Accurate and efficient modeling of the dynamics of binary black holes (BBHs) is crucial to their detection and parameter estimation through gravitational waves, both with LIGO/Virgo and LISA. General BBH configurations will have misaligned spins and eccentric orbits, eccentricity being particularly relevant at early times. Modeling these systems is both analytically and numerically challenging. Even though the 1.5 post-Newtonian (PN) order is Liouville integrable, numerical work has demonstrated chaos at 2PN order, which impedes the existence of an analytic solution. In this article we revisit integrability at both 1.5PN and 2PN orders. At 1.5PN, we construct four (out of five) action integrals. At 2PN, we show that the system is indeed integrable - but in a perturbative sense - by explicitly constructing five mutually-commuting constants of motion. Because of the KAM theorem, this is consistent with the past numerical demonstration of chaos. Our method extends to higher PN orders, opening the door for a fully analytical solution to the generic eccentric, spinning BBH problem.
Eccentricity evolution of spinning binaries and its dependence on the equation of state of the components
Gran Sasso Science Institute (GSSI)（格兰萨索科学研究所）；INFN, Laboratori Nazionali del Gran Sasso（意大利国家核物理研究所，格兰萨索国家实验室）；Max-Planck-Institut für Gravitationsphysik (Albert-Einstein-Institut)（马克斯·普朗克引力物理学研究所（爱因斯坦研究所））；Leibniz Universität Hannover（汉诺威莱布尼茨大学）
Several scenarios have been proposed in which the orbits of binary black holes enter the band of a gravitational wave detector with significant eccentricity. To avoid missing these signals or biasing parameter estimation it is important that we consider waveform models that account for eccentricity. The ingredients needed to compute post-Newtonian (PN) waveforms produced by spinning black holes inspiralling on quasi-eccentric orbits have been available for almost two decades at 2 PN order, and this work has recently been extended to 2.5 PN order. However, the computational cost of directly imp…
ESIGMAHM: An Eccentric, Spinning inspiral-merger-ringdown waveform model with Higher Modes for the detection and characterization of binary black holes
We derive fourth post-Newtonian (4PN) contributions to the Keplerian-type parametric solution associated with the conservative dynamics of eccentric, non-spinning compact binaries. The solution has been computed while ignoring certain zero-average, oscillatory terms arising due to 4PN tail effects. We provide explicit expressions for the parametric solution and various orbital elements in terms of the conserved energy, angular momentum and symmetric mass ratio. Canonical perturbation theory (along with the technique of Pade approximant) is used to incorporate the 4PN nonlocal-in-time tail effe…
Compact binary systems with black holes are the primary sources of interferometric advanced gravitational wave detectors. Astrophysical models suggest finite possibility of binary systems with appreciable non-zero eccentricity in the dense stellar environment like globular clusters and galactic nuclei. While most of the compact binary measurements have not shown appreciable eccentricity, constrains on the eccentricity have been placed on few detected events. With increasing sensitivity of the gravitational wave detectors, the eccentric binaries are plausible sources in the gravitational wave w…
The motion of a stellar compact object around a supermassive black hole can be approximated by the motion of a spinning test particle. The equations of motion describing such systems are in general non-integrable, and therefore, chaotic motion should be expected. This article discusses the integrability issue of the spinning particle for the cases of Schwarzschild and Kerr spacetime, and then it focuses on a canonical Hamiltonian formalism where the spin of the particle is included only up to the linear order.
