Parameterizing black hole orbits for adiabatic inspiral
Scott A. Hughes
备注：
10 pages, 5 figures, version to appear in Physical Review D. Incorporates helpful feedback from the referee and a few other comments received since this paper was originally posted. Posting includes Mathematica notebook and C++ code which implements the methods developed in this paper
Adiabatic binary inspiral in the small mass ratio limit treats the small body as moving along a geodesic of a large Kerr black hole, with the geodesic slowly evolving due to radiative backreaction. Up to initial conditions, geodesics are typically parameterized in two ways: using the integrals of motion energy $E$, axial angular momentum $L_z$, and Carter constant $Q$; or, using orbit geometry parameters semi-latus rectum $p$, eccentricity $e$, and (cosine of ) inclination $x_I \equiv \cos I$. The community has long known how to compute orbit integrals as functions of the orbit geometry parameters, i.e., as functions expressing $E(p, e, x_I)$, and likewise for $L_z$ and $Q$. Mappings in the other direction -- functions $p(E, L_z, Q)$, and likewise for $e$ and $x_I$ -- have not yet been developed in general. In this note, we develop generic mappings from ($E$, $L_z$, $Q$) to ($p$, $e$, $x_I$). The mappings are particularly simple for equatorial orbits ($Q = 0$, $x_I = \pm1$), and can be evaluated efficiently for generic cases. These results make it possible to more accurately compute adiabatic inspirals by eliminating the need to use a Jacobian which becomes singular as inspiral approaches the last stable orbit.
Extreme mass ratio inspirals (EMRIs) show a strong separation of timescales, with the time characterizing inspiral, $T_{\rm i}$, much longer than any time $T_{\rm o}$ characterizing orbital motions. The ratio of these timescales (which is essentially an EMRI's mass ratio) can be regarded as a parameter that controls a perturbative expansion. Here we describe the value and limitations of an "adiabatic" description of these binaries, which uses only the leading terms arising from such a two-timescale expansion. An adiabatic approach breaks down when orbits evolve through resonances, with importa…
Using two recent techniques giving non-perturbative re-summed estimates of the damping and of the conservative part of the dynamics of two-body systems, we describe the transition between adiabatic inspiral and plunge in binary non-spinning black holes moving along quasi-circular orbits.
The transition motion of a point particle around the last stable orbit of Kerr is described at leading order in the transition-timescale expansion. Taking systematically into account all self-force effects, we prove that the transition motion is still described by the Painlevé transcendent equation of the first kind. Using an asymptotically matched expansions scheme, we consistently match the quasi-circular adiabatic inspiral with the transition motion. The matching requires us to take into account the secular change of angular velocity due to radiation reaction during the adiabatic inspiral.
The inspiral of a stellar mass compact object falling into a massive Kerr black hole can be broken into three different regimes: An adiabatic inspiral phase, where the inspiral timescale is much larger than the orbital period; a late-time radial infall, which can be approximated as a plunging geodesic; and a regime where the body transitions from the inspiral to plunge. In earlier work, Ori and Thorne have outlined a method to compute the trajectory during this transition for a compact object in a circular, equatorial orbit. We generalize this technique to include inclination and eccentricity.
We investigate resonance crossings of a charged body moving around a Kerr black hole immersed in an external homogeneous magnetic field. This system can serve as an electromagnetic analogue of a weakly non-integrable extreme mass ratio inspiral (EMRI). In particular, the presence of the magnetic field renders the conservative part of the system non-integrable in the Liouville sense, while the electromagnetic self-force causes the charged body to inspiral. By studying the system without the self-force, we show the existence of an approximate Carter-like constant and discuss how resonances grow …
We use the post-Newtonian-Affine model to assess the validity of the adiabatic approximation in modeling tidal effects in the phase evolution of compact binary systems. We compute the dynamical evolution of the tidal tensor, which we estimate at the 2 PN order, and of the quadrupole tensor, finding that their ratio, i.e. the tidal deformability, increases in the last phases of the inspiral. We derive the gravitational wave phase corrections due to this phenomenon and quantify how they affect gravitational wave detectability.
