by Barausse, Enrico and Buonanno, Alessandra and Tiec, Alexandre Le
11 pages, 2 figures

Using the main result of a companion paper, in which the binding energy of a circular-orbit non-spinning compact binary system is computed at leading-order beyond the test-particle approximation, the exact expression of the effective-one-body (EOB) metric component $latex g^\text{eff}_{tt}$ is obtained through first order in the mass ratio. Combining these results with the recent gravitational self-force calculation of the periastron advance for circular orbits in the Schwarzschild geometry, the EOB metric component $latex g^\text{eff}_{rr}$ is also determined at linear order in the mass ratio. These results assume that the mapping between the real and effective Hamiltonians at the second and third post-Newtonian (PN) orders holds at all PN orders. Our findings also confirm the advantage of resumming the PN dynamics around the test-particle limit if the goal is to obtain a flexible model that can smoothly connect the test-mass and equal-mass limits.

by Tiec, Alexandre Le and Blanchet, Luc and Whiting, Bernard F.
43 pages, 3 figures

First laws of black hole mechanics, or thermodynamics, come in a variety of different forms. In this paper, from a purely post-Newtonian (PN) analysis, we obtain a first law for binary systems of point masses moving along an exactly circular orbit. Our calculation is valid through 3PN order and includes, in addition, the contributions of logarithmic terms at 4PN and 5PN orders. This first law of binary point-particle mechanics is then derived from first principles in general relativity, and analogies are drawn with the single and binary black hole cases. Some consequences of the first law are explored for PN spacetimes. As one such consequence, a simple relation between the PN binding energy of the binary system and Detweiler’s redshift observable is established. Through it, we are able to determine with high precision the numerical values of some previously unknown high order PN coefficients in the circular-orbit binding energy. Finally, we propose new gauge invariant notions for the energy and angular momentum of a particle in a binary system.

by Ohme, Frank
12 pages, 2 figures, 1 table, NRDA2011/Amaldi 9 proceedings

Models of gravitational waveforms from coalescing black-hole binaries play a crucial role in the efforts to detect and interpret those signatures in the data of large-scale interferometers. Here we summarize recent models that combine information both from analytical approximations and numerical relativity. We briefly lay out and compare the strategies employed to build such complete models and we recapitulate the errors associated with various aspects of the modelling process.

by Huerta, E. A. and Gair, Jonathan R. and Brown, Duncan A.
22 pages, 3 figures. Submitted to Phys. Rev. D. arXiv admin note: substantial text overlap with arXiv:1105.3567

We extend the numerical kludge waveform model introduced in [1] in two ways. We extend the equations of motion for spinning black hole binaries derived by Saijo et al. [2] using spin-orbit and spin-spin couplings taken from perturbative and post-Newtonian (PN) calculations at the highest order available. We also include first-order conservative self-force corrections for spin-orbit and spin-spin couplings, which are derived by comparison to PN results. We generate the inspiral evolution using fluxes that include the most recent calculations of small body spin corrections, spin-spin and spin-orbit couplings and higher-order fits to solutions of the Teukolsky equation. Using a simplified version of this model in [1], we found that small body spin effects may be measured through gravitational wave observations from intermediate-mass ratio inspirals (IMRIs) with mass ratio eta ~ 0.001, when both binary components are rapidly rotating. In this paper we study in detail how the spin of the small/big body affects parameter measurement using a variety of mass and spin combinations for typical IMRIs sources. We find that for IMRI events of a moderately rotating intermediate mass black hole (IMBH) of 10^4 solar masses, and a rapidly rotating central supermassive black hole (SMBH) of 10^6 solar masses, gravitational wave observations made with LISA at a fixed signal-to-noise ratio (SNR) of 1000 will be able to determine the inspiralling IMBH mass, the central SMBH mass, the SMBH spin magnitude, and the IMBH spin magnitude to within fractional errors of ~10^{-3}, 10^{-3}, 10^{-4}, and 9%, respectively. LISA can also determine the location of the source in the sky and the SMBH spin orientation to within ~10^{-4} steradians. We show that by including conservative corrections up to 2.5PN order, systematic errors no longer dominate over statistical errors for IMRIs with typical SNR ~1000.

by Damour, Thibault and Nagar, Alessandro and Pollney, Denis and Reisswig, Christian
4 pages, 2 figures

Using accurate numerical relativity simulations of (nonspinning) black-hole binaries with mass ratios 1:1, 2:1 and 3:1 we compute the gauge invariant relation between the (reduced) binding energy $latex E$ and the (reduced) angular momentum $latex j$ of the system. We show that the relation $latex E(j)$ is an accurate diagnostic of the dynamics of a black-hole binary in a highly relativistic regime. By comparing the numerical-relativity $latex E^{\rm NR} (j)$ curve with the predictions of several analytic approximation schemes, we find that, while the usual, non-resummed post-Newtonian-expanded $latex E^{\rm PN} (j)$ relation exhibits large and growing deviations from $latex E^{\rm NR} (j)$, the prediction of the effective one-body formalism, based purely on known analytical results (without any calibration to numerical relativity), agrees strikingly well with the numerical-relativity results.

by Hergt, Steven and Steinhoff, Jan and Schaefer, Gerhard
37 pages, submitted

The present paper addresses open questions regarding the handling of the spin supplementary condition within the effective field theory approach to the post-Newtonian approximation. In particular it is shown how the spin supplementary condition can be eliminated at the level of the potential and how the dynamics can be cast into a fully reduced Hamiltonian form. Two different methods are used and compared, one based on the well-known Dirac bracket and the other based on an action principle. It is discussed how the latter approach can be used to improve the Feynman rules by formulating them in terms of reduced canonical spin variables.

by Lovelace, Geoffrey and Boyle, Michael and Scheel, Mark A. and Szilagyi, Bela
17 pages, 7 figures, submitted to Classical and Quantum Gravity