The equations of motion for spinning compact binaries on eccentric orbits are treated perturbatively in powers of a fractional mass-difference ordering parameter. The solution is valid through first order in the mass-difference parameter. A canonical point transformation removes the leading order terms of the spin-orbit Hamiltonian which induce a wiggling precession of the orbital angular momentum around the conserved total angular momentum, a precession which disappears in the case of equal masses or one single spin. Action-angle variables are applied which make a canonical perturbation theor…
We present models of realistic globular clusters with post-Newtonian dynamics for black holes. By modeling the relativistic accelerations and gravitational-wave emission in isolated binaries and during three- and four-body encounters, we find that nearly half of all binary black hole mergers occur inside the cluster, with about 10% of those mergers entering the LIGO/Virgo band with eccentricities greater than 0.1. In-cluster mergers lead to the birth of a second generation of black holes with larger masses and high spins, which, depending on the black hole natal spins, can sometimes be retaine…
We analyze a new numerical relativity dataset of spinning but nonprecessing binary black holes on eccentric orbits, with eccentricities from approximately $0.1$ to $0.5$, with dimensionless spins up to $0.75$ included at mass ratios $q=m_1/m_2 = (1, 2)$, and further nonspinning binaries at mass ratios $q = (1.5, 3, 4)$. A comparison of the final mass and spin of these simulations with noneccentric data extends previous results in the literature on circularization of eccentric binaries to the spinning case. For the $(l,m)=(2,2)$ spin-weighted spherical harmonic mode we construct eccentric hybri…
Adding higher-order spherical harmonics in non-spinning eccentric binary black hole merger waveform models
Kavli Institute for Theoretical Physics, University of California Santa Barbara（加州大学圣塔芭芭拉分校卡弗里理论物理研究所）；Theoretical AstroPhysics Including Relativity and Cosmology, California Institute of Technology（加利福尼亚理工学院包括相对论和宇宙学的理论天体物理学）；Department of Physics and Center for Computational Research, University of Rhode Island（罗德岛大学物理系与计算研究中心）；Department of Physics, University of Massachusetts, Dartmouth（麻省大学达特茅斯分校物理系）；Center for Scientific Computing and Data Science Research, University of Massachusetts, Dartmouth（麻省大学达特茅斯分校科学计算与数据科学研究中心）；Department of Mathematics, University of Massachusetts, Dartmouth（麻省大学达特茅斯分校数学系）
We use the frequency and time domain Teukolsky formalism to calculate gravitational-wave fluxes from a spinning body on a bound eccentric equatorial orbit around a Kerr black hole. The spinning body is represented as a point particle following the pole-dipole approximation of the Mathisson-Papapetrou-Dixon equations. Reformulating these equations we are not only able to find the trajectory of a spinning particle in terms of its constants of motion, but also to provide a method to calculate the azimuthal and the radial frequency of this trajectory. Using these orbital quantities, we introduce t…
We study the motion of a classical spinning particle (with spin degrees of freedom described by a vector of Grassmann variables) in higher-dimensional general rotating black hole spacetimes with a cosmological constant. In all dimensions n we exhibit n bosonic functionally independent integrals of spinning particle motion, corresponding to explicit and hidden symmetries generated from the principal conformal Killing--Yano tensor. Moreover, we demonstrate that in 4-, 5-, 6-, and 7-dimensional black hole spacetimes such integrals are in involution, proving the bosonic part of the motion integrab…
We study dynamical capture binary black hole-neutron star (BH-NS) mergers focusing on the effects of the neutron star spin. These events may arise in dense stellar regions, such as globular clusters, where the majority of neutron stars are expected to be rapidly rotating. We initialize the BH-NS systems with positions and velocities corresponding to marginally unbound Newtonian orbits, and evolve them using general-relativistic hydrodynamical simulations. We find that even moderate spins can significantly increase the amount of mass in unbound material. In some of the more extreme cases, there…
