mass-ratio in the conservative orbital motion in the extreme-mass-ratio limit. Changes of frequencies due to the mass-ratio correction illustrate the test particle approximation should be invalid for constructing the EMRI's waveform templates.