Black hole binaries on non-eccentric orbits form an important subclass of gravitational wave sources, but it is a non-trivial issue to construct numerical initial data with minimal initial eccentricity for numerical simulations. We compute post-Newtonian orbital parameters for quasi-spherical orbits using the method of Buonanno, Chen and Damour (2006) and examine the resulting eccentricity in numerical simulations. Four different methods are studied resulting from the choice of Taylor-expanded or effective-one-body Hamiltonians, and from two choices for the energy flux. The eccentricity increases for unequal masses and for spinning black holes, but remains smaller than that obtained from previous post-Newtonian approaches. The effective-one-body Hamiltonian offers advantages for decreasing initial separation as expected, but in the context of this study also performs significantly better than the Taylor-expanded Hamiltonian for binaries with spin.
Large-Eccentricity Asymptotics and Fast Analytic Approximation for Fourier modes of Post-Newtonian Eccentric Waveforms
Instituto de Física Téorica UAM-CSIC, Universidad Autónoma de Madrid（马德里自治大学理论物理研究所）；Beijing Normal University（北京师范大学）；Hangzhou Institute for Advanced Study, UCAS（中国科学院大学杭州高等研究院）
Standard choices of quasi-circular orbit parameters for black-hole binary evolutions result in eccentric inspiral. We introduce a conceptually simple method, which is to integrate the post-Newtonian equations of motion through hundreds of orbits, and read off the values of the momenta at the separation at which we wish to start a fully general relativistic numerical evolution. For the particular case of non-spinning equal-mass inspiral with an initial coordinate separation of $D = 11M$ we show that this approach reduces the eccentricity by at least a factor of five to $e < 0.002$ as compared t…
We compare different methods of computing the orbital eccentricity of quasi-circular binary black hole systems using the orbital variables and gravitational wave phase and frequency. For eccentricities of about a per cent, most methods work satisfactorily. For small eccentricity, however, the gravitational wave phase allows a particularly clean and reliable measurement of the eccentricity. Furthermore, we measure the decay of the orbital eccentricity during the inspiral and find reasonable agreement with post-Newtonian results. Finally, we measure the periastron advance of non-spinning binary …
Post-Newtonian theory-inspired framework for characterizing eccentricity in gravitational waveforms
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（加州理工学院理论天体物理学（含相对论与宇宙学））；Department of Physics, University of California at Santa Barbara（加州大学圣塔芭芭拉分校物理系）；International Centre for Theoretical Sciences, Tata Institute of Fundamental Research（塔塔基础研究院国际理论科学中心）
We use equations of motion containing gravitational radiation-reaction terms through 4.5 post-Newtonian order to calculate the late-time eccentricities of inspiraling binary systems of non-spinning compact bodies as they cross the detection threshold of ground-based gravitational-wave interferometers. The initial eccentricities can be as large as 0.999. We find that the final eccentricities are systematically smaller than those predicted by the leading quadrupole approximation, by as much as 30 percent for a 300 solar mass binary crossing the LIGO/Virgo detection threshold at 10 Hz, or eight p…
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.
Using a post-Newtonian diagnostic tool developed by Mora and Will, we examine numerically generated quasiequilibrium initial data sets that have been used in recently successful numerical evolutions of binary black holes through plunge, merger and ringdown. We show that a small but significant orbital eccentricity is required to match post-Newtonian and quasiequilibrium calculations. If this proves to be a real eccentricity, it could affect the fine details of the subsequent numerical evolutions and the predicted gravitational waveforms.