An inspiralling object of mass $μ$ around a Kerr black hole of mass $M (\gg μ)$ experiences a continuous transition near the innermost stable circular orbit from adiabatic inspiral to plunge into the horizon as gravitational radiation extracts its energy and angular momentum. We investigate the collision of such an object with a generic counterpart around a Kerr black hole. We find that the angular momentum of the object is fine-tuned through gravitational radiation and that the high-velocity collision of the object with a generic counterpart naturally occurs around a nearly maximally rotating…
Scientific analysis for the gravitational-wave detector LISA will require theoretical waveforms from extreme-mass-ratio inspirals (EMRIs) that extensively cover all possible orbital and spin configurations around astrophysical Kerr black holes. However, on-the-fly calculations of these waveforms have not yet overcome the high dimensionality of the parameter space. To confront this challenge, we present a user-ready EMRI waveform model for generic (eccentric and inclined) orbits in Kerr spacetime, using an analytical self-force approach. Our model accurately covers all EMRIs with arbitrary incl…
The detection of gravitational waves from Extreme mass Ratio Inspirals (EMRIs) by the future space-based gravitational-wave detectors demands the generation of accurate enough waveform templates. Since the spin of the smaller secondary body cannot be neglected for the detection and parameter estimation of EMRIs, we study its influence on the phase of the gravitational waves from EMRIs with spinning secondary. We focus on generic eccentric equatorial orbits around a Kerr black hole. To model the spinning secondary object, we use the Mathisson-Papapetrou-Dixon equations in the pole-dipole approx…
We compute gravitational waves from inspiraling stellar-mass compact objects on the equatorial plane of a massive spinning black hole (BH). Our inspiral orbits are computed by taking into account the adiabatic change of orbital parameters due to gravitational radiation in the lowest order in mass ratio. We employ an interpolation method to compute the adiabatic change at arbitrary points inside the region of orbital parameter space computed in advance. Using the obtained inspiral orbits and associated gravitational waves, we compute power spectra of gravitational waves and the signal-to-noise …
We describe progress evolving an important limit of binary orbits in general relativity, that of a stellar mass compact object gradually spiraling into a much larger, massive black hole. These systems are of great interest for gravitational wave observations. We have developed tools to compute for the first time the radiated fluxes of energy and angular momentum, as well as instantaneous snapshot waveforms, for generic geodesic orbits. For special classes of orbits, we compute the orbital evolution and waveforms for the complete inspiral by imposing global conservation of energy and angular mo…
The significance of first post-adiabatic contributions for scalar charge measurements with intermediate and extreme mass ratio inspirals
University College Dublin（都柏林大学学院）；University of Glasgow（格拉斯哥大学）；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）；University of Nottingham（诺丁汉大学）
A test particle of mass mu on a bound geodesic of a Kerr black hole of mass M >> mu will slowly inspiral as gravitational radiation extracts energy and angular momentum from its orbit. This inspiral can be considered adiabatic when the orbital period is much shorter than the timescale on which energy is radiated, and quasi-circular when the radial velocity is much less than the azimuthal velocity. Although the inspiral always remains adiabatic provided mu << M, the quasi-circular approximation breaks down as the particle approaches the innermost stable circular orbit (ISCO). In this paper, we …
Using a semi-analytical approach recently developed to model the tidal deformations of neutron stars in inspiralling compact binaries, we study the dynamical evolution of the tidal tensor, which we explicitly derive at second post-Newtonian order, and of the quadrupole tensor. Since we do not assume a priori that the quadrupole tensor is proportional to the tidal tensor, i.e. the so called "adiabatic approximation", our approach enables us to establish to which extent such approximation is reliable. We find that the ratio between the quadrupole and tidal tensors (i.e., the Love number) increas…