Motivated by the possibility of observing gravitational waves from merging black holes whose spins are nearly extremal (i.e., 1 in dimensionless units), we present numerical waveforms from simulations of merging black holes with the highest spins simulated to date: (1) a 25.5-orbit inspiral, merger, and ringdown of two holes with equal masses and spins of magnitude 0.97 aligned with the orbital angular momentum; and (2) a previously reported 12.5-orbit inspiral, merger, and ringdown of two holes with equal masses and spins of magnitude 0.95 anti-aligned with the orbital angular momentum. First, we consider the horizon mass and spin evolution of the new aligned-spin simulation. During the inspiral, the horizon area and spin evolve in remarkably close agreement with Alvi’s analytic predictions, and the remnant hole’s final spin agrees reasonably well with several analytic predictions. We also find that the total energy emitted by a real astrophysical system with these parameters—almost all of which is radiated during the time included in this simulation—would be 10.952% of the initial mass at infinite separation. Second, we consider the gravitational waveforms for both simulations. After estimating their uncertainties, we compare the waveforms to several post-Newtonian approximants, finding significant disagreement well before merger, although the phase of the TaylorT4 approximant happens to agree remarkably well with the numerical prediction in the aligned-spin case. We find that the post-Newtonian waveforms have sufficient uncertainty that hybridized waveforms will require far longer numerical simulations (in the absence of improved post-Newtonian waveforms) for accurate parameter estimation of low-mass binary systems.

by Amaro-Seoane, Pau and Brem, Patrick and Cuadra, Jorge and Armitage, Philip J.
Submitted

Measurements of gravitational waves from the inspiral of a stellar-mass compact object into a massive black hole (MBH) are unique probes to test General Relativity (GR) and MBH properties, as well as the stellar distribution about these holes in galactic nuclei. Current data analysis techniques can provide us with parameter estimation with very narrow errors. However, an EMRI is not a two-body problem, since other stellar bodies orbiting nearby will influence the capture orbit. Any deviation from the isolated inspiral of the binary will induce a small, though observable deviation from the idealised waveform which could be misinterpreted as a failure of GR. Based on conservative analysis of mass segregation in a Milky Way like nucleus, we estimate that the possibility that a star has a semi-major axis comparable to that of the EMRI is non-negligible. This star introduces an observable perturbation in the orbit in the case in which we consider only loss of energy via gravitational radiation at periapsis. When considering the two first-order non-dissipative post-Newtonian contributions (the periapsis shift of the orbit) the evolution of the orbital elements of the EMRI turns out to be chaotic in nature. The implications of this study are twofold. From the one side, the application to testing GR and measuring MBHs parameters with the detection of EMRIs in galactic nuclei with a millihertz mission will be even more challenging than believed. From the other side, this behaviour could in principle be used as a signature of mass segregation in galactic nuclei.

by Galaviz, Pablo
15 pages, 15 figures and 4 tables

We study the stability and chaos of three compact objects using post-Newtonian (PN) equations of motion derived from the Arnowitt-Deser-Misner-Hamiltonian formulation, where we include terms up to 2.5 PN order in the orbital part and the leading order in spin corrections. We perform numerical simulations of a hierarchical configuration of three compact bodies in which a binary system is perturbed by a third, lighter body initially far away from the binary. The relative importance of the different PN orders is examined. The basin boundary method and the computation of Lyapunov exponent are employed to analyze the stability and chaotic properties of the system. The 1 PN terms produce a small but noticeable change in the stability regions of the parameters considered. On the other hand, the inclusion of spin or gravitational radiation does not produce a significant change with respect to the inclusion of the 1 PN terms.

by Favata, Marc
25 pages, 8 figures

The nonlinear gravitational-wave memory causes a time-varying but nonoscillatory correction to the gravitational-wave polarizations. It arises from gravitational-waves that are sourced by gravitational-waves. Previous considerations of the nonlinear memory effect have focused on quasicircular binaries. Here I consider the nonlinear memory from Newtonian orbits with arbitrary eccentricity. Expressions for the waveform polarizations and spin-weighted spherical-harmonic modes are derived for elliptic, hyperbolic, parabolic, and radial orbits. In the hyperbolic, parabolic, and radial cases the nonlinear memory provides a 2.5 post-Newtonian (PN) correction to the leading-order waveforms. This is in contrast to the elliptical and quasicircular cases, where the nonlinear memory corrects the waveform at leading (0PN) order. This difference in PN order arises from the fact that the memory builds up over a short “scattering” timescale in the hyperbolic case, as opposed to a much longer radiation-reaction timescale in the elliptical case. The nonlinear memory corrections presented here complete our knowledge of the leading-order (Peters-Mathews) waveforms for elliptical orbits. These calculations are also relevant for binaries with quasicircular orbits in the present epoch which had, in the past, large eccentricities. Because the nonlinear memory depends sensitively on the past evolution of a binary, I discuss the effect of this early-time eccentricity on the value of the late-time memory in nearly-circularized binaries. I also discuss the observability of large “memory jumps” in a binary’s past that could arise from its formation in a capture process. Lastly, I provide estimates of the signal-to-noise ratio of the linear and nonlinear memories from hyperbolic and parabolic binaries.

by Levi, Michele
24 pages, revtex4-1, 5 figures

We calculate the next-to-next-to-leading order (NNLO) spin1-spin2 dynamics of a compact binary evaluated at fourth post-Newtonian (PN) order. We use an effective field theory (EFT) approach, and first demonstrate here the ability of the EFT approach to go at NNLO in the PN corrections of spin effects. The NNLO spin1-spin2 interaction sector includes contributions from diagrams, which are not pure spin1-spin2 diagrams, as they arise from other sectors. These diagrams contribute through the leading order spin accelerations and precessions, that should be first taken into account here. The EFT calculation is carried out in terms of the nonrelativistic gravitational (NRG) fields. The fact that the spin is derivative-coupled adds significantly to the complexity of computations. In particular, for the irreducible two-loop diagrams, which are the most complicated in this sector, irreducible two-loop tensor integrals up to order 4 are encountered. Moreover, not all of the benefits of the NRG fields apply to spin interactions, as all possible diagram topologies are realized at each order of G included. Still, the NRG fields remain advantageous, and thus there was no use of automated computations in this work. Our final result can be reduced, and a NNLO spin1-spin2 Hamiltonian can be derived from it.

by Hartung, Johannes and Steinhoff, Jan
7 pages, submitted to AdP

We present the next-to-next-to-leading order post-Newtonian (PN) spin(1)-spin(2) Hamiltonian for two self-gravitating spinning compact objects. If both objects are rapidly rotating, then the corresponding interaction is comparable in strength to a 4PN effect. The Hamiltonian is checked via the global Poincare algebra with the center-of-mass vector uniquely determined by an ansatz.