While there has been much success in understanding the orbital dynamics and gravitational wave emission of eccentric compact binaries in the post-Newtonian formalism, some problems still remain. The largest of these concerns hereditary effects: non-linear phenomena related to the scattering off of the background curved spacetime (tails) and to the generation of gravitational waves by gravitational waves (memory). Currently, these hereditary effects are only known numerically for arbitrary eccentricity through infinite sums of Bessel functions, with closed-form, analytic results only available …
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…
We study the secular, hierarchical three-body problem to first-order in a post-Newtonian expansion of General Relativity. We expand the first-order post-Newtonian Hamiltonian to leading-order in the ratio of the semi-major axis of the two orbits. In addition to the well-known terms that correspond to the GR precession of the inner and outer orbits, we find a new secular post-Newtonian interaction term that can affect the long-term evolution of the triple. We explore the parameter space for highly inclined and eccentric systems, where the Kozai-Lidov mechanism can produce large-amplitude oscill…
We integrate the third and a half post-Newtonian equations of motion for a fully generic binary black hole system, allowing both for non-circular orbits, and for one or both of the black holes to spin, in any orientation. Using the second post-Newtonian order expression beyond the leading order quadrupole formula, we study the gravitational waveforms produced from such systems. Our results are validated by comparing to Taylor T4 in the aligned-spin circular cases, and the additional effects and modulations introduced by the eccentricity and the spins are analyzed. We use the framework to evalu…
Gravitational waves radiated during binary black hole coalescence is a perfect probe for studying the characteristics of strong gravity. Advanced techniques for creating numerical relativity substitute models for eccentric binary black hole systems are presumed to become more crucial in existing and anticipated gravitational wave detectors. The imprint on the observation data of the gravitational wave emitted by the binary coalescence enhances the two body system studies. The aim of this study is to present an overview of the change in characteristics behaviours of hierarchical massive astroph…
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…
The inspiral and merger of supermassive black hole binary systems with high orbital eccentricity are among the promising sources of the advanced gravitational wave observatories. In this paper we derive analytic ready-to-use first post-Newtonian eccentric waveform in Fourier domain with the use of Hansen coefficients. Introducing generic perturbations of celestial mechanics we have generalized the Hansen expansion to the first post-Newtonian order which are then used to express the waveforms. Taking into account the high eccentricity of the orbit leads to the appearance of secular terms in the…
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…
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…
Spin precession effects in the phasing formula of eccentric compact binary inspirals up to the second post-Newtonian order
Indian Institute of Technology Madras（印度马德拉斯理工学院）；Centre for Strings, Gravitation and Cosmology, Department of Physics, Indian Institute of Technology Madras（印度马德拉斯理工学院物理系弦、引力和宇宙学中心）；Center for Interdisciplinary Exploration and Research in Astrophysics (CIERA), Northwestern University（西北大学天体物理学交叉探索与研究中心）
In a previous paper we derived a post-Newtonian approximation to cosmology which, in contrast to former Newtonian and post-Newtonian cosmological theories, has a well-posed initial value problem. In this paper, this new post-Newtonian theory is compared with the fully general relativistic theory, in the context of the k = 0 Friedmann Robertson Walker cosmologies. It is found that the post-Newtonian theory reproduces the results of its general relativistic counterpart, whilst the Newtonian theory does not.
We present simple procedures to construct quasi-circular initial data for numerical evolutions of binary black hole spacetimes. Our method consists of using Post-Newtonian theory in three ways: first to provide an initial guess for the initial momenta at 3.5PN order that implies low residual eccentricity, second to measure the resulting eccentricity, and third to calculate corrections to the momenta or initial separation which further reduce the eccentricity. Regarding the initial guess, we compare numerical evolutions in post-Newtonian theory to the post-circular and post-post-circular analyt…
We raise the analytical knowledge of the eccentricity-expansion of the Detweiler-Barack-Sago redshift invariant in a Schwarzschild spacetime up to the 9.5th post-Newtonian order (included) for the $e^2$ and $e^4$ contributions, and up to the 4th post-Newtonian order for the higher eccentricity contributions through $e^{20}$. We convert this information into an analytical knowledge of the effective-one-body radial potentials $\bar d(u)$, $ρ(u)$ and $q(u)$ through the 9.5th post-Newtonian order. We find that our analytical results are compatible with current corresponding numerical self-force da…
With the post-Newtonian approxination, the energy of Schwarzschild spacetime in the Weinberg prescription is obtained. The energy for the first post-Newtonian approximation $E^{(1)} = m$ gives the Newtonian treatment of Schwarzschild spacetime. However, for the second post-Newtonian approximation, the erergy is shown that $E^{(1)}$ adds extra terms $E^{(2)}$ which consist of the energy stored in the configuration $E_{\rm config}$, in the gravitational field $E_{\rm field}$. and a term of surface integral. These extra terms gives post-Newtonian corrections to the Newtonian treatment.
