Effective one body approach to the dynamics of two spinning black holes with next-to-leading order spin-orbit coupling

TL;DR

This work extends the Effective One Body (EOB) framework to spinning black-hole binaries by incorporating next-to-leading order (NLO) spin-orbit coupling. It builds a spin-dependent EOB Hamiltonian using a canonical transformation to an effective Kerr-like metric with an effective spin ${f S}_0$, plus a supplementary spin-orbit term $oldsymbol{ abla H}_{ m so}$ involving a test-spin vector $oldsymbol{oldsymbol{\sigma}}$, and maps to a real Hamiltonian via the standard EOB relation. The authors show that NLO spin-orbit terms significantly moderate the previously strong LO spin effects on circular-orbit energetics, especially for large parallel spins, yielding more physically reasonable last stable circular orbits and a smoother dependence of the LSO Kerr parameter on spin. They explore energy–frequency curves, LSO properties, and the sensitivity to a4PN-like deformation parameter $a_5$, and discuss the potential alignment with numerical relativity results for final black-hole spins in spinning binaries. The work lays groundwork for improved waveform templates and motivates further study of merger outcomes using the refined EOB description.

Abstract

Using a recent, novel Hamiltonian formulation of the gravitational interaction of spinning binaries, we extend the Effective One Body (EOB) description of the dynamics of two spinning black holes to next-to-leading order (NLO) in the spin-orbit interaction. The spin-dependent EOB Hamiltonian is constructed from four main ingredients: (i) a transformation between the ``effective'' Hamiltonian and the ``real'' one, (ii) a generalized effective Hamilton-Jacobi equation involving higher powers of the momenta, (iii) a Kerr-type effective metric (with Padé-resummed coefficients) which depends on the choice of some basic ``effective spin vector'' $\bf{S}_{\rm eff}$, and which is deformed by comparable-mass effects, and (iv) an additional effective spin-orbit interaction term involving another spin vector $\bsigma$. As a first application of the new, NLO spin-dependent EOB Hamiltonian, we compute the binding energy of circular orbits (for parallel spins) as a function of the orbital frequency, and of the spin parameters. We also study the characteristics of the last stable circular orbit: binding energy, orbital frequency, and the corresponding dimensionless spin parameter $\hat{a}_{\rm LSO}\equiv c J_{\rm LSO}/\boldsymbol(G(H_{\rm LSO}/c^2)^2\boldsymbol)$. We find that the inclusion of NLO spin-orbit terms has a significant ``moderating'' effect on the dynamical characteristics of the circular orbits for large and parallel spins.

Figures (6)

• Figure 1: Binding energy curves for circular orbits of symmetric non-spinning binaries ($m_1=m_2$ and $\hat{\mathbf{a}}_1=\hat{\mathbf{a}}_2=\mathbf{0}$): dimensionless "non relativistic" energy $e$ versus dimensionless angular frequency $\hat{\Omega}$. The notation E$(n,*)$ means computation of the energy using the EOB-improved real Hamiltonian \ref{['hreal']} with the $n$PN-accurate metric function $\Delta_t(R)$; the function $\Delta_t(R)$ was computed by means of Eq. \ref{['deltat']} using the $(1,n)$ Padé approximant at the $n$PN order. Here $n=1,2,3,4$, where $n=4$ refers to the "4PN" case where a term $+a_5\nu\hat{u}^5$ is added to the function $A(\hat{u})$. For the curves labelled by T$(n,*)$ the computation was done with the direct PN-expanded (ADM-coordinates) orbital Hamiltonian \ref{['horb']} with the terms up to the $n$PN order included.
• Figure 3: Energy $e$ versus angular frequency $\hat{\Omega}$ along circular orbits for various values of the parameter $\hat{a}_0$, as predicted by the EOB Hamiltonian. We have assumed $m_1=m_2$, $a_1=a_2$, and $\theta_0=\pi/2$. As before E$(n,s)$ refers to an EOB Hamiltonian, with $n$PN accuracy in the orbital terms, and an accuracy in the spin-orbit coupling equal to the LO one if $s=0$, and the NLO one if $s=1$. In all cases, we include the full LO spin-spin coupling.
• Figure 4: Binding energy of the Last Stable (circular) Orbit (LSO) predicted by the EOB approach. We study the effect of including NLO spin-orbit terms by contrasting the LO and NLO predictions. We plot the dimensionless energy $e_\mathrm{LSO}$ of the LSO versus $\hat{a}_0$. We have assumed $m_1=m_2$, $a_1=a_2$, and $\theta_0=\pi/2$. For E$(3,0)$ a LSO exists up to $\hat{a}_0\le+0.9$.
• Figure 5: The dimensionless total angular momentum Kerr parameter $\hat{a}_J^{\mathrm{LSO}}$, Eq. \ref{['abh3']}, at the LSO, versus $\hat{a}_0$. We have assumed $m_1=m_2$, $a_1=a_2$, and $\theta_0=\pi/2$. The parameter $\hat{a}_J^\mathrm{LSO}$ is computed from Eq. \ref{['abh3']} with $\hat{j}_\mathrm{LSO}=\hat{\ell}_\mathrm{LSO}+\hat{a}_1 + \hat{a}_2= \hat{\ell}_\mathrm{LSO}+2\hat{a}_0$. We compare the various EOB predictions obtained either by improving the accuracy of spin-orbit terms [E$(3,1)$ versus E$(3,0)$], or by improving the accuracy of orbital terms [E$(4,1)$ versus E$(3,1)$]. We use two representative values of the 4PN parameter $a_5 = + 25$ and $a_5 = + 60$. For comparison, we also include a fit to recent numerical estimates of the final Kerr parameter of the black hole resulting from the coalescence of the two constituent black holes.
• Figure 6: The dimensionless total angular momentum Kerr parameter $\hat{a}_J^{\mathrm{LSO}}$ at the E(3,1) LSO versus $\hat{a}_2$ for various values of the parameter $\hat{a}_1$. Here we consider spin-dissymmetric systems with $a_1\neq a_2$ (but still $m_1=m_2$ and $\theta_0=\pi/2$). The parameter $\hat{a}_J^\mathrm{LSO}$ is computed from Eq. \ref{['abh3']} with $\hat{j}_\mathrm{LSO}=\hat{\ell}_\mathrm{LSO}+\hat{a}_1+\hat{a}_2$.