by Nagar, Alessandro
11 pages, no figures. Submitted to Phys. Rev. D

Building on the recently computed next-to-next-to-leading order (NNLO) post-Newtonian (PN) spin-orbit Hamiltonian for spinning binaries \cite{Hartung:2011te} we extend the effective-one-body (EOB) description of the dynamics of two spinning black-holes to NNLO in the spin-orbit interaction. The calculation that is presented extends to NNLO the next-to-leading order (NLO) spin-orbit Hamiltonian computed in Ref. \cite{Damour:2008qf}. The present EOB Hamiltonian reproduces the spin-orbit coupling through NNLO in the test-particle limit case. In addition, in the case of spins parallel or antiparallel to the orbital angular momentum, when circular orbits exist, we find that the inclusion of NNLO spin-orbit terms moderates the effect of the NLO spin-orbit coupling.

by Tiec, Alexandre Le and Mroué, Abdul H. and Barack, Leor and Buonanno, Alessandra and Pfeiffer, Harald P. and Sago, Norichika and Taracchini, Andrea
5 pages, 3 figures

The general relativistic periastron advance of non-spinning black hole binaries on quasi-circular orbits has been computed using black hole perturbation theory, post-Newtonian expansions, and the effective-one-body formalism. We compare these approximations with accurate numerical relativity simulations of mass ratios 1/8 < m1/m2 m1m2/(m1+m2)^2. The effective-one-body prediction also agrees very well over the entire mass-ratio range considered.

by Blanchet, Luc and Buonanno, Alessandra and Faye, Guillaume
Submitted to Physical Review D

Gravitational waves contain tail effects which are due to the back-scattering of linear waves in the curved space-time geometry around the source. In this paper we improve the knowledge and accuracy of the two-body inspiraling post-Newtonian (PN) dynamics and gravitational-wave signal by computing the spin-orbit terms induced by tail effects. Notably, we derive those terms at 3PN order in the gravitational-wave energy flux, and 2.5PN and 3PN orders in the wave polarizations. This is then used to derive the spin-orbit tail effects in the phasing through 3PN order. Our results can be employed to carry out more accurate comparisons with numerical-relativity simulations and to improve the accuracy of analytical templates aimed at describing the all process of inspiral, merger and ringdown.

by Fujita, Ryuichi
4 pages, 2 figures

We derive gravitational waveforms needed to compute the 14th post-Newtonian (14PN) order energy flux, i.e. $latex v^{28}$ beyond Newtonian approximation where $latex v$ is the orbital velocity of a test particle, in a circular orbit around a Schwarzschild black hole. We exhibit clearly the convergence of the energy flux in the PN expansion and suggest the fitting formula which can be used for more general case. The phase difference between the 14PN waveforms and numerical waveforms after two years inspiral becomes about $latex 10^{-7}$ for $latex \mu/M=10^{-4}$ and $latex 10^{-3}$ for $latex \mu/M=10^{-5}$ where $latex \mu$ and $latex M$ are the masses of a compact object and a supermassive black hole at the centers of galaxies respectively. The 14PN expressions will lead to the parameter estimation comparable to numerical waveforms for extreme mass ratio inspirals, which are one of the main targets of Laser Interferometer Space Antenna.

by Zhang, Zhongyang and Yunes, Nicolas and Berti, Emanuele
9 pages, 8 figures

We study the effect of black hole spin on the accuracy of the post-Newtonian approximation. We focus on the gravitational energy flux for the quasicircular, equatorial, extreme mass-ratio inspiral of a compact object into a Kerr black hole of mass M and spin J. For a given dimensionless spin a=J/M^2 (in geometrical units), the energy flux depends only on the orbital velocity v or (equivalently) on the Boyer-Lindquist orbital radius r. We investigate the formal region of validity of the Taylor post-Newtonian expansion of the energy flux (which is known up to order v^8 beyond the quadrupole formula), generalizing previous work by two of us. The “error function” used to determine the region of validity of the post-Newtonian expansion can have two qualitatively different kinds of behavior, and we deal with these two cases separately. We find that, at any fixed post-Newtonian order, the edge of the region of validity (as measured by v/v_{ISCO}, where v_{ISCO} is the orbital velocity at the innermost stable circular orbit) is only weakly dependent on a. Unlike in the nonspinning case, the lack of sufficiently high order terms does not allow us to determine if there is a convergent to divergent transition at order v^6. Independently of a, the inclusion of angular multipoles up to and including l=5 in the numerical flux is necessary to achieve the level of accuracy of the best-known (N=8) PN expansion of the energy flux.

by Will, Clifford M.
9 pages, 2 figures, submitted to Proceedings of the National Academy of Sciences (US)

The post-Newtonian approximation is a method for solving Einstein’s field equations for physical systems in which motions are slow compared to the speed of light and where gravitational fields are weak. Yet it has proven to be remarkably effective in describing certain strong-field, fast-motion systems, including binary pulsars containing dense neutron stars and binary black hole systems inspiraling toward a final merger. The reasons for this effectiveness are largely unknown. When carried to high orders in the post-Newtonian sequence, predictions for the gravitational-wave signal from inspiraling compact binaries will play a key role in gravitational-wave detection by laser-interferometric observatories.

by Sperhake, Ulrich and Bruegmann, Bernd and Mueller, Doreen and Sopuerta, Carlos F.
Submitted to CQG, Special Edition for NRDA/CAPRA 2010, 13 pages, 7 figures

We analyse an eleven-orbit inspiral of a non-spinning black-hole binary with mass ratio q=M1/M2=4. The numerically obtained gravitational waveforms are compared with post-Newtonian (PN) predictions including several sub-dominant multipoles up to multipolar indices (l=5,m=5). We find that (i) numerical and post-Newtonian predictions of the phase of the (2,2) mode accumulate a phase difference of about 0.35 rad at the PN cut off frequency 0.1 for the Taylor T1 approximant; (ii) in contrast to previous studies of equal-mass and specific spinning binaries, we find the Taylor T4 approximant to agree less well with numerical results, provided the latter are extrapolated to infinite extraction radius; (iii) extrapolation of gravitational waveforms to infinite extraction radius is particularly important for subdominant multipoles with l unequal m; (iv) 3PN terms in post-Newtonian multipole expansions significantly improve the agreement with numerical predictions for sub-dominant multipoles.

