Spinning test-body orbiting around Schwarzschild black hole: circular dynamics and gravitational-wave fluxes

TL;DR

This paper compares different formalisms for modeling a spinning test-body orbiting a Schwarzschild black hole, analyzing their effects on orbital dynamics and gravitational-wave fluxes, with implications for waveform modeling in gravitational wave astronomy.

Contribution

It provides the first detailed comparison of multiple spin-dynamics prescriptions, including the Mathisson-Papapetrou and effective-one-body formalisms, in the context of circular orbits around black holes.

Findings

Different prescriptions agree at low frequencies and small spins.

Significant discrepancies appear at higher frequencies and positive spins.

ISCO shifts can differ up to a factor of two for large positive spins.

Abstract

We consider a spinning test-body in circular motion around a nonrotating black hole and analyze different prescriptions for the body's dynamics. We compare, for the first time, the Mathisson-Papapetrou formalism under the Tulczyjew spin-supplementary-condition (SSC), the Pirani SSC and the Ohashi-Kyrian-Semerak SSC, and the spinning particle limit of the effective-one-body Hamiltonian of [Phys.~Rev.~D.90,~044018(2014)]. We analyze the four different dynamics in terms of the ISCO shifts and in terms of the coordinate invariant binding energies, separating higher-order spin contributions from spin-orbit contributions. The asymptotic gravitational wave fluxes produced by the spinning body are computed by solving the inhomogeneous $(2+1)D$ Teukolsky equation and contrasted for the different cases. For small orbital frequencies $\Omega$, all the prescriptions reduce to the same dynamics and the same radiation fluxes. For large frequencies, ${x \equiv (M \Omega)^{2/3} >0.1 }$, where $M$ is the black hole mass, and especially for positive spins (aligned with orbital angular momentum) a significant disagreement between the different dynamics is observed. The ISCO shifts can differ up to a factor two for large positive spins; for the Ohashi-Kyrian-Semerak and the Pirani SSC the ISCO diverges around dimensionless spins $\sim0.52$ and $\sim0.94$ respectively. In the spin-orbit part of the energetics the deviation from the Hamiltonian dynamics is largest for the Ohashi-Kyrian-Semerak SSC; it exceeds $10\%$ for $x>0.17$. The Tulczyjew and the Pirani SSCs behave compatible across almost the whole spin and frequency range. Our results will have direct application in including spin effects to effective-one-body waveform models for circularized binaries in the extreme-mass-ratio limit.