Extending the effective-one-body Hamiltonian of black-hole binaries to include next-to-next-to-leading spin-orbit couplings

TL;DR

Extends the effective-one-body (EOB) Hamiltonian to include next-to-next-to-leading spin-orbit couplings at 3.5PN for arbitrary mass ratio, addressing accurate modeling of spinning binary black holes. Builds two spin-mapping schemes—one dynamical-variable dependent and one independent—via a Lie-generated canonical transformation and a PN-expanded spinning test-particle in a deformed Kerr spacetime, ensuring the correct test-particle limit. Provides explicit expressions for the effective gyromagnetic coefficients $g^{\rm eff}_{\sigma}$ and $g^{\rm eff}_{\sigma^*}$ up to 3.5PN, and analyzes equatorial dynamics including the ISCO and a plunge-frequency peak, with gauge choices to improve convergence. The framework supports calibration to numerical relativity and self-force results, enabling more accurate analytical waveform templates for LIGO/Virgo searches and reliable parameter estimation in the strong-field regime.

Abstract

In the effective-one-body (EOB) approach the dynamics of two compact objects of masses m1 and m2 and spins S1 and S2 is mapped into the dynamics of one test particle of mass mu = m1 m2/(m1+m2) and spin S* moving in a deformed Kerr metric with mass M = m1+m2 and spin Skerr. In a previous paper we computed an EOB Hamiltonian for spinning black-hole binaries that (i) when expanded in post-Newtonian orders, reproduces the leading order spin-spin coupling and the leading and next-to-leading order spin-orbit couplings for any mass ratio, and (iii) reproduces all spin-orbit couplings in the test-particle limit. Here we extend this EOB Hamiltonian to include next-to-next-to-leading spin-orbit couplings for any mass ratio. We discuss two classes of EOB Hamiltonians that differ by the way the spin variables are mapped between the effective and real descriptions. We also investigate the main features of the dynamics when the motion is equatorial, such as the existence of the innermost stable circular orbit and of a peak in the orbital frequency during the plunge subsequent to the inspiral.

Figures (7)

• Figure 1: The spin parameter of the binary at the ISCO given by Eq. \ref{['chiISCO']} for the 2.5PN and 3.5PN EOB models with dynamical mapping of the spins, for binaries having spins parallel to $\boldsymbol{L}$, mass ratio $q = m_2/m_1$ and spin-parameter projections onto the direction of $\boldsymbol{L}$ given by $\chi_1 = \chi_2 = \chi$.
• Figure 2: The same as in Fig. \ref{['fig:finalspin']} but for the binding energy of the binary at the ISCO.
• Figure 3: The same as in Fig. \ref{['fig:finalspin']} but for the ISCO frequency.
• Figure 4: The ISCO frequency for the 3.5PN EOB models with dynamical (dyn) and non-dynamical (non-dyn) mapping of the spins, for binaries having spins parallel to $\boldsymbol{L}$, mass ratio $q = m_2/m_1$ and spin-parameter projections onto the direction of $\boldsymbol{L}$ given by $\chi_1 = \chi_2 = \chi$.
• Figure 5: The same as in Fig. \ref{['fig:finalspin']}, but for the maximum of the orbital frequency during the plunge.