In the small mass ratio expansion and on the equatorial plane, the two-body problem for point particles in general relativity admits a quasi-circular inspiral motion followed by a transition-to-plunge motion. We first derive the equations governing the quasi-circular inspiral in the Kerr background at adiabatic, post-adiabatic and post-post-adiabatic orders in the slow-timescale expansion in terms of the self-force and we highlight the structure of the equations of motion at higher subleading orders. We derive in parallel the equations governing the transition-to-plunge motion to any subleadin…
Extreme-mass-ratio inspirals, in which a stellar-mass compact object spirals into a supermassive black hole in a galactic core, are expected to be key sources for LISA. Modelling these systems with sufficient accuracy for LISA science requires going to second (or {\em post-adiabatic}) order in gravitational self-force theory. Here we present a practical two-timescale framework for achieving this and generating post-adiabatic waveforms. The framework comprises a set of frequency-domain field equations that apply on the fast, orbital timescale, together with a set of ordinary differential equati…
We revisit the problem of the emission of gravitational waves from a test mass orbiting and thus perturbing a Kerr black hole. The source term of the Teukolsky perturbation equation contains a Dirac delta function which represents a point particle. We present a technique to effectively model the delta function and its derivatives using as few as four points on a numerical grid. The source term is then incorporated into a code that evolves the Teukolsky equation in the time domain as a (2+1) dimensional PDE. The waveforms and energy fluxes are extracted far from the black hole. Our comparisons …
Inspirals of stellar mass compact objects into massive black holes are an important source for future gravitational wave detectors such as Advanced LIGO and LISA. Detection of these sources and extracting information from the signal relies on accurate theoretical models of the binary dynamics. We cast the equations describing binary inspiral in the extreme mass ratio limit in terms of action angle variables, and derive properties of general solutions using a two-timescale expansion. This provides a rigorous derivation of the prescription for computing the leading order orbital motion. As shown…
A key source for LISA will be the inspiral of compact objects into supermassive black holes. Recently Mino has shown that in the adiabatic limit, gravitational waveforms for these sources can be computed using for the radiation reaction force the gradient of one half the difference between the retarded and advanced metric perturbations. Using post-Newtonian expansions, we argue that the resulting waveforms should be sufficiently accurate for signal detection with LISA. Data-analysis templates will require higher accuracy, going beyond adiabaticity; this remains a significant challenge. We desc…
This is the second in a series of papers whose aim is to generate ``adiabatic'' gravitational waveforms from the inspiral of stellar-mass compact objects into massive black holes. In earlier work, we presented an accurate (2+1)D finite-difference time-domain code to solve the Teukolsky equation, which evolves curvature perturbations near rotating (Kerr) black holes. The key new ingredient there was a simple but accurate model of the singular source term based on a discrete representation of the Dirac-delta function and its derivatives. Our earlier work was intended as a proof of concept, using…
We utilize the dynamical renormalization group formalism to calculate the real space trajectory of a compact binary inspiral for long times via a systematic resummation of secularly growing terms. This method generates closed form solutions without orbit averaging, and the accuracy can be systematically improved. The expansion parameter is $v^5 νΩ(t-t_0)$ where $t_0$ is the initial time, $t$ is the time elapsed, and $Ω$ and $v$ are the angular orbital frequency and initial speed, respectively, and $ν$ is the binary's symmetric mass ratio. We demonstrate how to apply the renormalization group m…
We compare the performances of the templates defined by three different types of approaches: traditional post-Newtonian templates (Taylor-approximants), ``resummed'' post-Newtonian templates assuming the adiabatic approximation and stopping before the plunge (P-approximants), and further ``resummed'' post-Newtonian templates going beyond the adiabatic approximation and incorporating the plunge with its transition from the inspiral (Effective-one-body approximants). The signal to noise ratio is significantly enhanced (mainly because of the inclusion of the plunge signal) by using these new effe…
