The gravitational-wave memory effect

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.Comment: 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 Gravit

Nonlinear gravitational-wave memory from binary black hole mergers

Some astrophysical sources of gravitational waves can produce a "memory effect," which causes a permanent displacement of the test masses in a freely falling gravitational-wave detector. The Christodoulou memory is a particularly interesting nonlinear form of memory that arises from the gravitational-wave stress-energy tensor's contribution to the distant gravitational-wave field. This nonlinear memory contributes a nonoscillatory component to the gravitational-wave signal at leading (Newtonian-quadrupole) order in the waveform amplitude. Previous computations of the memory and its detectability considered only the inspiral phase of binary black hole coalescence. Using an "effective-one-body" (EOB) approach calibrated to numerical relativity simulations, as well as a simple fully analytic model, the Christodoulou memory is computed for the inspiral, merger, and ringdown. The memory will be very difficult to detect with ground-based interferometers, but is likely to be observable in supermassive black hole mergers with LISA out to a redshift of two. Detection of the nonlinear memory could serve as an experimental test of the ability of gravity to "gravitate."Comment: 4 pages, 3 figures. v2: minor changes to text and references; published in ApJ Letter

Conservative self-force correction to the innermost stable circular orbit: comparison with multiple post-Newtonian-based methods

[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 "nonresummed" 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 numerical-relativity calculations.Comment: 27 pages, 2 figures, 2 tables. v2: some changes to Sec. VI in response to referee comments; references added; other minor changes to match published versio

Energy Localization Invariance of Tidal Work in General Relativity

It is well known that, when an external general relativistic (electric-type) tidal field E(t) interacts with the evolving quadrupole moment I(t) of an isolated body, the tidal field does work on the body (``tidal work'') -- i.e., it transfers energy to the body -- at a rate given by the same formula as in Newtonian theory: dW/dt = -1/2 E dI/dt. Thorne has posed the following question: In view of the fact that the gravitational interaction energy between the tidal field and the body is ambiguous by an amount of order E(t)I(t), is the tidal work also ambiguous by this amount, and therefore is the formula dW/dt = -1/2 E dI/dt only valid unambiguously when integrated over timescales long compared to that for I(t) to change substantially? This paper completes a demonstration that the answer is no; dW/dt is not ambiguous in this way. More specifically, this paper shows that dW/dt is unambiguously given by -1/2 E dI/dt independently of one's choice of how to localize gravitational energy in general relativity. This is proved by explicitly computing dW/dt using various gravitational stress-energy pseudotensors (Einstein, Landau-Lifshitz, Moller) as well as Bergmann's conserved quantities which generalize many of the pseudotensors to include an arbitrary function of position. A discussion is also given of the problem of formulating conservation laws in general relativity and the role played by the various pseudotensors.Comment: 15 pages, no figures, revtex. Submitted to Phys. Rev.

Conservative corrections to the innermost stable circular orbit (ISCO) of a Kerr black hole: a new gauge-invariant post-Newtonian ISCO condition, and the ISCO shift due to test-particle spin and the gravitational self-force

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 (nonprecessing) 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.Comment: 17 pages, 2 figures, 1 table. v2: references added; minor changes to match published versio

Gravitational-Wave Memory Revisited: Memory from the Merger and Recoil of Binary Black Holes

Gravitational-wave memory refers to the permanent displacement of the test masses in an idealized (freely-falling) gravitational-wave interferometer. Inspiraling binaries produce a particularly interesting form of memory - the Christodoulou memory. Although it originates from nonlinear interactions at 2.5 post-Newtonian order, the Christodoulou memory affects the gravitational-wave amplitude at leading (Newtonian) order. Previous calculations have computed this non-oscillatory amplitude correction during the inspiral phase of binary coalescence. Using an effective-one-body description calibrated with the results of numerical relativity simulations, the evolution of the memory during the inspiral, merger, and ringdown phases, as well as the memory\u27s final saturation value, are calculated. Using this model for the memory, the prospects for its detection are examined, particularly for supermassive black hole binary coalescences that LISA will detect with high signal-to-noise ratios. Coalescing binary black holes also experience center-of-mass recoil due to the anisotropic emission of gravitational radiation. These recoils can manifest themselves in the gravitational-wave signal in the form of a linear memory and a Doppler shift of the quasi-normal-mode frequencies. The prospects for observing these effects are also discussed

Gravitational-wave phasing for low-eccentricity inspiralling compact binaries to 3PN order