by Favata, Marc
17 pages, 2 figures, 1 table

The innermost stable circular orbit (ISCO) delimits the transition from circular orbits to those that plunge into a black hole. In the test-mass limit, well-defined ISCO conditions exist for the Kerr and Schwarzschild spacetimes. In the finite-mass case, there are a large variety of ways to define an ISCO in a post-Newtonian (PN) context. Here I generalize the gauge-invariant ISCO condition of Blanchet & Iyer (2003) to the case of spinning (non-precessing) binaries. The Blanchet-Iyer ISCO condition has two desirable and unexpected properties: (1) it exactly reproduces the Schwarzschild ISCO in the test-mass limit, and (2) it accurately approximates the recently-calculated shift in the Schwarzschild ISCO frequency due to the conservative-piece of the gravitational self-force [Barack & Sago (2009)]. The generalization of this ISCO condition to spinning binaries has the property that it also exactly reproduces the Kerr ISCO in the test-mass limit (up to the order at which PN spin corrections are currently known). The shift in the ISCO due to the spin of the test-particle is also calculated. Remarkably, the gauge-invariant PN ISCO condition exactly reproduces the ISCO shift predicted by the Papapetrou equations for a fully-relativistic spinning particle. It is surprising that an analysis of the stability of the standard PN equations of motion is able (without any form of “resummation”) to accurately describe strong-field effects of the Kerr spacetime. The ISCO frequency shift due to the conservative self-force in Kerr is also calculated from this new ISCO condition, as well as from the effective-one-body Hamiltonian of Barausse & Buonanno (2010). These results serve as a useful point-of-comparison for future gravitational self-force calculations in the Kerr spacetime.

by Damour, T. and Trias, M. and Nagar, A.
29 pages, 7 figures, 1 table

The coalescences of binary black hole (BBH) systems, here taken to be non-spinning, are among the most promising sources for gravitational wave (GW) ground-based detectors, such as LIGO and Virgo. To detect the GW signals emitted by BBHs, and measure the parameters of the source, one needs to have in hand a bank of GW templates that are both effectual (for detection), and accurate (for measurement). We study the effectualness and the accuracy of the two types of parametrized banks of templates that are directly defined in the frequency-domain by means of closed-form expressions, namely ‘post-Newtonian’ (PN) and ‘phenomenological’ models. In absence of knowledge of the exact waveforms, our study assumes as fiducial, target waveforms the ones generated by the most accurate version of the effective one body (EOB) formalism. We find that, for initial GW detectors the use, at each point of parameter space, of the best closed-form template (among PN and phenomenological models) leads to an effectualness >97% over the entire mass range and >99% in an important fraction of parameter space; however, when considering advanced detectors, both of the closed-form frequency-domain models fail to be effectual enough in significant domains of the two-dimensional [total mass and mass ratio] parameter space. Moreover, we find that, both for initial and advanced detectors, the two closed-form frequency-domain models fail to satisfy the minimal required accuracy standard in a very large domain of the two-dimensional parameter space. In addition, a side result of our study is the determination, as a function of the mass ratio, of the maximum frequency at which a frequency-domain PN waveform can be ‘joined’ onto a NR-calibrated EOB waveform without undue loss of accuracy.

by Favata, Marc
26 pages, 2 figures, 2 tables

[abridged] Barack & Sago have recently computed the shift of the innermost stable circular orbit (ISCO) due to the conservative self-force that arises from the finite-mass of an orbiting test-particle. This is one of the first concrete results of the self-force program, and provides an exact point of comparison with approximate post-Newtonian (PN) computations of the ISCO. Here this exact ISCO shift is compared with nearly all known PN-based methods. These include both “non-resummed” and “resummed” approaches (the latter reproduce the test-particle limit by construction). The best agreement with the exact result is found from effective-one-body (EOB) calculations that are fit to numerical relativity simulations. However, if one considers uncalibrated methods based only on the currently-known 3PN-order conservative dynamics, the best agreement is found from the gauge-invariant ISCO condition of Blanchet and Iyer (2003). This method reproduces the exact test-particle limit without any resummation. A comparison of PN methods with the equal-mass ISCO is also performed. The results of this study suggest that the EOB approach—while exactly incorporating the conservative test-particle dynamics—does not (in the absence of calibration) incorporate conservative self-force effects more accurately than standard PN methods. I also consider how the conservative self-force ISCO shift, combined with numerical relativity computations of the ISCO, can be used to constrain our knowledge of (1) the EOB effective metric, (2) phenomenological inspiral-merger-ringdown templates, and (3) 4PN and 5PN order terms in the PN orbital energy. These constraints could help in constructing better gravitational-wave templates. Lastly, I suggest a new method to calibrate unknown PN-terms in inspiral templates using “low-cost” numerical-relativity calculations.

by Hannam, Mark and Husa, Sascha and Ohme, Frank and Ajith, P.
13 pages, 11 figures, 2 tables

One way to produce complete inspiral-merger-ringdown gravitational waveforms from black-hole-binary systems is to connect post-Newtonian (PN) and numerical-relativity (NR) results to create “hybrid” waveforms. Hybrid waveforms are central to the construction of some phenomenological models for GW search templates, and for tests of GW search pipelines. The dominant error source in hybrid waveforms arises from the PN contribution, and can be reduced by increasing the number of NR GW cycles that are included in the hybrid. Hybrid waveforms are considered sufficiently accurate for GW detection if their mismatch error is below 3% (i.e., a fitting factor about 0.97). We address the question of the length requirements of NR waveforms such that the final hybrid waveforms meet this requirement, considering nonspinning binaries with q = M_2/M_1 \in [1,4] and equal-mass binaries with \chi = S_i/M_i^2 \in [-0.5,0.5]. We conclude that for the cases we study simulations must contain between three (in the equal-mass nonspinning case) and ten (the \chi = 0.5 case) orbits before merger, but there is also evidence that these are the regions of parameter space for which the least number of cycles will be needed.