We derive Keplerian-type parametrization for the solution of post-Newtonian (PN) accurate conservative dynamics of spinning compact binaries moving in eccentric orbits. The PN accurate dynamics that we consider consists of the third post-Newtonian accurate conservative orbital dynamics influenced by the leading order spin effects, namely the leading order spin-orbit interactions. The orbital elements of the representation are explicitly given in terms of the conserved orbital energy, angular momentum and a quantity that characterizes the leading order spin-orbit interactions in Arnowitt, Deser…
Including higher-order modes in a quadrupolar eccentric numerical relativity surrogate using universal eccentric modulation functions
Kavli Institute for Theoretical Physics, University of California Santa Barbara（加州大学圣塔芭芭拉分校卡弗里理论物理研究所）；Department of Mathematics, Center for Scientific Computing and Data Science Research, University of Massachusetts, Dartmouth（马萨诸塞大学达特茅斯分校数学系科学计算与数据科学研究中心）；Max Planck Institute for Gravitational Physics (Albert Einstein Institute)（马克斯·普朗克引力物理研究所（爱因斯坦研究所））；Nicholas and Lee Begovich Center for Gravitational-Wave Physics and Astronomy, California State University Fullerton（加州州立大学富勒顿分校尼古拉斯和李贝戈维奇引力波物理与天文学中心）；The Grainger College of Engineering, Department of Physics & Illinois Center for Advanced Studies of the Universe, University of Il（伊利诺伊大学格瑞纳工程学院物理系及伊利诺伊宇宙高级研究中心）
We use the open source, community-driven, numerical relativity software, the Einstein Toolkit to study the physics of eccentric, spinning, nonprecessing binary black hole mergers with mass-ratios $q=\{2, 4, 6\}$, individual dimensionless spin parameters $χ_{1z}=\pm0.6$, $χ_{2z}=\pm0.3$, that include higher order gravitational wave modes $\ell\leq4$, except for memory modes. Assuming stellar mass binary black hole mergers that may be detectable by the advanced LIGO detectors, we find that including modes up to $\ell=4$ increases the signal-to-noise of compact binaries between $3.5\%$ to $35\%$,…
We calculate the eccentricity dependence of the high-order post-Newtonian (PN) series for the generalized redshift invariant $\langle u^t \rangle_τ$ for eccentric-orbit extreme-mass-ratio inspirals on a Schwarzschild background. These results are calculated within first-order black hole perturbation theory (BHPT) using Regge-Wheeler-Zerilli (RWZ) gauge. Our \textsc{Mathematica} code is based on a familiar procedure, using PN expansion of the Mano-Suzuki-Takasugi (MST) analytic function formalism for $l$ modes up to a certain maximum and then using a direct general-$l$ PN expansion of the RWZ e…
This paper is to introduce a new software called CBwaves which provides a fast and accurate computational tool to determine the gravitational waveforms yielded by generic spinning binaries of neutron stars and/or black holes on eccentric orbits. This is done within the post-Newtonian (PN) framework by integrating the equations of motion and the spin precession equations while the radiation field is determined by a simultaneous evaluation of the analytic waveforms. In applying CBwaves various physically interesting scenarios have been investigated. In particular, we have studied the appropriate…
Analytical Solution of Spinning, Eccentric Binary Black Hole Dynamics at the Second Post-Newtonian Order
Laboratoire d’étude de l’Univers et des phénomènes eXtrêmes (LUX), Observatoire de Paris, Université PSL, Sorbonne Université, CNRS（极端宇宙与现象实验室（LUX），巴黎天文台，巴黎文理研究大学，索邦大学，法国国家科学研究中心）；Universidad de Ingenieria y Tecnologia – UTEC（工程与技术大学 - UTEC）
We provide a systematic analysis of the multipolar gravitational waveform, energy and angular momentum fluxes emitted by a nonspinning test particle orbiting a Kerr black hole along equatorial, eccentric orbits. These quantities are computed by numerically solving the Teukolsky equation in the time domain and are then used to test and improve the radiation reaction (and waveform) of an effective-one-body (EOB) model. Eccentricity is incorporated into EOB by replacing the quasi-circular Newtonian (or leading-order) prefactors in the EOB-factorized multipolar waveform (and fluxes) with their gen…
Although the gravitational waves observed by advanced LIGO and Virgo are consistent with compact binaries in a quasi-circular inspiral prior to coalescence, eccentric inspirals are also expected to occur in Nature. Due to their complexity, we currently lack ready-to-use, analytic waveforms in the Fourier domain valid for sufficiently high eccentricity, and such models are crucial to coherently extract weak signals from the noise. We here take the first steps to derive and properly validate an analytic waveform model in the Fourier domain that is valid for inspirals of arbitrary orbital eccentr…