The post-Newtonian hydrodynamic equations for a non-perfect fluid are developed within the framework of a post-Newtonian Boltzmann equation. The post-Newtonian components of the energy-momentum tensor are determined by considering the relativistic Eckart decomposition for a viscous and heat conducting fluid. From the relativistic Grad distribution function its post-Newtonian expression is derived. The hydrodynamic equations for the mass density, mass-energy density and momentum density are determined from a post-Newtonian transfer equation and Grad's distribution function. In the non-relativis…
Inspiralling compact binaries are expected to circularize before their gravitational-wave signals reach the sensitive frequency band of ground-based detectors. Current searches for gravitational waves from compact binaries using the LIGO and Virgo detectors therefore use circular templates to construct matched filters. Binary formation models have been proposed which suggest that some systems detectable by the LIGO--Virgo network may have non-negligible eccentricity. We investigate the ability of the restricted 3.5 post-Newtonian order TaylorF2 template bank, used by LIGO and Virgo to search f…
The post-Newtonian Jeans equation for stationary self-gravitating systems is derived from the post-Newtonian Boltzmann equation in spherical coordinates. The Jeans equation is coupled with the three Poisson equations from the post-Newtonian theory. The Poisson equations are functions of the energy-momentum tensor components which are determined from the post-Newtonian Maxwell--Jüttner distribution function. As an application, the effect of a central massive black hole on the velocity dispersion profile of the host galaxy is investigated and the influence of the post-Newtonian corrections are d…
Jeans instability within the framework of post-Newtonian Boltzmann and Poisson equations are analyzed. The components of the energy-momentum tensor are calculated from a post-Newtonian Maxwell-Jüttner distribution function. The perturbations of the distribution function and gravitational potentials from their background states with the representation of the perturbations as plane waves lead to a dispersion relation with post-Newtonian corrections. The influence of the post-Newtonian approximation on the Jeans mass is determined and it was shown that the mass necessary for an overdensity to beg…
High-order post-Newtonian expansion of the generalized redshift invariant for eccentric-orbit, equatorial extreme-mass-ratio inspirals with a spinning primary
We estimate virial masses of galaxy clusters using the parametrized post-Newtonian (PPN) virial theorem. Also, we show explicitly that post-Newtonian corrections can not address the mass discrepancy in the galaxy clusters.
The energy radiated (without the 1.5PN tail contribution which requires a different treatment) by a binary system of compact objects moving in a hyperboliclike orbit is computed in the frequency domain through the second post-Newtonian level as an expansion in the large-eccentricity parameter up to next-to-next-to-leading order, completing the time domain corresponding information (fully known in closed form at the second post-Newtonian of accuracy). The spectrum contains quadratic products of the modified Bessel functions of the first kind (Bessel K functions) with frequency-dependent order (…
Post-Newtonian expansions of the Brill-Lindquist and Misner-Lindquist solutions of the time-symmetric two-black-hole initial value problem are derived. The static Hamiltonians related to the expanded solutions, after identifying the bare masses in both solutions, are found to differ from each other at the third post-Newtonian approximation. By shifting the position variables of the black holes the post-Newtonian expansions of the three metrics can be made to coincide up to the fifth post-Newtonian order resulting in identical static Hamiltonians up the third post-Newtonian approximation. The c…
The second post-Newtonian (2PN) contribution to the `plus' and `cross' gravitational wave polarizations associated with gravitational radiation from non-spinning, compact binaries moving in elliptic orbits is computed. The computation starts from our earlier results on 2PN generation, crucially employs the 2PN accurate generalized quasi-Keplerian parametrization of elliptic orbits by Damour, Schäfer and Wex and provides 2PN accurate expressions modulo the tail terms for gravitational wave polarizations incorporating effects of eccentricity and periastron precession.
We present an elementary derivation of the planetary advance of the perihelion for a general spherically symmetric line element in the post- newtonian approximation.
The second post-Newtonian hydrodynamic equations are analyzed within the framework of a plane wave solution. The hydrodynamic equations for the mass and momentum density are coupled with six Poisson equations for the Newtonian and post-Newtonian gravitational potentials. Perturbations of the basic fields and gravitational potentials from a background state by assuming plane wave representations lead to a dispersion relation where the Jeans instability condition emerges. The influence of the first and second post-Newtonian approximations on the Jeans mass is determined. It was shown that the re…
Contents: I. Introduction; II. Summary of optimal signal filtering; III. Newtonian binary polarization waveforms; IV. Newtonian orbital phase evolution; V. Post-Newtonian wave-generation; VI. Inspiral binary waveform.
We discuss various aspects of the post-Newtonian approximation in general relativity. After presenting the foundation based on the Newtonian limit, we use the (3+1) formalism to formulate the post-Newtonian approximation for the perfect fluid. As an application we show the method for constructing the equilibrium configuration of nonaxisymmetric uniformly rotating fluid. We also discuss the gravitational waves including tail from post-Newtonian systems.