[abridged] Although gravitational radiation causes inspiralling compact binaries to circularize, a variety of astrophysical scenarios suggest that binaries might have small but nonnegligible orbital eccentricities when they enter the low-frequency bands of ground and space-based gravitational-wave detectors. If not accounted for, even a small orbital eccentricity can cause a potentially significant systematic error in the mass parameters of an inspiralling binary. Gravitational-wave search templates typically rely on the quasi-circular approximation, which provides relatively simple expressions for the gravitational-wave phase to 3.5 post-Newtonian (PN) order. The quasi-Keplerian formalism provides an elegant but complex description of the post-Newtonian corrections to the orbits and waveforms of inspiralling binaries with any eccentricity. Here we specialize the quasi-Keplerian formalism to binaries with low eccentricity. In this limit the non-periodic contribution to the gravitational-wave phasing can be expressed explicitly as simple functions of frequency or time, with little additional complexity beyond the well-known formulas for circular binaries. These eccentric phase corrections are computed to 3PN order and to leading order in the eccentricity for the standard PN approximants. For a variety of systems these eccentricity corrections cause significant corrections to the number of gravitational wave cycles that sweep through a detector's frequency band. This is evaluated using several measures, including a modification of the useful cycles. We also evaluate the role of periodic terms that enter the phasing and discuss how they can be incorporated into some of the PN approximants. While the eccentric extension of the PN approximants is our main objective, this work collects a variety of results that may be of interest to others modeling eccentric relativistic binaries.Comment: 49 pages, 4 figures. Submitted to Phys. Rev. D. Supplementary materials available at http://link.aps.org/supplemental/10.1103/PhysRevD.93.124061. V2: minor updates to match published versio

Systematic Parameter Errors in Inspiraling Neutron Star Binaries

The coalescence of two neutron stars is an important gravitational wave source for LIGO and other detectors. Numerous studies have considered the precision with which binary parameters (masses, spins, Love numbers) can be measured. Here I consider the accuracy with which these parameters can be determined in the presence of systematic errors due to waveform approximations. These approximations include truncation of the post-Newtonian (PN) series and neglect of neutron star (NS) spin, tidal deformation, or orbital eccentricity. All of these effects can yield systematic errors that exceed statistical errors for plausible parameter values. In particular, neglecting spin, eccentricity, or high-order PN terms causes a significant bias in the NS Love number. Tidal effects will not be measurable with PN inspiral waveforms if these systematic errors are not controlled

Kicking Black Holes, Crushing Neutron Stars, and the Validity of the Adiabatic Approximation for Extreme-Mass-Ratio Inspirals

Current experiments hope to detect gravitational waves--oscillations of space and time predicted by Einstein. The strongest sources of gravitational waves are compact object binaries--orbiting neutron stars or black holes. Gravitational waves carry away energy, linear momentum, and angular momentum until the binary merges to form a single black hole. This thesis concerns three distinct projects regarding binary coalescence. The linear momentum radiated when binaries merge imparts a recoil or "kick" to the final black hole. Black hole recoils have important astrophysical consequences: black holes can be displaced or ejected from their host galaxies or globular clusters, affecting black hole growth, quasar activity, and the density structure of galaxies. We compute the kick velocity using black hole perturbation theory, treating the binary as a small mass spiraling into a massive, spinning black hole. We find that the recoil can easily reach ~100-200 km/s but probably does not exceed 500 km/s. Binary neutron stars are another important source of gravitational waves. Understanding the final coalescence phase of the gravitational wave signal requires computer simulations. Some numerical simulations have shown that the neutron stars are subject to a crushing force late in the inspiral. This crushing effect has had no explanation and is disputed. We show that a compressive force arises due to a coupling of gravitomagnetic tidal fields to the current-quadrupole moment of the neutron star. However, except in special circumstances, this gravitomagnetic crushing effect is overwhelmed by stabilizing Newtonian tidal interactions. A small compact object orbiting a massive black hole will be a strong source for space-based gravitational wave detectors. Accurate waveforms for these systems will require computing the self-force on the compact object. The tools to do this do not yet exist. But when the inspiral time is much longer than the orbital period (the adiabatic approximation), approximate waveforms for generic orbits can be computed. We estimate the error in the adiabatic approximation by computing the gravitational wave phase using post-Newtonian theory. We find that, for orbits with small eccentricity, the adiabatic waveforms will be good enough for detection but not for parameter extraction