We calculate the eccentricity dependence of the high-order post-Newtonian (PN) expansion of the spin-precession invariant $ψ$ for eccentric-orbit extreme-mass-ratio inspirals with a Schwarzschild primary. The series is calculated in first-order black hole perturbation theory through direct analytic expansion of solutions in the Regge-Wheeler-Zerilli formalism, using a code written in \textsc{Mathematica}. Modes with small values of $l$ are found via the Mano-Suzuki-Takasugi (MST) analytic function expansion formalism for solutions to the Regge-Wheeler equation. Large-$l$ solutions are found by…
Closed-form solutions of spinning, eccentric binary black holes at 1.5 post-Newtonian order
Observatoire de Paris, Université PSL, Sorbonne Université, CNRS（巴黎天文台，巴黎文理研究大学，索邦大学，法国国家科学研究中心）；Indian Statistical Institute（印度统计研究所）；Birla Institute of Technology and Science Pilani（比拉理工皮拉尼学院）；Department of Physics and Astronomy, The University of Mississippi（密西西比大学物理与天文学系）
The Laser Interferometer Space Antenna (LISA) is designed to detect gravitational wave signals from astrophysical sources, including those from coalescing binary systems of compact objects such as black holes. Colliding galaxies have central black holes that sink to the center of the merged galaxy and begin to orbit one another and emit gravitational waves. Some galaxy evolution models predict that the binary black hole system will enter the LISA band with significant orbital eccentricity, while other models suggest that the orbits will already have circularized. Using a full seventeen paramet…
The rotation of the bodies and the eccentricity of the orbit have significant effects on the emitted gravitational radiation of binary systems. This work focuses on the evaluation of the gravitational wave polarization states for spinning compact binaries. We consider binaries on eccentric orbits and the spin-orbit interaction up to the 1.5 post-Newtonian order in a way which is independent of the parameterization of the orbit. The equations of motion for angular variables are included. The formal expressions of the polarization states are given with the inclusion of higher order corrections t…
While current gravitational wave observations with ground based detectors have been consistent with compact binaries in quasi-circular orbits, eccentric binaries may be detectable by ground-based and space-based instruments in the near future. Eccentricity significantly complicates the gravitational wave signal, and we currently lack fast and accurate models that are valid in the moderate to large eccentricity range. In a previous paper, we built a Fourier domain, eccentric waveform model at leading order in the post-Newtonian approximation, i.e. as an expansion in small velocities and weak fi…
The secular evolution of a spinning, massive binary system in eccentric orbit is analyzed, expanding and generalizing our previous treatments of the Lense-Thirring motion and the one-spin limit. The spin-orbit and spin-spin effects up to the 3/2 post-Newtonian order are considered, both in the equations of motion and in the radiative losses. The description of the orbit in terms of the true anomaly parametrization provides a simple averaging technique, based on the residue theorem, over eccentric orbits. The evolution equations of the angle variables characterizing the relative orientation of …
Symplectic mechanics of relativistic spinning compact bodies. III. quadratic-in-spin integrability in Type-D Einstein spacetimes: persistence and breakdown
Université de Caen Normandie, ENSICAEN, CNRS/IN2P3, LPC Caen UMR6534（卡昂诺曼底大学）；Department of Physics, National University of Singapore（新加坡国立大学物理系）；Université Libre de Bruxelles, International Solvay Institutes（布鲁塞尔自由大学）
High-order post-Newtonian expansion of the generalized redshift invariant for eccentric-orbit, equatorial extreme-mass-ratio inspirals with a spinning primary
We study eccentric equatorial orbits of a test-body around a Kerr black hole under the influence of gravitational radiation reaction. We have adopted a well established two-step approach: assuming that the particle is moving along a geodesic (justifiable as long as the orbital evolution is adiabatic) we calculate numerically the fluxes of energy and angular momentum radiated to infinity and to the black hole horizon, via the Teukolsky-Sasaki-Nakamura formalism. We can then infer the rate of change of orbital energy and angular momentum and thus the evolution of the orbit. The orbits are fully …