It is shown that the recently found non-uniqueness of the third post-Newtonian binary point-mass ADM-Hamiltonian is related to the non-uniqueness at the third post-Newtonian approximation of the applied ADM-coordinate conditions.
In this paper we explore how far the post-Newtonian theory goes in overcoming the difficulties associated with anisotropic homogeneous cosmologies in the Newtonian approximation. It will be shown that, unlike in the Newtonian case, the cosmological equations of the post-Newtonian approximation are much more in the spirit of general relativity with regard to the nine Bianchi types and issues of singularities. The situations of vanishing rotation and vanishing shear are treated separately. The homogeneous Bianchi I model is considered as an example of a rotation-free cosmology with anisotropy. I…
We study a solution of the field equations for dilatonic gravity and obtain its post-post-newtonian limit. It turns out that terms to this and higher orders in the expansion may become important in strong gravitational fields, even though the post-newtonian limit coincides with that of General Relativity. This suggests that strong gravitational fields can only be studied by exact solutions of the field equations.
We use post-Newtonian (PN) approximations to determine the initial orbital and spin parameters of black hole binaries that lead to low-eccentricity inspirals when evolved with numerical relativity techniques. In particular, we seek initial configurations that lead to very small eccentricities at small separations, as is expected for astrophysical systems. We consider three cases: (i) quasicircular orbits with no radial velocity, (ii) quasicircular orbits with an initial radial velocity determined by radiation reaction, and (iii) parameters obtained form evolution of the PN equations of motion …
We compute the tail contributions to the gravitational-wave mode amplitudes for compact binaries in eccentric orbits at the third post-Newtonian order of general relativity. We combine them with the already available instantaneous pieces and include the post-adiabatic corrections required to fully account for the effects of radiation-reaction forces on the motion. We compare the resulting waveform in the small eccentricity limit to the circular one, finding perfect agreement.
In 2002, Poujade and Blanchet succeeded in matching the post-Newtonian solution to the Einstein field equation to the post-Minkowskian field up to any arbitrary order as well as reproducing, in a different way, the results of the 1998 paper by Blanchet in which he showed how to match the post-Minkowskian series to the post-Newtonian expansion. Comparing these two papers, it might be asked whether it is possible to match the post-Newtonian field to the post-Minkowskian one by means of a method similar to the one used in the 1998 paper. The answer is affirmative, and in the present paper we prov…
The set up of matched filters for the detection of gravitational waves from in-spiraling compact binaries is usually carried out using the restricted post-Newtonian approximation: the filter phase is modelled including post-Newtonian corrections, whereas the amplitude is retained at the Newtonian order. Here we investigate the effects of the introduction of post-Newtonian corrections also to the amplitude and we discuss some of the implications for signal detection and parameter estimation.
We derive the equations of motion for binary systems of compact bodies in the post-Newtonian (PN) approximation to general relativity. Results are given through 2PN order (order (v/c)^4 beyond Newtonian theory), and for gravitational radiation reaction effects at 2.5PN and 3.5PN orders. The method is based on a framework for direct integration of the relaxed Einstein equations (DIRE) developed earlier, in which the equations of motion through 3.5PN order can be expressed in terms of Poisson-like potentials that are generalizations of the instantaneous Newtonian gravitational potential, and in …
Using post-Newtonian equations of motion for fluid bodies valid to the second post-Newtonian order, we derive the equations of motion for binary systems with finite-sized, non-spinning but arbitrarily shaped bodies. In particular we study the contributions of the internal structure of the bodies (such as self-gravity) that would diverge if the size of the bodies were to shrink to zero. Using a set of virial relations accurate to the first post-Newtonian order that reflect the stationarity of each body, and redefining the masses to include 1PN and 2PN self-gravity terms, we demonstrate the comp…
In this paper, second post-Newtonian approximation of Einstein-aether theory is obtained by Chandrasekhar's approach. Five parameterized post-Newtonian parameters in first post-Newtonian approximation are presented after a time transformation and they are identical with previous works, in which $γ=1$, $β=1$ and two preferred-frame parameters remain. Meanwhile, in second post-Newtonian approximation, a parameter, which represents third order nonlinearity for gravity, is zero the same as in general relativity. For an application for future deep space laser ranging missions, we reduce the metric …
We study the dynamics of merging compact binaries near a rotating supermassive black hole (SMBH) in a hierarchical triple configuration. We include various general relativistic effects that couple the inner orbit, the outer orbit and the spin of the SMBH. During the orbital decay due to gravitational radiation, the inner binary can encounter an "apsidal precession resonance" and experience eccentricity excitation. This resonance occurs when the apsidal precession rate of the inner binary matches that of the outer binary, with the precessions driven by both Newtonian interactions and various po…
The increasing sophistication and accuracy of numerical simulations of compact binaries (especially binary black holes) presents the opportunity to test the regime in which post-Newtonian (PN) predictions for the emitted gravitational waves are accurate. In order to confront numerical results with those of post-Newtonian theory, it is convenient to compare multipolar decompositions of the two waveforms. It is pointed out here that the individual modes can be computed to higher post-Newtonian order by examining the radiative multipole moments of the system, rather than by decomposing the 2.5PN …
The post-Newtonian approximation is useful to discuss gravitational waveforms for binary inspirals. In this note, for retrograde circular orbits in the equatorial plane, we discuss the region of validity of the post-Newtonian approximation by using the gravitational energy flux derived numerically and analytically in the black hole perturbation approach. It is found that the edge of the allowable region behaves quite intricately, including prograde orbits even if we treat higher post-Newtonian orders.
