Using Full Information When Computing Modes of Post-Newtonian Waveforms From Inspiralling Compact Binaries in Circular Orbit

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 polarization waveforms. In particular, the dominant (l = 2, m = 2) mode can be computed to 3PN order. Individual modes are computed to as high a post-Newtonian order as possible given previous post-Newtonian results.Comment: 15 page

Evolution systems for non-linear perturbations of background geometries

The formulation of the initial value problem for the Einstein equations is at the heart of obtaining interesting new solutions using numerical relativity and still very much under theoretical and applied scrutiny. We develop a specialised background geometry approach, for systems where there is non-trivial a priori knowledge about the spacetime under study. The background three-geometry and associated connection are used to express the ADM evolution equations in terms of physical non-linear deviations from that background. Expressing the equations in first order form leads naturally to a system closely linked to the Einstein-Christoffel system, introduced by Anderson and York, and sharing its hyperbolicity properties. We illustrate the drastic alteration of the source structure of the equations, and discuss why this is likely to be numerically advantageous.Comment: 12 pages, 3 figures, Revtex v3.0. Revised version to appear in Physical Review

Ruling Out Chaos in Compact Binary Systems

We investigate the orbits of compact binary systems during the final inspiral period before coalescence by integrating numerically the second-order post-Newtonian equations of motion. We include spin-orbit and spin-spin coupling terms, which, according to a recent study by Levin [J. Levin, Phys. Rev. Lett. 84, 3515 (2000)], may cause the orbits to become chaotic. To examine this claim, we study the divergence of initially nearby phase-space trajectories and attempt to measure the Lyapunov exponent gamma. Even for systems with maximally spinning objects and large spin-orbit misalignment angles, we find no chaotic behavior. For all the systems we consider, we can place a strict lower limit on the divergence time t_L=1/gamma that is many times greater than the typical inspiral time, suggesting that chaos should not adversely affect the detection of inspiral events by upcoming gravitational-wave detectors.Comment: 8 pages, 4 figures, submitted to Phys. Rev. Let

Coalescence of Two Spinning Black Holes: An Effective One-Body Approach

We generalize to the case of spinning black holes a recently introduced ``effective one-body'' approach to the general relativistic dynamics of binary systems. The combination of the effective one-body approach, and of a Pad\'e definition of some crucial effective radial functions, is shown to define a dynamics with much improved post-Newtonian convergence properties, even for black hole separations of the order of $6 GM / c^2$. We discuss the approximate existence of a two-parameter family of ``spherical orbits'' (with constant radius), and, of a corresponding one-parameter family of ``last stable spherical orbits'' (LSSO). These orbits are of special interest for forthcoming LIGO/VIRGO/GEO gravitational wave observations. It is argued that for most (but not all) of the parameter space of two spinning holes the effective one-body approach gives a reliable analytical tool for describing the dynamics of the last orbits before coalescence. This tool predicts, in a quantitative way, how certain spin orientations increase the binding energy of the LSSO. This leads to a detection bias, in LIGO/VIRGO/GEO observations, favouring spinning black hole systems, and makes it urgent to complete the conservative effective one-body dynamics given here by adding (resummed) radiation reaction effects, and by constructing gravitational waveform templates that include spin effects. Finally, our approach predicts that the spin of the final hole formed by the coalescence of two arbitrarily spinning holes never approaches extremality.Comment: 26 pages, two eps figures, accepted in Phys. Rev. D, minor updating of the text, clarifications added and inclusion of a few new reference

Improved filters for gravitational waves from inspiralling compact binaries

The order of the post-Newtonian expansion needed, to extract in a reliable and accurate manner the fully general relativistic gravitational wave signal from inspiralling compact binaries, is explored. A class of approximate wave forms, called P-approximants, is constructed based on the following two inputs: (a) The introduction of two new energy-type and flux-type functions e(v) and f(v), respectively, (b) the systematic use of Pade approximation for constructing successive approximants of e(v) and f(v). The new P-approximants are not only more effectual (larger overlaps) and more faithful (smaller biases) than the standard Taylor approximants, but also converge faster and monotonically. The presently available O(v/c)^5-accurate post-Newtonian results can be used to construct P-approximate wave forms that provide overlaps with the exact wave form larger than 96.5% implying that more than 90% of potential events can be detected with the aid of P-approximants as opposed to a mere 10-15 % that would be detectable using standard post-Newtonian approximants.Comment: Latex ([prd,aps,eqsecnum,epsf]{revtex}), 40 pages including 12 encapsulated figures. (The paper, together with all the figures and tables is available from ftp://carina.astro.cf.ac.uk/pub/incoming/sathya/dis97.uu

Einstein boundary conditions for the 3+1 Einstein equations

In the 3+1 framework of the Einstein equations for the case of vanishing shift vector and arbitrary lapse, we calculate explicitly the four boundary equations arising from the vanishing of the projection of the Einstein tensor along the normal to the boundary surface of the initial-boundary value problem. Such conditions take the form of evolution equations along (as opposed to across) the boundary for certain components of the extrinsic curvature and for certain space-derivatives of the intrinsic metric. We argue that, in general, such boundary conditions do not follow necessarily from the evolution equations and the initial data, but need to be imposed on the boundary values of the fundamental variables. Using the Einstein-Christoffel formulation, which is strongly hyperbolic, we show how three of the boundary equations should be used to prescribe the values of some incoming characteristic fields. Additionally, we show that the fourth one imposes conditions on some outgoing fields.Comment: Revtex 4, 6 pages, text and references added, typos corrected, to appear in Phys. Rev.