The binary black holes (BBHs) formed near the supermassive black holes (SMBHs) in the galactic nuclei would undergo eccentricity excitation due to the gravitational perturbations from the SMBH and therefore merger more efficiently. In this paper, we study the coupling of the three body 1st post-Newtonian (PN) effects with the spin effects from the SMBH in the hierarchical triple system. We extend previous work by including the coupling between the de Sitter precession and the Lense-Thirring precession from the SMBH spin. This coupling includes both the precessions of the inner orbit angular mo…
In this paper, we continue our study of the motion of spinning test bodies orbiting Kerr black holes. Non-spinning test bodies follow geodesics of the spacetime in which they move. A test body's spin couples to the curvature of that spacetime, introducing a "spin-curvature force" which pushes the body's worldline away from a geodesic trajectory. The spin-curvature force is an important example of a post-geodesic effect which must be modeled carefully in order to accurately characterize the motion of bodies orbiting black holes. One motivation for this work is to understand how to include such …
Symplectic mechanics of relativistic spinning compact bodies. I. linear-in-spin integrability under Killing-Yano symmetry
Université de Caen Normandie, ENSICAEN, CNRS/IN2P3, LPC Caen UMR6534（法国卡昂诺曼底大学、ENSICAEN、CNRS/IN2P3、LPC Caen UMR6534）；Department of Physics, National University of Singapore（新加坡国立大学物理系）
We present $\texttt{ENIGMA}$, a time domain, inspiral-merger-ringdown waveform model that describes non-spinning binary black holes systems that evolve on moderately eccentric orbits. The inspiral evolution is described using a consistent combination of post-Newtonian theory, self-force and black hole perturbation theory. Assuming eccentric binaries that circularize prior to coalescence, we smoothly match the eccentric inspiral with a stand-alone, quasi-circular merger, which is constructed using machine learning algorithms that are trained with quasi-circular numerical relativity waveforms. W…
We develop new high-order results for the post-Newtonian (PN) expansions of the energy and angular momentum fluxes at infinity for eccentric-orbit extreme-mass-ratio inspirals (EMRIs) on a Schwarzschild background. The series are derived through direct expansion of the MST solutions within the RWZ formalism for first-order black hole perturbation theory (BHPT). By utilizing factorization and a few computational simplifications, we are able to compute the fluxes to 19PN, with each PN term calculated as a power series in (Darwin) eccentricity to $e^{10}$. This compares favorably with the numeric…
The Fast and the Frame-Dragging: Efficient waveforms for asymmetric-mass eccentric equatorial inspirals into rapidly-spinning black holes
University of Birmingham（伯明翰大学）；University of Glasgow（格拉斯哥大学）；European Space Agency (ESA)（欧洲航天局（ESA））；University of Southampton（南安普顿大学）；Université de Toulouse（图卢兹大学）；CNRS（法国国家科学研究中心）；NASA Marshall Space Flight Center（美国宇航局马歇尔太空飞行中心）；Max Planck Institute for Gravitational Physics (Albert Einstein Institute)（马克斯·普朗克引力物理研究所（爱因斯坦研究所））；National University of Singapore（新加坡国立大学）
本文提出了一种高效方法，用于生成非对称质量偏心赤道 inspirals 进入快速旋转黑洞的波形，支持偏心率 e < 0.9 和半焦距参数 p < 200，生成长时程 IMRI 波形，且在 100 毫秒内完成，具有 10^-5 的匹配度。
We present the first analytic computation of the Detweiler-Barack-Sago gauge-invariant redshift function for a small mass in {\it eccentric} orbit around a {\it spinning} black hole. Our results give the redshift contributions that mix eccentricity and spin effects, through second order in eccentricity, second order in spin parameter, and the eight-and-a-half post-Newtonian order.
Ultralight bosons can be abundantly produced through superradiance process by a spinning black hole and form a bound state with hydrogen-like spectrum. We show that such a "gravitational atom" typically possesses anomalously large mass quadrupole and leads to significant orbital precession when it forms an eccentric binary with a second compact object. Dynamically formed black hole binaries or pulsar-black hole binaries are typically eccentric during their early inspirals. We show that the large orbital precession can generate distinct and observable signature in their gravitational wave or pu…
We derive a fully analytic Keplerian-type parametrization solution to conservative motion of spinning binary in ADM gauge. This solution is able to describe three dimensional motion of binaries of arbitrary eccentricity, mass ratio and initial configuration of spin angular momentum up to the leading order of post-Newtonian(PN) approximation and a linear order in spin. Based on our results waveforms can be quickly computed with high accuracy.