Extreme mass-ratio inspirals detectable by the future Laser Interferometer Space Antenna provide a unique way to test general relativity and fundamental physics. Motivated by this possibility, here we study in detail the EMRI dynamics in the presence of a spinning secondary, collecting and extending various results that appeared in previous work and also providing useful intermediate steps and new relations for the first time. We present the results of a frequency-domain code that computes gravitational-wave fluxes and the adiabatic orbital evolution for the case of circular, equatorial orbits…
We develop the first model for extreme mass-ratio inspirals (EMRIs) into a rotating massive black hole driven by the gravitational self-force. Our model is based on an action angle formulation of the method of osculating geodesics for eccentric, equatorial (i.e., spin-aligned) motion in Kerr spacetime. The forcing terms are provided by an efficient spectral interpolation of the first-order gravitational self-force in the outgoing radiation gauge. We apply a near-identity (averaging) transformation to eliminate all dependence of the orbital phases from the equations of motion, while maintaining…
We provide a method for analytically constructing high-accuracy templates for the gravitational wave signals emitted by compact binaries moving in inspiralling eccentric orbits. By contrast to the simpler problem of modeling the gravitational wave signals emitted by inspiralling {\it circular} orbits, which contain only two different time scales, namely those associated with the orbital motion and the radiation reaction, the case of {\it inspiralling eccentric} orbits involves {\it three different time scales}: orbital period, periastron precession and radiation-reaction time scales. By using …
Combining recent techniques giving non-perturbative re-summed estimates of the damping and conservative parts of the two-body dynamics, we describe the transition between the adiabatic phase and the plunge, in coalescing binary black holes with comparable masses moving on quasi-circular orbits. We give initial dynamical data for numerical relativity investigations, with a fraction of an orbit left, and provide, for data analysis purposes, an estimate of the gravitational wave-form emitted throughout the inspiral, plunge and coalescence phases.
We model the adiabatic inspiral of relativistic binary neutron stars in a quasi-equilibrium (QE) approximation, and compute the gravitational wavetrain from the late phase of the inspiral. We compare corotational and irrotational sequences and find a significant difference in the inspiral rate, which is almost entirely caused by differences in the binding energy. We also compare our results with those of a point-mass post-Newtonian calculation. We illustrate how the late inspiral wavetrain computed with our QE numerical scheme can be matched to the subsequent plunge and merger waveform calcula…
We investigate the dynamics and gravitational-wave (GW) emission in the binary merger of equal-mass black holes as obtained from numerical relativity simulations. Results from the evolution of three sets of initial data are explored in detail, corresponding to different initial separations of the black holes. We find that to a good approximation the inspiral phase of the evolution is quasi-circular, followed by a "blurred, quasi-circular plunge", then merger and ring down. We present first-order comparisons between analytical models of the various stages of the merger and the numerical results…
There are three regimes of gravitational-radiation-reaction-induced inspiral for a compact body with mass mu, in a circular, equatorial orbit around a Kerr black hole with mass M>>mu: (i) The "adiabatic inspiral regime", in which the body gradually descends through a sequence of circular, geodesic orbits. (ii) A "transition regime", near the innermost stable circular orbit (isco). (iii) The "plunge regime", in which the body travels on a geodesic from slightly below the isco into the hole's horizon. This paper gives an analytic treatment of the transition regime and shows that, with some luck,…
Computationally efficient waveforms are of central importance for gravitational wave data analysis of inspiralling and coalescing compact binaries. We show that the post-adiabatic (PA) approximation to the effective-one-body (EOB) description of the binary dynamics, when pushed to high-order, allows one to accurately and efficiently compute the waveform of coalescing binary neutron stars (BNSs) or black holes (BBHs) up to a few orbits before merger. This is accomplished bypassing the usual need of numerically solving the relative EOB dynamics described by a set of ordinary differential equatio…