by Barack, Leor and Damour, Thibault and Sago, Norichika
25 pages, 5 eps figures

Using a recently presented numerical code for calculating the Lorenz-gauge gravitational self-force (GSF), we compute the $latex O(m)$ conservative correction to the precession rate of the small-eccentricity orbits of a particle of mass $latex m$ moving around a Schwarzschild black hole of mass $latex {\mathsf M}\gg m$. Specifically, we study the gauge-invariant function $latex \rho(x)$, where $latex \rho$ is defined as the $latex O(m)$ part of the dimensionless ratio $latex (\hat\Omega_r/\hat\Omega_{\varphi})^2$ between the squares of the radial and azimuthal frequencies of the orbit, and where $latex x=[Gc^{-3}({\mathsf M}+m)\hat\Omega_{\varphi}]^{2/3}$ is a gauge-invariant measure of the dimensionless gravitational potential (mass over radius) associated with the mean circular orbit. Our GSF computation of the function $latex \rho(x)$ in the interval $latex 0<x\leq 1/6$ determines, for the first time, the {\em strong-field behavior} of a combination of two of the basic functions entering the Effective One Body (EOB) description of the conservative dynamics of binary systems. We show that our results agree well in the weak-field regime (small $latex x$) with the 3rd post-Newtonian (PN) expansion of the EOB results, and that this agreement is improved when taking into account the analytic values of some of the logarithmic-running terms occurring at higher PN orders. Furthermore, we demonstrate that GSF data give access to higher-order PN terms of $latex \rho(x)$ and can be used to set useful new constraints on the values of yet-undetermined EOB parameters. Most significantly, we observe that an {\em excellent global representation} of $latex \rho(x)$ can be obtained using a simple `two-point' Pad\'{e} approximant which combines 3PN knowledge at $latex x=0$ with GSF information at a single strong-field point (say, $latex x=1/6$).

by Blanchet, Luc and Detweiler, Steven and Tiec, Alexandre Le and Whiting, Bernard F.
29 pages, 3 figures; to appear in the book “Mass and Motion in General Relativity”, proceedings of the C.N.R.S. School in Orleans, France, eds. L. Blanchet, A. Spallicci and B. F. Whiting

The relativistic motion of a compact binary system moving in circular orbit is investigated using the post-Newtonian (PN) approximation and the perturbative self-force (SF) formalism. A particular gauge-invariant observable quantity is computed as a function of the binary’s orbital frequency. The conservative effect induced by the gravitational SF is obtained numerically with high precision, and compared to the PN prediction developed to high order. The PN calculation involves the computation of the 3PN regularized metric at the location of the particle. Its divergent self-field is regularized by means of dimensional regularization. The poles proportional to 1/(d-3) which occur within dimensional regularization at the 3PN order disappear from the final gauge-invariant result. The leading 4PN and next-to-leading 5PN conservative logarithmic contributions originating from gravitational-wave tails are also obtained. Making use of these exact PN results, some previously unknown PN coefficients are measured up to the very high 7PN order by fitting to the numerical self-force data. Using just the 2PN and new logarithmic terms, the value of the 3PN coefficient is also confirmed numerically with very high precision. The consistency of this cross-cultural comparison provides a crucial test of the very different regularization methods used in both SF and PN formalisms, and illustrates the complementarity of these approximation schemes when modelling compact binary systems.

by Nichols, David A. and Chen, Yanbei
13 pages, 6 figures, 1 table, revtex4 format

Black-hole-binary coalescence is often divided into three stages: inspiral, merger and ringdown. The post-Newtonian (PN) approximation treats the inspiral phase, black-hole perturbation (BHP) theory describes the ringdown, and the nonlinear dynamics of spacetime characterize the merger. In this paper, we introduce a hybrid method that incorporates elements of PN and BHP theories, and we apply it to the head-on collision of black holes with transverse, anti-parallel spins. We compare our approximation technique with a full numerical-relativity simulation, and we find good agreement between the gravitational waveforms and the radiated energy and momentum. Our results suggest that PN and BHP theories may suffice to explain the main features of outgoing gravitational radiation for head-on mergers. This would further imply that linear perturbations to exact black-hole solutions can capture the nonlinear aspects of head-on binary-black-hole mergers accessible to observers far from the collision.

by Tessmer, Manuel and Schaefer, Gerhard
28 pages

The article provides ready-to-use 1pN-accurate frequency-domain gravitational wave forms for eccentric nonspinning compact binaries of arbitrary mass ratio including the first post-Newtonian (1pN) point particle corrections to the far-zone gravitational wave amplitude, given in terms of tensor spherical harmonics. The averaged equations for the decay of the eccentricity and growth of radial frequency due to radiation reaction are used to provide stationary phase approximations to the frequency-domain wave forms.

by Galley, Chad R. and Herrmann, Frank and Silberholz, John and Tiglio, Manuel and Guerberoff, Gustavo

We perform a statistical analysis of the binary black hole problem in the post-Newtonian approximation by systematically sampling and evolving the parameter space of initial configurations for quasi-circular inspirals. Through a principal component analysis of spin and orbital angular momentum variables we systematically look for uncorrelated quantities and find three of them which are highly conserved in a statistical sense, both as functions of time and with respect to variations in initial spin orientations. We also look for and find the variables that account for the largest variations in the problem. We present binary black hole simulations of the full Einstein equations analyzing to what extent these results might carry over to the full theory in the inspiral and merger regimes. Among other applications these results should be useful both in semi-analytical and numerical building of templates of gravitational waves for gravitational wave detectors.

by Gergely, László Á. and Biermann, Peter L. and Caramete, Laurenţiu I.
11 pages, 2 figures

During post-Newtonian evolution of a compact binary, a mass ratio different from one provides a second small parameter, which can lead to unexpected results. We present a statistics of supermassive black hole candidates, which enables us first to derive their mass distribution, then to establish a logarithmically even probability of the mass ratios at their encounter. In the mass ratio range (1/30,1/3) of supermassive black hole mergers representing 40% of all possible cases, the combined effect of spin-orbit precession and gravitational radiation leads to a spin-flip of the dominant spin during the inspiral phase of the merger. This provides a mechanism for explaining a large set of observations on X-shaped radio galaxies. In another 40%, with mass ratios (1/30,1/1000) a spin-flip never happens, while in the remaining 20% of mergers with mass ratios (1/3,1) it may occur during the plunge. We analyze the magnitude of the spin-flip angle occurring during the inspiral as function of the mass ratio and original relative orientation of the spin and orbital angular momentum. We also derive a formula for the final spin at the end of the inspiral in this mass ratio range.