Eccentricity in Disguise? Insights from GW231123 and Numerically Simulated Binary Black Hole Merger Signals
Max Planck Institute for Gravitational Physics (Albert Einstein Institute)（马克斯·普朗克引力物理研究所（爱因斯坦研究所））；Institute for Gravitation & the Cosmos, Department of Physics, Penn State University（宾夕法尼亚州立大学引力与宇宙学研究所物理学系）；Indian Institute of Technology, Bombay（印度理工学院孟买分校）；Gran Sasso Science Institute (GSSI)（格兰萨索科学研究所）；INFN, Laboratori Nazionali del Gran Sasso（意大利国家核物理研究所格兰萨索国家实验室）；School of Physics and Astronomy and Institute for Gravitational Wave Astronomy, University of Birmingham（伯明翰大学物理与天文学院及引力波天文学研究所）
We analytically compute the gravitational self-force correction to the gyroscope precession along slightly eccentric equatorial orbits in the Kerr spacetime, generalizing previous results for the Schwarzschild spacetime. Our results are accurate through the 9.5 post-Newtonian order and to second order in both eccentricity and rotation parameter. We also provide a post-Newtonian check of our results based on the currently known Hamiltonian for spinning binaries.
The radiation of linear momentum imparts a recoil (or "kick") to the center of mass of a merging black hole binary system. Recent numerical relativity calculations have shown that eccentricity can lead to an approximate 25% increase in recoil velocities for equal-mass, spinning binaries with spins lying in the orbital plane ("superkick" configurations) [U Sperhake et al. Phys. Rev. D 101 (2020) 024044 (arXiv:1910.01598)]. Here we investigate the impact of nonzero eccentricity on the kick magnitude and gravitational-wave emission of nonspinning, unequal-mass black hole binaries. We confirm that…
We investigate for order and chaos the dynamical system of a spinning test particle of mass $m$ moving in the spacetime background of a Kerr black hole of mass M. This system is approximated in our investigation by the linear in spin Hamiltonian function provided in [E. Barausse, and A. Buonanno, Phys.Rev. D 81, 084024 (2010)]. We study the corresponding phase space by using 2D projections on a surface of section and the method of color and rotation on a 4D Poincaré section. Various topological structures coming from the non-integrability of the linear in spin Hamiltonian are found and discuss…
The phase space of a Hamiltonian system is symplectic. However, the post-Newtonian Hamiltonian formulation of spinning compact binaries in existing publications does not have this property, when position, momentum and spin variables $[X, P, S_1, S_2]$ compose its phase space. This may give a convenient application of perturbation theory to the derivation of the post-Newtonian formulation, but also makes classic theories of a symplectic Hamiltonian system be a serious obstacle in application, especially in diagnosing integrability and nonintegrability from a dynamical system theory perspective.…
Mergers of stellar-mass black holes on highly eccentric orbits are among the targets for ground-based gravitational-wave detectors, including LIGO, VIRGO, and KAGRA. These sources may commonly form through gravitational-wave emission in high velocity dispersion systems or through the secular Kozai-Lidov mechanism in triple systems. Gravitational waves carry information about the binaries' orbital parameters and source location. Using the Fisher matrix technique, we determine the measurement accuracy with which the LIGO-VIRGO-KAGRA network could measure the source parameters of eccentric binari…
We present the first analytical computation of the (conservative) gravitational self-force correction to the periastron advance around a spinning black hole. Our result is accurate to the second order in the rotational parameter and through the 9.5 post-Newtonian level. It has been obtained as the circular limit of the correction to the gyroscope precession invariant along slightly eccentric equatorial orbits in the Kerr spacetime. The latter result is also new and we anticipate here the first few terms only of the corresponding post-Newtonian expansion.
The Detweiler-Barack-Sago redshift function for particles moving along slightly eccentric equatorial orbits around a Kerr black hole is currently known up to the second order in eccentricity, second order in spin parameter, and the 8.5 post-Newtonian order. We improve the analytical computation of such a gauge-invariant quantity by including terms up to the fourth order in eccentricity at the same post-Newtonian approximation level. We also check that our results agrees with the corresponding post-Newtonian expectation of the same quantity, calculated by using the currently known Hamiltonian f…
