Coalescence of Two Spinning Black Holes: An Effective One-Body Approach

TL;DR

This work extends the effective-one-body (EOB) approach to binary systems of spinning black holes by embedding spin effects into a deformed Kerr-like metric and applying Padé resummation to improve post-Newtonian convergence near the last stable orbit (LSO).A compact energy-momentum mapping and an effective spin $S_{\rm eff}$ link the real two-body dynamics to an effective one-body geodesic problem, enabling tractable analysis of spin-orbit and spin-spin interactions within a largely conservative framework.The authors analyze spherical orbits and the last stable spherical orbits (LSSO), deriving how the LSSO binding energy depends on the symmetric mass ratio, projected spin $\widehat{a}_p$, and spin misalignment, and they show moderate robustness to 3PN ambiguities for moderate spins.They predict that the final black hole formed by coalescence remains sub-extremal (typically $\widehat{a}_{\rm BH} \lesssim 0.87$ for representative spins) and argue that the EOB approach provides reliable analytical templates for gravitational waves in many astrophysically relevant spinning configurations, while highlighting the need to include radiation reaction for complete waveform modeling.

Abstract

We generalize to the case of spinning black holes a recently introduced ``effective one-body'' approach to the general relativistic dynamics of binary systems. The combination of the effective one-body approach, and of a Padé definition of some crucial effective radial functions, is shown to define a dynamics with much improved post-Newtonian convergence properties, even for black hole separations of the order of $6 GM / c^2$. We discuss the approximate existence of a two-parameter family of ``spherical orbits'' (with constant radius), and, of a corresponding one-parameter family of ``last stable spherical orbits'' (LSSO). These orbits are of special interest for forthcoming LIGO/VIRGO/GEO gravitational wave observations. It is argued that for most (but not all) of the parameter space of two spinning holes the effective one-body approach gives a reliable analytical tool for describing the dynamics of the last orbits before coalescence. This tool predicts, in a quantitative way, how certain spin orientations increase the binding energy of the LSSO. This leads to a detection bias, in LIGO/VIRGO/GEO observations, favouring spinning black hole systems, and makes it urgent to complete the conservative effective one-body dynamics given here by adding (resummed) radiation reaction effects, and by constructing gravitational waveform templates that include spin effects. Finally, our approach predicts that the spin of the final hole formed by the coalescence of two arbitrarily spinning holes never approaches extremality.

Figures (2)

• Figure 1: Dependence of the binding energy $e \equiv ({\cal E}_{\rm real} / M) - 1$ of the LSSO on the effective spin parameter $\widehat{a} \equiv S_{\rm eff} / M^2$ (taken with the sign of $\cos \theta_{\rm LS}$). We consider an equal-mass system with a typical misalignment angle $\cos \theta_{\rm LS} = \pm \, 1 / \sqrt 3$. All three curves used the Padé-resummed Effective-one-body approach. The lower curves use a 3 PN-level Hamiltonian: the lowest one uses $\omega_s = 0 = \omega_s^{\rm DJS}$$[10]$, while the middle one uses $\omega_s = - 2.365 = \omega_s^{\rm BF}$$[17]$. The upper curve uses $\omega_s = - 9.344 = \omega_s^*$, which is (essentially) equivalent to using a 2 PN-level Hamiltonian $[8]$.
• Figure 2: Approximate prediction for the spin parameter $\widehat{a}_{\rm BH} \simeq \vert \hbox{\boldmath$J$} \vert^{\rm LSSO} / M^2$ of the black hole formed by the coalescence of two identical spinning holes (with spins parallel or antiparallel to the orbital angular momentum). The horizontal axis is the effective spin parameter $\widehat{a} = \frac{7}{8} \, \widehat{a}_1 = \frac{7}{8} \, \widehat{a}_2$. The three curves correspond to the three cases plotted in Fig. \ref{['Fig1']}. Note the prediction (robust under changing the 3 PN contribution to the effective potential by 25 %) that the final spin parameter is always sub-extremal, and reaches a maximum $\widehat{a}_{\rm BH} \simeq 0.87$ for $\widehat{a} \simeq + \, 0.3$.