An improved effective-one-body Hamiltonian for spinning black-hole binaries

TL;DR

The authors construct an improved effective-one-body Hamiltonian for spinning black-hole binaries by deriving a spinning test-particle Hamiltonian in axisymmetric spacetimes, specializing to Kerr, and then mapping the two-body problem to a test particle in a deformed Kerr geometry. They combine canonical transformations of the ADM Hamiltonian with a deformed-Kerr metric, ensuring the PN expansion reproduces leading spin-spin and spin-orbit couplings through 2.5PN and 2PN respectively, while maintaining the correct test-particle limit. A deformed metric with a controllable horizon structure via a parameter $K(oldsymbol{ abla})$ yields a physically sensible ISCO and a plunging-frequency peak near the light ring, enabling robust matching to merger-ringdown. The resulting H_real^{ m improved} provides a practical, resummed description of spinning binaries across regimes and offers a flexible framework for including higher-order spin effects and calibrations to numerical-relativity data.

Abstract

Building on a recent paper in which we computed the canonical Hamiltonian of a spinning test particle in curved spacetime, at linear order in the particle's spin, we work out an improved effective-one-body (EOB) Hamiltonian for spinning black-hole binaries. As in previous descriptions, we endow the effective particle not only with a mass m, but also with a spin S*. Thus, the effective particle interacts with the effective Kerr background (having spin S_Kerr) through a geodesic-type interaction and an additional spin-dependent interaction proportional to S*. When expanded in post-Newtonian (PN) orders, the EOB Hamiltonian reproduces the leading order spin-spin coupling and the spin-orbit coupling through 2.5PN order, for any mass-ratio. Also, it reproduces all spin-orbit couplings in the test-particle limit. Similarly to the test-particle limit case, when we restrict the EOB dynamics to spins aligned or antialigned with the orbital angular momentum, for which circular orbits exist, the EOB dynamics has several interesting features, such as the existence of an innermost stable circular orbit, a photon circular orbit, and a maximum in the orbital frequency during the plunge subsequent to the inspiral. These properties are crucial for reproducing the dynamics and gravitational-wave emission of spinning black-hole binaries, as calculated in numerical relativity simulations.

Figures (9)

• Figure 1: The frequency at the EOB ISCO for binaries having spins parallel to $\boldsymbol{L}$, with mass ratio $q=m_2/m_1$ and with spin-parameter projections onto the direction of $\boldsymbol{L}$ given by $\chi_1=\chi_2=\chi$. As expected, the frequency increases with $\chi$ for a given mass ratio, while for fixed $\chi$ it increases with $q$ if $\chi\lesssim 0.9$, while it decreases with $q$ if $\chi$ is almost extremal (see text for details).
• Figure 2: The final spin parameter $\chi_{\rm fin}$ as inferred at the EOB ISCO, for binaries having spins parallel to $\boldsymbol{L}$, with mass ratio $q=m_2/m_1$ and with spin-parameter projections onto the direction of $\boldsymbol{L}$ given by $\chi_1=\chi_2=\chi$. As expected, $\chi_{\rm fin}$ flattens for large $\chi$ in the comparable mass case (see text for details).
• Figure 3: The final spin parameter $\chi_{\rm fin}$ as inferred at the EOB ISCO for binaries having spins parallel to $\boldsymbol{L}$, with mass ratio $q=m_2/m_1$ and with spin-parameter projections onto the direction of $\boldsymbol{L}$ given by $\chi_1=\chi_2=\chi$, compared to the remnant's final spin parameter predicted by the formula presented in Ref. spin_formula ("BR09"), which accurately reproduces numerical-relativity results. The EOB model and the BR09 formula agree when the mass ratio is small ($q=0.1$), because the emission during the plunge, merger and ringdown is negligible in this case. For $q=0.5$ and $q=1$, there is an offset, because the EOB result, at this stage, neglects the gravitational-wave emission during the plunge, merger and ringdown (see text for details).
• Figure 4: The mass loss inferred at the EOB ISCO for binaries having spins parallel to $\boldsymbol{L}$, with mass ratio $q=m_2/m_1$ and with spin-parameter projections onto the direction of $\boldsymbol{L}$ given by $\chi_1=\chi_2=\chi$, compared to the total mass lost during the inspiral, merger and ringdown, as predicted by the formulas presented in Ref. aei_aligned_spins ("AEI09") and in Ref. RITfit ("RIT09"), which reproduce numerical-relativity results, although with different accuracies because of the different parameter regions they cover (see the text for details). The EOB model and the AEI09 and RIT09 fits agree when the mass ratio is small ($q=0.1$), while there is an offset for $q=0.5$ and $q=1$. The reason is that the ringdown emission, which is negligible for small mass-ratios, is not taken into account by our EOB model at this stage.
• Figure 5: The frequency at the EOB ISCO for binaries having spins parallel to $\boldsymbol{L}$, with mass ratio $q=m_2/m_1$ and with spin-parameter projections onto the direction of $\boldsymbol{L}$ given by $\chi_1=-\chi_2=\chi$. As expected, the frequency is constant in the equal-mass case, because the spins of the two black holes cancel out (see text for details).