The application of CCE to binary black-hole simulations was first carried out in [243, 244
] 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
, two separate
choices of outer Cauchy boundary were located at
and
, with the
corresponding characteristic extraction worldtubes ranging from
to
,
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 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
to
. It is notable
that the results in [243
] 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 Weyl component obtained at
via CCE was compared to
the corresponding value extracted at finite radii and then extrapolated to infinity [244
]. 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
, 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
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
. 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
mode. Since CCE supplies either
the news function or its time derivative
, 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
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
mode is small compared to the dominant
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.
http://www.livingreviews.org/lrr-2012-2 |
Living Rev. Relativity 15, (2012), 2
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