Spin effects in gravitational radiation backreaction III. Compact binaries with two spinning components

The secular evolution of a spinning, massive binary system in eccentric orbit is analyzed, expanding and generalizing our previous treatments of the Lense-Thirring motion and the one-spin limit. The spin-orbit and spin-spin effects up to the 3/2 post-Newtonian order are considered, both in the equations of motion and in the radiative losses. The description of the orbit in terms of the true anomaly parametrization provides a simple averaging technique, based on the residue theorem, over eccentric orbits. The evolution equations of the angle variables characterizing the relative orientation of the spin and orbital angular momenta reveal a speed-up effect due to the eccentricity. The dissipative evolutions of the relevant dynamical and angular variables is presented in the form of a closed system of differential equations.Comment: 10 pages, 1 figur

The Innermost Stable Circular Orbit of Binary Black Holes

We introduce a new method to construct solutions to the constraint equations of general relativity describing binary black holes in quasicircular orbit. Black hole pairs with arbitrary momenta can be constructed with a simple method recently suggested by Brandt and Bruegmann, and quasicircular orbits can then be found by locating a minimum in the binding energy along sequences of constant horizon area. This approach produces binary black holes in a "three-sheeted" manifold structure, as opposed to the "two-sheeted" structure in the conformal-imaging approach adopted earlier by Cook. We focus on locating the innermost stable circular orbit and compare with earlier calculations. Our results confirm those of Cook and imply that the underlying manifold structure has a very small effect on the location of the innermost stable circular orbit.Comment: 8 pages, 3 figures, RevTex, submitted to PR

On the determination of the last stable orbit for circular general relativistic binaries at the third post-Newtonian approximation

We discuss the analytical determination of the location of the Last Stable Orbit (LSO) in circular general relativistic orbits of two point masses. We use several different ``resummation methods'' (including new ones) based on the consideration of gauge-invariant functions, and compare the results they give at the third post-Newtonian (3PN) approximation of general relativity. Our treatment is based on the 3PN Hamiltonian of Jaranowski and Sch\"afer. One of the new methods we introduce is based on the consideration of the (invariant) function linking the angular momentum and the angular frequency. We also generalize the ``effective one-body'' approach of Buonanno and Damour by introducing a non-minimal (i.e. ``non-geodesic'') effective dynamics at the 3PN level. We find that the location of the LSO sensitively depends on the (currently unknown) value of the dimensionless quantity \oms which parametrizes a certain regularization ambiguity of the 3PN dynamics. We find, however, that all the analytical methods we use numerically agree between themselves if the value of this parameter is \oms\simeq-9. This suggests that the correct value of \oms is near -9 (the precise value \oms^*\equiv-{47/3}+{41/64}\pi^2=-9.3439... seems to play a special role). If this is the case, we then show how to further improve the analytical determination of various LSO quantities by using a ``Shanks'' transformation to accelerate the convergence of the successive (already resummed) PN estimates.Comment: REVTeX, 25 pages, 3 figures, submitted to Phys. Rev.

Spin effects in gravitational radiation backreaction II. Finite mass effects

A convenient formalism for averaging the losses produced by gravitational radiation backreaction over one orbital period was developed in an earlier paper. In the present paper we generalize this formalism to include the case of a closed system composed from two bodies of comparable masses, one of them having the spin S. We employ the equations of motion given by Barker and O'Connell, where terms up to linear order in the spin (the spin-orbit interaction terms) are kept. To obtain the radiative losses up to terms linear in the spin, the equations of motion are taken to the same order. Then the magnitude L of the angular momentum L, the angle kappa subtended by S and L and the energy E are conserved. The analysis of the radial motion leads to a new parametrization of the orbit. From the instantaneous gravitational radiation losses computed by Kidder the leading terms and the spin-orbit terms are taken. Following Apostolatos, Cutler, Sussman and Thorne, the evolution of the vectors S and L in the momentary plane spanned by these vectors is separated from the evolution of the plane in space. The radiation-induced change in the spin is smaller than the leading-order spin terms in the momentary angular momentum loss. This enables us to compute the averaged losses in the constants of motion E, L and L_S=L cos kappa. In the latter, the radiative spin loss terms average to zero. An alternative description using the orbital elements a,e and kappa is given. The finite mass effects contribute terms, comparable in magnitude, to the basic, test-particle spin terms in the averaged losses.Comment: 12 pages, 1 figure, Phys.Rev.D15, March, 199