Spinning particles near Kerr black holes: Orbits and gravitational-wave fluxes through the Hamilton-Jacobi formalism

TL;DR

The paper develops a semi-analytical framework to model extreme mass ratio inspirals with a spinning secondary in Kerr spacetime by solving the linear-in-spin MPD equations via a Hamilton-Jacobi approach, reducing dynamics to first-order form and expressing deviations from referential geodesics in Mino-time. It provides analytic expressions for spin-induced shifts to constants of motion and frequencies, and couples the orbital solutions to a Teukolsky-based solver to compute gravitational-wave energy and angular-momentum fluxes and waveform snapshots, including fully precessing secondary spins. This yields fast, accurate 1PA building blocks for EMRI waveforms and offers flexible parametrizations (FC, DH, FF) to ensure phase coherence and applicability to data-analysis frameworks. The work represents a significant advance in modeling eccentric, precessing spinning binaries at large mass ratios, with direct implications for precise LISA waveform templates and EMRI physics.

Abstract

Extreme mass-ratio inspirals are among the key sources of gravitational waves for the Laser Interferometer Space Antenna space-based gravitational-wave detector. Achieving sufficient accuracy in the gravitational-wave template for these binaries requires modeling the effects of the spin of the comparably light secondary compact object. In this work, we employ the solution of the Hamilton-Jacobi equations for the motion of spinning bodies in Kerr space-time for the first time to obtain general bound orbits. Specifically, we implement a new solver for the Mathisson-Papapetrou-Dixon equations of motion reduced to first-order form. Our approach provide novel semianalytical expressions for the spin corrections to the orbital motion and frequencies, valid for any choice of referential geodesics, and new analytic expressions for the constants of motion shifts. Then, using the Teukolsky formalism, we compute gravitational-wave energy and angular-momentum fluxes sourced by these orbits valid to linear order in secondary spin and provide waveform snapshots corresponding to the motion. The solver and the novel method we have developed substantially improve on previous studies in terms of speed and accuracy. Additionally, we include the full effect of a general precessing secondary spin in the waveform for the first time. As such, it provides a breakthrough building block for the modeling of waveforms of precessing compact binaries at large mass ratios.

Figures (11)

• Figure 1: Plots of trajectories in $z$-$r$ space for increasing spin alignment angle $\varphi_{\mathrm{s}}$ in the "FC" parametrization. The top left plot shows geodesic trajectories for comparison. At $\varphi_{\rm s} =0$ the spin is fully misaligned and oscillating with an independent frequency that externally drives the $r,z$ librations. For $\varphi_{\rm s} = 90^\circ$, the spin is fully aligned and the dynamics are fully determined by the $r,z$ frequencies. Additionally, the radial libration increases while the polar libration decreases as the spin becomes more aligned.
• Figure 2: Same as Fig.\ref{['fig:z_r_phis_FC']}, but for the "DH" parametrization. In this case, both radial and polar libration have oscillations that average to zero by definition.
• Figure 3: Top panel: the evolution of $r_{\mathrm{g}}+q \delta r^{\text{FC}}$ for different values of $\varphi_{\mathrm{s}}$ and fixed secondary spin in the "FC" scheme. Bottom panel: same plot as above but in the "DH" scheme.
• Figure 4: Comparison of the radial and polar trajectories for a perturbed and geodesic orbit with $\varphi_{\mathrm{s}}=45^{\circ}$ and in the "FC" scheme. We evaluate the geodesic quantities using the perturbed mean-anomalies to remove dephasing.
• Figure 5: Comparison of the radial and polar trajectories for a perturbed and geodesic orbit with $\varphi_{\mathrm{s}}=45^{\circ}$ using the fixed "DH" parametrization. We evaluate the geodesic quantities using the geodesic mean-anomalies as is done in Fig. 3 in Ref. Drummond:2022xej.