Binary black hole coalescence in the large-mass-ratio limit: The hyperboloidal layer method and waveforms at null infinity

Abstract

We compute and analyze the gravitational waveform emitted to future null infinity by a system of two black holes in the large-mass-ratio limit. We consider the transition from the quasiadiabatic inspiral to plunge, merger, and ringdown. The relative dynamics is driven by a leading order in the mass ratio, 5PN-resummed, effective-one-body (EOB), analytic-radiation reaction. To compute the waveforms, we solve the Regge-Wheeler-Zerilli equations in the time-domain on a spacelike foliation, which coincides with the standard Schwarzschild foliation in the region including the motion of the small black hole, and is globally hyperboloidal, allowing us to include future null infinity in the computational domain by compactification. This method is called the hyperboloidal layer method, and is discussed here for the first time in a study of the gravitational radiation emitted by black hole binaries. We consider binaries characterized by five mass ratios, ¼ 10 À2;À3;À4;À5;À6 , that are primary targets of space-based or thirdgeneration gravitational wave detectors. We show significative phase differences between finite-radius and null-infinity waveforms. We test, in our context, the reliability of the extrapolation procedure routinely applied to numerical relativity waveforms. We present an updated calculation of the final and maximum gravitational recoil imparted to the merger remnant by the gravitational wave emission, v end kick =ðc 2 Þ ¼ 0:04474 AE 0:00007 and v max kick =ðc 2 Þ ¼ 0:05248 AE 0:00008. As a self-consistency test of the method, we show an excellent fractional agreement (even during the plunge) between the 5PN EOB-resummed mechanical angular momentum loss and the gravitational wave angular momentum flux computed at null infinity. New results concerning the radiation emitted from unstable circular orbits are also presented. The high accuracy waveforms computed here could be considered for the construction of template banks or for calibrating analytic models such as the effective-one-body model.

Caustic echoes from a Schwarzschild black hole

“…To avoid loss of resolution near infinity, we combine this spatial compactification with a time transformation. The time transformation is constructed requiring invariance of outgoing characteristic speeds [12,42], or equivalently, of the outgoing null surfaces in local coordinates [43,44]. Outgoing null surfaces satisfy u = t − (r + 4M log(r − 2M )) .…”

Sources of Gravitational Waves: Theory and Observations

“…Furthermore, at leading order in the computation of the radiation field, one can assume that the small test mass moves along an adiabatic sequence of geodesics of the fixed background spacetime and compute the gravitational radiation numerically solving the RWZ and Teukolsky equations [55,56,[206][207][208]. Much progress has been made in the last twenty years to evolve those equations in a robust, accurate and fast way [209][210][211][212][213][214][215][216], and compute the gravitational waveform h (1) αβ in the wave zone. Today, time-domain RWZ and Teukolsky equations can compute not only the waveform emitted during the very long inspiral stage, but also the plunge, merger and ringdown stages [216][217][218][219][220][221].…”

“…To gain more insight and improve the transition from merger to ringdown [216,217,[219][220][221] combined the EOB approch to numerical studies in BH perturbation theory. Concretely, they used the EOB formalism to compute the trajectory followed by an object spiraling and plunging into a much larger BH, and then used that trajectory in the source term of either the time-domain RWZ [202,203] or Teukolsky equation [204].…”

“…Much progress has been made in the last twenty years to evolve those equations in a robust, accurate and fast way [209][210][211][212][213][214][215][216], and compute the gravitational waveform h (1) αβ in the wave zone. Today, time-domain RWZ and Teukolsky equations can compute not only the waveform emitted during the very long inspiral stage, but also the plunge, merger and ringdown stages [216][217][218][219][220][221]. Table 6.2 State-of-the-art of PN calculations in BH perturbation theory (i.e., for an extreme mass-ratio compact binary) in the case of quasi-circular orbits.…”

Complete nonspinning effective-one-body metric at linear order in the mass ratio

“…Furthermore, the EOB construction has grown to include a model for the gravitational waveforms [56][57][58][59][60], allowing detailed comparisons (and calibrations of the EOB model's unknown parameters) with NR waveforms for non-spinning and spinning comparable-mass binaries [61][62][63][64][65][66][67][68][69][70][71], as well as with Regge-Wheeler-Zerilli [57,[72][73][74][75] and Teukolsky waveforms [60,[76][77][78] for small and extreme mass-ratios.…”