Improved effective-one-body Hamiltonian for spinning black-hole binaries

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 $\mu$, but also with a spin ${\mathbf{S}}_{*}$. Thus, the effective particle interacts with the effective Kerr background (having spin ${\mathbf{S}}_{\mathrm{Kerr}}$) through a geodesic-type interaction and an additional spin-dependent interaction proportional to ${\mathbf{S}}_{*}$. When expanded in post-Newtonian orders, the EOB Hamiltonian reproduces the leading order spin-spin coupling and the spin-orbit coupling through 2.5 post-Newtonian 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.

Figure 1

The frequency at the EOB ISCO for binaries having spins parallel to $\mathit{L}$, with mass ratio $q=m_2/m_1$ and with spin-parameter projections onto the direction of $\mathit{L}$ 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 ${\chi }_{\mathrm{fin}}$ as inferred at the EOB ISCO, for binaries having spins parallel to $\mathit{L}$, with mass ratio $q=m_2/m_1$ and with spin-parameter projections onto the direction of $\mathit{L}$ given by ${\chi }_1={\chi }_2=\chi$. As expected, ${\chi }_{\mathrm{fin}}$ flattens for large $\chi$ in the comparable mass case (see text for details).

Figure 3

The final spin parameter ${\chi }_{\mathrm{fin}}$ as inferred at the EOB ISCO for binaries having spins parallel to $\mathit{L}$, with mass ratio $q=m_2/m_1$ and with spin-parameter projections onto the direction of $\mathit{L}$ given by ${\chi }_1={\chi }_2=\chi$, compared to the remnant’s final spin parameter predicted by the formula presented in Ref. 45 (“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 $\mathit{L}$, with mass ratio $q=m_2/m_1$ and with spin-parameter projections onto the direction of $\mathit{L}$ 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. 53 (AEI09) and in Ref. 46 (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 $\mathit{L}$, with mass ratio $q=m_2/m_1$ and with spin-parameter projections onto the direction of $\mathit{L}$ 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).

Figure 6

The final spin parameter ${\chi }_{\mathrm{fin}}$ as inferred at the EOB ISCO, for binaries having spins parallel to $\mathit{L}$, with mass ratio $q=m_2/m_1$ and with spin-parameter projections onto the direction of $\mathit{L}$ given by ${\chi }_1=-{\chi }_2=\chi$. The results are the same for all equal-mass binaries, for which the spins of the two black holes cancel out (see text for details).

Figure 7

The maximum of the EOB orbital frequency during the plunge, for binaries having spins parallel to $\mathit{L}$, with mass ratio $q=m_2/m_1$ and with spin-parameter projections onto the direction of $\mathit{L}$ 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 $q$ is $\chi$ is almost extremal (see text for details).

Figure 8

For binaries having spins parallel to $\mathit{L}$, with mass ratio $q=m_2/m_1=1$ (left panel) and $q=m_2/m_1=0.1$ (right panel) and with spin-parameter projections onto the direction of $\mathit{L}$ given by ${\chi }_1={\chi }_2=\chi$, we plot twice the maximum of the EOB orbital frequency during the plunge against the frequencies of the first 8 overtones of the $\ell =2$, $m=2$ quasinormal mode of a Kerr black hole. The quasinormal mode frequency is computed using the final spin and final mass of the remnant. The final spin is estimated by applying the formula of Ref. 45, while for the final mass we use the formula of Ref. 53 (AEI09) in the $q=1$ case and that of Ref. 46 (RIT09) in the $q=0.1$ case. As can be seen, in the $q=0.1$ case the peak frequencies lie among the high overtones of the $\ell =2$, $m=2$ mode, while in the $q=1$ case they are generally lower than them. In the $q=1$ case we also mark with a square the numerical gravitational-wave frequency at the peak of the $h_{22}$ mode when $\chi =0$. This gravitational-wave frequency coincides with (twice) the maximum of the EOB orbital frequency at the time when the matching of the quasinormal modes is performed in the nonspinning EOB model of Ref. 7. The numerical gravitational-wave frequency is computed from the numerical simulation of Refs. 58 (“Caltech-Cornell”).

Figure 9

The maximum of the EOB orbital frequency during the plunge, for binaries having spins parallel to $\mathit{L}$, with mass ratio $q=m_2/m_1$ and with spin-parameter projections onto the direction of $\mathit{L}$ given by ${\chi }_1=-{\chi }_2=\chi$. The results are the same for all equal-mass binaries, for which the spins of the two black holes cancel out (see text for details).