by Klein, Antoine and Jetzer, Philippe
7 pages, 1 figure; Accepted for publication in Phys. Rev. D

We compute here the spin-orbit and spin-spin couplings needed for an accurate computation of the phasing of gravitational waves emitted by comparable-mass binaries on eccentric orbits at the second post-Newtonian (PN) order. We use a quasi-Keplerian parametrization of the orbit free of divergencies in the zero eccentricity limit. We find that spin-spin couplings induce a residual eccentricity for coalescing binaries at 2PN, of the order of $latex 10^{-4}$-$latex 10^{-3}$ for supermassive black hole binaries in the LISA band. Spin-orbit precession also induces a non-trivial pattern in the evolution of the eccentricity, which could help to reduce the errors on the determination of the eccentricity and spins in a gravitational wave measurement.

by Cornish, Neil J. and Key, Joey Shapiro
9 pages

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 the 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 implementing these waveforms is high, requiring many steps per orbit to evolve the system of coupled differential equations. Here we employ a separation of timescales and a generalized Keplarian parameterization of the orbits to produce efficient waveforms describing spinning black hole binaries with arbitrary spin orientations on quasi-eccentric orbits to 1.5 PN order. Our solution includes the spin contributions to the decay of the semi-major axis and eccentricity. We outline a scheme for extending our approach to higher post-Newtonian order.

by Hergt, Steven and Steinhoff, Jan and Schaefer, Gerhard
11 pages, submitted to CQG

Within the post Newtonian framework the fully reduced Hamiltonian (i.e., with eliminated spin supplementary condition) for the next-to-leading order spin-squared dynamics of general compact binaries is presented. The Hamiltonian is applicable to the spin dynamics of all kinds of binaries with self-gravitating components like black holes and/or neutron stars taking into account spin-induced quadrupolar deformation effects in second post-Newtonian order perturbation theory of Einstein’s field equations. The corresponding equations of motion for spin, position and momentum variables are given in terms of canonical Poisson brackets. Comparison with a nonreduced potential calculated within the Effective Field Theory approach is made.

by Lubich, Christian and Walther, Benny and Bruegmann, Bernd
9 pages, 6 figures

We present a non-canonically symplectic integration scheme tailored to numerically computing the post-Newtonian motion of a spinning black-hole binary. Using a splitting approach we combine the flows of orbital and spin contributions. In the context of the splitting, it is possible to integrate the individual terms of the spin-orbit and spin-spin Hamiltonians analytically, exploiting the special structure of the underlying equations of motion. The outcome is a symplectic, time-reversible integrator, which can be raised to arbitrary order by composition. A fourth-order version is shown to give excellent behavior concerning error growth and conservation of energy and angular momentum in long-term simulations. Favorable properties of the integrator are retained in the presence of weak dissipative forces due to radiation damping in the full post-Newtonian equations.

by Favata, Marc
11 pages, 2 figures; proceedings of the 8th Amaldi Conference on Gravitational Waves (New York, June 2009); accepted for publication in special issue of Classical and Quantum Gravity

The nonlinear memory effect is a slowly-growing, non-oscillatory contribution to the gravitational-wave amplitude. It originates from gravitational waves that are sourced by the previously emitted waves. In an ideal gravitational-wave interferometer a gravitational-wave with memory causes a permanent displacement of the test masses that persists after the wave has passed. Surprisingly, the nonlinear memory affects the signal amplitude starting at leading (Newtonian-quadrupole) order. Despite this fact, the nonlinear memory is not easily extracted from current numerical relativity simulations. After reviewing the linear and nonlinear memory I summarize some recent work, including: (1) computations of the memory contribution to the inspiral waveform amplitude (thus completing the waveform to third post-Newtonian order); (2) the first calculations of the nonlinear memory that include all phases of binary black hole coalescence (inspiral, merger, ringdown); and (3) realistic estimates of the detectability of the memory with LISA.

by Tessmer, Manuel and Hartung, Johannes and Schaefer, Gerhard
26 pages, 1 figure

A quasi-Keplerian parameterisation for the solutions of second post-Newtonian (PN) accurate equations of motion for spinning compact binaries is obtained including leading order spin-spin and next-to-leading order spin-orbit interactions. Rotational deformation of the compact objects is incorporated. For arbitrary mass ratios the spin orientations are taken to be parallel or anti-parallel to the orbital angular momentum vector. The emitted gravitational wave forms are given in analytic form up to 2PN point particle, 1.5PN spin orbit and 1PN spin-spin contributions, where the spins are counted of 0PN order.

by Rothe, Tilman J. and Schäfer, Gerhard
18 pages, no figures

The recently constructed Hamiltonians for spinless binary black holes through third post-Newtonian order and for spinning ones through formal second post-Newtonian order, where the spins are counted of zero post-Newtonian order, are transformed into fully canonical center-of-mass and rest-frame variables. The mixture terms in the Hamiltonians between center-of-mass and rest-frame variables are in accordance with the relation between the total linear momentum and the center-of-mass velocity as demanded by global Lorentz invariance. The various generating functions for the center-of-mass and rest-frame canonical variables are explicitly given in terms of the single-particle canonical variables. The no-interaction theorem does not apply because the world-line condition of Lorentz covariant position variables is not imposed.

by Kesden, Michael and Sperhake, Ulrich and Berti, Emanuele
20 pages, 16 figures, revtex

The inspiral of binary black holes is governed by gravitational radiation reaction at binary separations r 10 M. Fortunately, binary evolution between these separations is well described by post-Newtonian equations of motion. We examine how this post-Newtonian evolution affects the distribution of spin orientations at separations r near 10 M where numerical-relativity simulations typically begin. Although isotropic spin distributions at r =1000 M remain isotropic at r = 10 M, distributions that are initially partially aligned with the orbital angular momentum can be significantly distorted during the post-Newtonian inspiral. Spin-orbit resonances tend to align (anti-align) the binary black hole spins with each other if the spins were initially partially aligned (anti-aligned) with respect to the orbital angular momentum, thus increasing (decreasing) the average final spin. Resonant effects are stronger for comparable-mass binaries, and they could produce significant spin alignment in massive black hole mergers at high redshifts and in stellar-mass black hole binaries. We also point out that precession induces an intrinsic accuracy limitation of 0.03 in the dimensionless spin magnitude, and about 20 degrees in the direction in predicting the final spin resulting from widely separated binary configurations.