We calculate the real-space trajectory and spin precession of a generic spinning compact binary inspiral at any time instant using the dynamical renormalization group formalism. This method leads to closed-form analytic solutions to the binary motion through treating radiation reaction as perturbations and resumming the secular growth of perturbative terms. We consider the spin-orbit effects at leading order and the 2.5PN radiation reaction without orbit averaging or precession averaging for arbitrary individual masses and spin magnitudes and orientations. The solutions are written in a moving…
Consistency of spin effects between numerical relativity and perturbation theory for inspiraling comparable-mass black hole binaries
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）；University of Rhode Island（罗德岛大学）；University of Massachusetts, Dartmouth（马萨诸塞大学达特茅斯分校）
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。
Extreme mass ratio inspirals into black holes surrounded by matter: Resonance crossings
Astronomical Institute of the Czech Academy of Sciences（捷克科学院天文研究所）；Institute of Theoretical Physics, Faculty of Mathematics and Physics, Charles University（查理大学数学物理学院理论物理研究所）
We show that transient resonances occur in the two body problem in general relativity, in the highly relativistic, extreme mass-ratio regime for spinning black holes. These resonances occur when the ratio of polar and radial orbital frequencies, which is slowly evolving under the influence of gravitational radiation reaction, passes through a low order rational number. At such points, the adiabatic approximation to the orbital evolution breaks down, and there is a brief but order unity correction to the inspiral rate. Corrections to the gravitational wave signal's phase due to resonance effect…
In this work we analyze some judiciously chosen solutions of Kerr Black Holes with Scalar Hair (KBHsSH) of special interest for Gravitational Wave (GW) events originated from Extreme Mass Ratio Inspirals (EMRIs). Because of the off-center distribution of energy density, these spacetimes are warped in such a way that not all metric functions behave monotonically on the equatorial plane as in the exterior region of Kerr black holes (KBHs). This has great impact on the orbital parameters, which in turn affects the imprints on signals descendant from EMRIs in a adiabatic evolution. By investigatin…
The largest part of any gravitational-wave inspiral of a compact binary can be understood as a slow, adiabatic drift between the trajectories of a certain referential conservative system. In many contexts, the phase space of this conservative system is smooth and there are no "topological transitions" in the phase space, meaning that there are no sudden qualitative changes in the character of the orbital motion during the inspiral. However, in this chapter we discuss the cases where this assumption fails and non-linear and/or non-smooth transitions come into play. In integrable conservative sy…
[abridged] Chern-Simons (CS) modified gravity is a 4D effective theory that descends both from string theory and loop quantum gravity, and that corrects the Einstein-Hilbert action by adding the product of a scalar field and the parity-violating, Pontryagin density. In this theory, the gravitational field of spinning black holes is described by a modified Kerr geometry whose multipole moments deviate from the Kerr ones only at the fourth multipole, l = 4. We investigate possible signatures of this theory in the gravitational wave emission produced in the inspiral of stellar compact objects int…
Extreme-mass-ratio-inspiral (EMRI) is one of the most important sources for the future space-borne gravitational wave detectors. In such kind of systems, the compact objects usually orbit around the central supermassive black holes with complicated trajectories. Usually, the trajectory is approximated as geodesic of a test-particle in Kerr space-time, and the orbital evolution are simulated with the help of adiabatic approximation. However, this omits the influence of the compact object on the back ground. In the present paper, employing effective-one-body formalism, we analytically calculate …
We extend the Ori and Thorne (OT) procedure to compute the transition from an adiabatic inspiral into a geodesic plunge for any spin, with emphasis on near-extremal ones. Our analysis revisits the validity of the approximations made in OT. In particular, we discuss possible effects coming from eccentricity and non-geodesic past-history of the orbital evolution. We find three different scaling regimes according to whether the mass ratio is much smaller, of the same order or much larger than the near extremal parameter describing how fast the primary black hole rotates. Eccentricity and non-geod…