We investigated hydrodynamical post-Newtonian models of selfgravitating stationary black-hole-disk systems. The post-Newtonian scheme presented here and also in our recent paper is a continuation of previous, purely Newtonian studies of selfgravitating hydrodynamical disks rotating according to the Keplerian rotation law. The post-Newtonian relativistic corrections are significant even at the 1PN level. The 1PN correction to the angular velocity can be of the order of 10% of its Newtonian value. It can be expressed as a combination of geometric and hydrodynamical terms. Moreover, in contrast t…
Two long-standing problems with the post-Newtonian approximation for isolated slowly-moving systems in general relativity are: (i) the appearance at high post-Newtonian orders of divergent Poisson integrals, casting a doubt on the soundness of the post-Newtonian series; (ii) the domain of validity of the approximation which is limited to the near-zone of the source, and prevents one, a priori, from incorporating the condition of no-incoming radiation, to be imposed at past null infinity. In this article, we resolve the problem (i) by iterating the post-Newtonian hierarchy of equations by means…
While the expectation is that the majority of gravitational wave events observable by ground-based detectors will be emitted by compact binaries in quasi-circular orbits, the growing number of detections suggests the possibility of detecting waves from binaries with non-negligible orbital eccentricity in the near future. Several gravitational wave models incorporate the effects of small orbital eccentricities ($e \lesssim 0.2$), but these models may not be sufficient to analyze waves from systems with moderate eccentricity. We recently developed a gravitational wave model that faithfully accou…
We construct a set of post-Newtonian reference frames for a comprehensive study of the orbital dynamics and rotational motion of the Moon and Earth by means of lunar laser ranging (LLR) with the precision of one millimeter. We also derive the post-Newtonian coordinate transformations between the frames and analyze the residual gauge freedom, which is used for removing spurious post-Newtonian effects from the equations of motion of the solar system bodies.
Newtonian Cosmology is commonly used in astrophysical problems, because of its obvious simplicity when compared with general relativity. However it has inherent difficulties, the most obvious of which is the non-existence of a well-posed initial value problem. In this paper we investigate how far these problems are met by using the post-Newtonian approximation in cosmology.
We study the Jeans gravitational instability for a mixture of baryonic and dark matter particles, in the post-Newtonian approximation. We adopt a kinetic model consisting of a coupled system of post-Newtonian collisionless Boltzmann equations, for each species, coupled to the post-Newtonian Poisson equations. We derive the stability criterion, accounting for both post-Newtonian corrections and the presence of dark matter. It is shown that both effects give rise to smaller Jeans masses, in comparison with the standard Jeans criterion, meaning that a smaller mass is needed to begin the gravitati…
We introduce two new eccentricity measures to analyze numerical simulations. Unlike earlier definitions these eccentricity measures do not involve any free parameters which makes them easy to use. We show how relatively inexpensive grid setups can be used to estimate the eccentricity during the early inspiral phase. Furthermore, we compare standard puncture data and post-Newtonian data in ADMTT gauge. We find that both use different coordinates. Thus low eccentricity initial momentum parameters for a certain separation measured in ADMTT coordinates are hard to use in puncture data, because it …