by Blanchet, Luc and Detweiler, Steven and Tiec, Alexandre Le and Whiting, Bernard F.
32 pages, 2 figures

We continue a previous work on the comparison between the post-Newtonian (PN) approximation and the gravitational self-force (SF) analysis of circular orbits in a Schwarzschild background. We show that the numerical SF data contain physical information corresponding to extremely high PN approximations. We find that knowing analytically determined appropriate PN parameters helps tremendously in allowing the numerical data to be used to obtain higher order PN coefficients. Using standard PN theory we compute analytically the leading 4PN and the next-to-leading 5PN logarithmic terms in the conservative part of the dynamics of a compact binary system. The numerical perturbative SF results support well the analytic PN calculations through first order in the mass ratio, and are used to accurately measure the 4PN and 5PN non-logarithmic coefficients in a particular gauge invariant observable. Furthermore we are able to give estimates of higher order contributions up to the 7PN level. In our best fit we also confirm with high precision the value of the 3PN coefficient. This interplay between PN and SF efforts is important for the synthesis of template waveforms of extreme mass ratio inspirals to be analysed by the space-based gravitational wave instrument LISA. Our work will also have an impact on efforts that combine numerical results in a quantitative analytical framework so as to generate complete inspiral waveforms for the ground-based detection of gravitational waves by instruments such as LIGO and Virgo.

by Campanelli, Manuela and Lousto, Carlos O. and Mundim, Bruno C. and Nakano, Hiroyuki and Zlochower, Yosef and Bischof, Hans-Peter
12 pages, 5 figures, Prepared for 8th Edoardo Amaldi Conference on Gravitational Waves (Amaldi8)

We review some of the recent dramatic developments in the fully nonlinear simulation of generic, highly-precessing, black-hole binaries, and introduce a new approach for generating hybrid post-Newtonian / Numerical waveforms for these challenging systems.

by Itoh, Yousuke
38 pages, no figures. Accepted for publication in Phys. Rev. D

We report our rederivation of the equations of motion for relativistic compact binaries through the third-and-a-half post-Newtonian (3.5 PN) order approximation to general relativity using the strong field point particle limit to describe self-gravitating stars instead of the Dirac delta functional. The computation is done in harmonic coordinates. Our equations of motion describe the orbital motion of the binary consisting of spherically symmetric non-rotating stars. The resulting equations of motion fully agree with the 3.5 PN equations of motion derived in the previous works. We also show that the locally defined energy of the star has a simple relation with its mass up to the 3.5 PN order.

by Damour, Thibault and Nagar, Alessandro
21 pages, 5 figures. Submitted to Phys. Rev. D

The late part of the gravitational wave signal of binary neutron star inspirals can in principle yield crucial information on the nuclear equation of state via its dependence on relativistic tidal parameters. In the hope of analytically describing the gravitational wave phasing during the late inspiral (essentially up to contact) we propose an extension of the effective one body (EOB) formalism which includes tidal effects. We compare the prediction of this tidal-EOB formalism to recently computed nonconformally flat quasi-equilibrium circular sequences of binary neutron star systems. Our analysis suggests the importance of higher-order (post-Newtonian) corrections to tidal effects, even beyond the first post-Newtonian order, and their tendency to {\it significantly} increase the “effective tidal polarizability” of neutron stars. We compare the EOB predictions to some recently advocated, nonresummed, post-Newtonian based (“Taylor-T4”) description of the phasing of inspiralling systems. This comparison shows the strong sensitivity of the late-inspiral phasing to the choice of the analytical model, but raises the hope that a sufficiently accurate numerical–relativity–“calibrated” EOB model might give us a reliable handle on the nuclear equation of state

by Gergely, László Árpád
12 pages, 2 figures

We establish the set of independent variables suitable to monitor the complicated evolution of the spinning compact binary during the inspiral. Our approach is valid up to the second post-Newtonian order, including spin and mass quadrupolar effects, for generic (noncircular, nonspherical) orbits. Then we analyze the conservative spin dynamics in terms of these variables. We prove that the only precessing and spinning black hole or neutron star binary configuration which is preserved by the post-Newtonian evolution with spin-spin and quadrupole-monopole contributions included is the equal mass, equal and identically oriented spin configuration. This analytic result puts severe limitations on what particular configurations can be selected in numerical investigations of compact binary evolutions, even in those including only the last orbits of the inspiral.

by Barausse, Enrico
5 pages, 3 figures. Submitted as proceeding of the 8th Amaldi International Conference on Gravitational Waves, NYC, 21-26 June 2009

The prediction of the spin of the black hole resulting from the merger of a generic black-hole binary system is of great importance to study the cosmological evolution of supermassive black holes. Several attempts have been recently made to model the spin via simple expressions exploiting the results of numerical-relativity simulations. Here I compare the results of all the simulations appeared so far in the literature with various formulas for the final spin magnitude and direction. I show that although all the formulas give reasonable results for the final spin magnitude, only the formula that I recently proposed in (Barausse & Rezzolla, Apj 704 L40) accurately predicts the final spin direction when applied to binaries with separations of hundred or thousands of gravitational radii. This makes my formula particularly suitable for cosmological merger-trees and N-body simulations, which provide the spins and angular momentum of the two black holes when their separation is of thousands of gravitational radii, and happens because my formula takes into account the post-Newtonian precession of the spins in a consistent manner.

