6.2 Application of CCE to binary black hole inspirals

The emission of gravitational waves from the inspiral and merger of binary black holes is the most likely source for detection by gravitational wave observatories. The post-Newtonian regime of the inspiral can be accurately modeled by the chirp waveforms obtained by perturbation theory and the final ringdown waveform can be accurately modeled by the known quasi-normal modes. This places special importance on reliable waveforms for the nonlinear inspiral and merger waveform, which interpolates between these early and late time phases. Here CCE plays the important role of providing an unambiguous waveform at + ℐ, which can be used to avoid the error introduced by perturbative extraction techniques.

The application of CCE to binary black-hole simulations was first carried out in [243Jump To The Next Citation Point, 244Jump To The Next Citation Point] using an implementation of the PITT code for the characteristic evolution. The Cauchy evolution was carried out using a variant of the BSSN formulation [268, 38]. Simulations of inspiral and merger were carried out for equal-mass non-spinning black holes and for equal-mass black holes with spins aligned with the orbital angular momentum. For a binary of mass M, two separate choices of outer Cauchy boundary were located at R = 3600 M and R = 2000 M, with the corresponding characteristic extraction worldtubes ranging from RE = 100 M to RE = 250 M, sufficient to causally isolate the characteristic extraction from the outer boundary during the simulation of eight orbits prior to merger and ringdown. The difference between CCE waveforms in this range of extraction radii was found to be of comparable size to the numerical error. In particular, for the grid resolutions used, the dominant numerical error was due to the Cauchy evolution.

The CCE waveforms at + ℐ were also used to evaluate the quality of perturbative waveforms based upon Weyl tensor extraction. In order to reduce finite extraction effects, the perturbative waveforms were extrapolated to infinity by extraction at six radii in the range R = 280 M to R = 1000 M. It is notable that the results in [243Jump To The Next Citation Point] indicate that the systematic error in perturbative extraction had, previously, been underestimated.

The lack of reflection symmetry in the spinning case leads to a recoil, or “kick”, due to the linear momentum carried off by the gravitational waves. The astrophysical consequence of this kick to the evolution of a galactic core has accentuated the important role of CCE waveforms to supply the energy, momentum and angular momentum radiated during binary black hole inspirals. The radiated energy and momentum obtained from the ψ4 Weyl component obtained at + ℐ via CCE was compared to the corresponding value extracted at finite radii and then extrapolated to infinity [244Jump To The Next Citation Point]. The extrapolated value was found to be of comparable accuracy to the CCE result for the large extraction radii used. For extraction at a single radius of R = 100 M, commonly used in numerical relativity, this was no longer true and the error was 1 to 2 orders of magnitude larger. The CCE energy loss obtained via ψ4 was also found to be consistent, within numerical error, to the recoil computed from the news function. The work emphasizes the need for an accurate description of the astrophysical consequences of gravitational radiation, which CCE is designed to provide.

In addition to the dominant oscillatory gravitational-wave signals produced during binary inspirals, there are also memory effects described by the long time scale change in the strain Δh = h (t,𝜃,ϕ ) − h (− ∞, 𝜃,ϕ). In a follow-up to the work in [243, 244], these were studied by means of CCE [247] for the inspiral of spinning black holes. It was found that the memory effect was greatest for the case of spins aligned with the orbital angular momentum, as might be expected since this case also produces the strongest radiation. The largest spherical harmonic mode for the effect was found to be the (ℓ = 2,m = 0) mode. Since CCE supplies either the news function or its time derivative ψ 4, a major difficulty in measuring the memory is the proper setting of the integration constants in determining the strain. This was done by matching the numerical evolution to a post-Newtonian precursor. There is a slow monotonic growth of Δh during the inspiral followed by a rapid rise during the merger phase, which over the time scale of the simulation leads to a step-like behavior modulated by the final ringdown. The simulations showed that the largest memory offset occurs for highly-spinning black holes, with an estimated value of 0.24 in the maximally-spinning case. These results are central to determining the detectability of the memory effect by observations of gravitational waves. Since the size of the (ℓ = 2,m = 0) mode is small compared to the dominant (ℓ = 2,m = 2) radiation mode, the memory effect is unlikely to be observable in LIGO signals. However, the long period behavior of the effect might make it more conducive to detection by proposed pulsar timing arrays designed to measure the residual times-of-arrival caused by intervening gravitational waves.

Another application of CCE has been to the study of gravitational waves from precessing binary black holes with spins aligned or anti-aligned to the orbital angular momentum [245]. It was found that binaries with spin aligned with the orbital angular momentum are more powerful sources than the corresponding binaries with anti-aligned spins. The results were confirmed by comparing the waveforms obtained using perturbative extraction at finite radius to those obtained using CCE. The comparisons showed that the difference between the two approaches was within the numerical error of the simulation.

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