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

In the effective-one-body (EOB) approach, the dynamics of two compact objects of masses $m_1$ and $m_2$ and spins ${\mathbf{S}}_1$ and ${\mathbf{S}}_2$ is mapped into the dynamics of one test particle of mass $\mu =m_1{m}_2/(m_1+m_2)$ and spin ${\mathbf{S}}_{*}$ moving in a deformed Kerr metric with mass $M=m_1+m_2$ and spin ${\mathbf{S}}_{\mathrm{Kerr}}$. 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.

Figure 1

The spin parameter of the binary at the ISCO given by Eq. (67) for the 2.5PN and 3.5PN EOB models with dynamical mapping of the spins, for binaries having spins parallel to $\mathit{L}$, mass ratio $q=m_2/m_1$, and spin-parameter projections onto the direction of $\mathit{L}$ given by ${\chi }_1={\chi }_2=\chi$.

Figure 2

The same as in Fig. 1 but for the binding energy of the binary at the ISCO.

Figure 3

The same as in Fig. 1 but for the ISCO frequency.

Figure 4

The ISCO frequency for the 3.5PN EOB models with dynamical (dyn) and nondynamical (non-dyn) mapping of the spins, for binaries having spins parallel to $\mathit{L}$, mass ratio $q=m_2/m_1$, and spin-parameter projections onto the direction of $\mathit{L}$ given by ${\chi }_1={\chi }_2=\chi$.

Figure 5

The same as in Fig. 1, but for the maximum of the orbital frequency during the plunge.

Figure 6

The same as in Fig. 4, but for the maximum frequency during the plunge.

Figure 7

The shift of the ISCO frequency $c_{\Omega }$, defined in Eq. (71), for the 2.5PN and 3.5PN EOB models with dynamical mapping of the spins, for a binary having spins parallel to $\mathit{L}$, mass ratio $q=m_2/m_1={10}^{-3}$, and spin-parameter projections onto the direction of $\mathit{L}$ given by ${\chi }_1=\chi$ and ${\chi }_2=0$.