by Damour, Thibault
44 pages

We discuss various ways in which the computation of conservative Gravitational Self Force (GSF) effects on a point mass moving in a Schwarzschild background can inform us about the basic building blocks of the Effective One-Body (EOB) Hamiltonian. We display the information which can be extracted from the recently published GSF calculation of the first-GSF-order shift of the orbital frequency of the last stable circular orbit, and we combine this information with the one recently obtained by comparing the EOB formalism to high-accuracy numerical relativity (NR) data on coalescing binary black holes. The information coming from GSF data helps to break the degeneracy (among some EOB parameters) which was left after using comparable-mass NR data to constrain the EOB formalism. We suggest various ways of obtaining more information from GSF computations: either by studying eccentric orbits, or by focussing on a special zero-binding zoom-whirl orbit. We show that logarithmic terms start entering the post-Newtonian expansions of various (EOB and GSF) functions at the fourth post-Newtonian (4PN) level, and we analytically compute the first logarithm entering a certain, gauge-invariant “redshift” GSF function (defined along the sequence of circular orbits).

by Tiec, Alexandre Le and Blanchet, Luc and Will, Clifford M.
9 pages, 5 figures

The gravitational recoil or “kick” of a black hole formed from the merger of two orbiting black holes, and caused by the anisotropic emission of gravitational radiation, is an astrophysically important phenomenon. We combine (i) an earlier calculation, using post-Newtonian theory, of the kick velocity accumulated up to the merger of two non-spinning black holes, (ii) a “close-limit approximation” calculation of the radiation emitted during the ringdown phase, and based on a solution of the Regge-Wheeler and Zerilli equations using initial data accurate to second post-Newtonian order. We prove that ringdown radiation produces a significant “anti-kick”. Adding the contributions due to inspiral, merger and ringdown phases, our results for the net kick velocity agree with those from numerical relativity to 10-15 percent over a wide range of mass ratios, with a maximum velocity of 180 km/s at a mass ratio of 0.38.

by Tiec, Alexandre Le and Blanchet, Luc
34 pages, 6 figures

The ringdown phase of a black hole formed from the merger of two orbiting black holes is described by means of the close-limit (CL) approximation starting from second-post-Newtonian (2PN) initial conditions. The 2PN metric of point-particle binaries is formally expanded in CL form and identified with that of a perturbed Schwarzschild black hole. The multipolar coefficients describing the even-parity (or polar) and odd-parity (axial) components of the linear perturbation consistently satisfy the 2PN-accurate perturbative field equations. We use these coefficients to build initial conditions for the Regge-Wheeler and Zerilli wave equations, which we then evolve numerically. The ringdown waveform is obtained in two cases: head-on collision with zero-angular momentum, composed only of even modes, and circular orbits, for which both even and odd modes contribute. In a separate work, this formalism is applied to the study of the gravitational recoil produced during the ringdown phase of coalescing binary black holes.

by Schaefer, Gerhard
44 pages, to appear in the book “Mass and Motion in General Relativity”, proceedings of the CNRS School in Orleans/France, eds. L. Blanchet, A. Spallicci, and B. Whiting

A detailed account is given on approximation schemes to the Einstein theory of general relativity where the iteration starts from the Newton theory of gravity. Two different coordinate conditions are used to represent the Einstein field equations, the generalized isotropic ones of the canonical formalism of Arnowitt, Deser, and Misner and the harmonic ones of the Lorentz-covariant Fock-de Donder approach. Conserved quantities of isolated systems are identified and the Poincar\’e algebra is introduced. Post-Newtonian expansions are performed in the near and far (radiation) zones. The natural fitting of multipole expansions to post-Newtonian schemes is emphasized. The treated matter models are ideal fluids, pure point masses, and point masses with spin and mass-quadrupole moments modelling rotating black holes. Various Hamiltonians of spinning binaries are presented in explicit forms to higher post-Newtonian orders. The delicate use of black holes in post-Newtonian expansion calculations and of the Dirac delta function in general relativity find discussions.

by Blanchet, Luc and Detweiler, Steven and Tiec, Alexandre Le and Whiting, Bernard F.
36 pages, 3 figures

The problem of a compact binary system whose components move on circular orbits is addressed using two different approximation techniques in general relativity. The post-Newtonian (PN) approximation involves an expansion in powers of v/c<<1, and is most appropriate for small orbital velocities v. The perturbative self-force (SF) analysis requires an extreme mass ratio m1/m2<<1 for the components of the binary. A particular coordinate-invariant observable is determined as a function of the orbital frequency of the system using these two different approximations. The post-Newtonian calculation is pushed up to the third post-Newtonian (3PN) order. It involves the metric generated by two point particles and evaluated at the location of one of the particles. We regularize the divergent self-field of the particle by means of dimensional regularization. We show that the poles proportional to 1/(d-3) appearing in dimensional regularization at the 3PN order cancel out from the final gauge invariant observable. The 3PN analytical result, through first order in the mass ratio, and the numerical SF calculation are found to agree well. The consistency of this cross cultural comparison confirms the soundness of both approximations in describing compact binary systems. In particular, it provides an independent test of the very different regularization procedures invoked in the two approximation schemes.

by Ajith, P. and Hannam, M. and Husa, S. and Chen, Y. and Bruegmann, B. and Dorband, N. and Mueller, D. and Ohme, F. and Pollney, D. and Reisswig, C. and Santamaria, L. and Seiler, J.
4 pages, 5 figures

We present the first analytical inspiral-merger-ringdown gravitational waveforms from black-hole (BH) binaries with non-precessing spins. By matching a post-Newtonian description of the inspiral to a set of numerical calculations performed in full general relativity, we obtain a waveform family with a conveniently small number of physical parameters. The physical content of these waveforms includes the “orbital hang-up” effect, when BHs are spinning rapidly along the direction of the orbital angular momentum. These waveforms will allow us to detect a larger parameter space of BH binary coalescence, to explore various scientific questions related to GW astronomy, and could dramatically improve the expected detection rates of GW detectors.

by Gergely, László Á. and Biermann, Peter L. and Mikóczi, Balázs and Keresztes, Zoltán
10 pages, to appear in Class. Quantum Grav. GWDAW13 Proceedings Special Issue, v2: no typos conjecture

We analyze galactic black hole mergers and their emitted gravitational waves. Such mergers have typically unequal masses with mass ratio of the order 1/10. The emitted gravitational waves carry the inprint of spins and mass quadrupoles of the binary components. Among these contributions, we consider here the quasi-precessional evolution of the spins. A method of taking into account these third post-Newtonian (3PN) effects by renormalizing (redefining) the 1.5 PN and 2PN accurate spin contributions to the accumulated orbital phase is developed.