- The simplest technique was via the quadrupole formula, which estimates the waveform in terms of the second time derivative of the mass quadrupole tensor but does not take into account the effects of curvature or relativistic motion.
- Extraction at a finite radius via the Newman–Penrose component.
- Perturbative extraction at a finite radius based upon the Regge–Wheeler–Zerilli–Moncrief (RWZM) formalism.
- CCE, utilizing the PITT code, which avoids the near field or perturbative approximations of the above techniques..

Historically, the quadrupole formula, which is computed in the inner region where the numerical grid is most accurate, has been the predominant extraction tool used in stellar collapse. The metric or curvature based methods suffer from numerical error in extracting a signal, which is many orders of magnitude weaker than that from a binary inspiral from the numerical noise. This is especially pertinent to RWZM and extraction, where the signal must be extracted in the far field. In addition, the radiation is dominant in the spherical harmonic mode, in which the memory effect complicates the relationship between and the strain at low frequencies.

CCE was used as the benchmark in comparing the various extraction techniques. For all three choices of initial stellar configurations, extraction via RWZM yielded the largest discrepancy and showed a large spurious spike at core bounce and other spurious high-frequency contributions. Quadrupole and extraction only led to small differences with CCE. It was surprising that the quadrupole technique gave such good agreement, given its simplistic assumptions. Overall, quadrupole extraction performed slightly better than extraction when compared to CCE. One reason is that the double time integration of to produce the strain introduces low-frequency errors. Also, extraction led to larger peak amplitudes compared to either quadrupole extraction or CCE.

Several important observations emerged from this study. (i) extraction and CCE converge properly with extraction worldtube radius. RWZM produces spurious high-frequency effects, which no other method reproduces. (ii) Waveforms from CCE, extraction and quadrupole extraction agree well in phase. The high-frequency contamination of RWZM makes phase comparisons meaningless. (iii) Compared to CCE, the maximum amplitudes at core bounce differ by 1 to 7%, depending on initial stellar parameters, for extraction and by 5 to 11% for quadrupole extraction. (iv) Only quadrupole extraction is free of low frequency errors. (v) For use in gravitational wave data analysis, except for RWZM, the three other extraction techniques yield results, which are equivalent up to the uncertainties intrinsic to matched-filter searches.

Certain technical issues cloud the above observations. CCE, and RWZM extraction are based upon vacuum solutions at the extraction worldtube, which is not the case for those simulations in which the star extends over the entire computational grid. This could be remedied by the inclusion of matter terms in the CCE technique, which might also improve the low frequency behavior. In any case, this work represents a milestone in showing that CCE has important relevance to waveform extraction from astrophysically-realistic collapse models.

The above study [246] employed a sufficiently stiff equation of state to produce core bounce after collapse. In subsequent work, CCE was utilized to study the gravitational radiation from a collapsar model [220], in which a rotating star collapses to form a black hole with accretion disk. The simulations tracked the initial collapse and bounce, followed by a post bounce phase leading to black-hole formation. At bounce, there is a burst of gravitational waves similar to the above study, followed by a turbulent post bounce with weak gravitational radiation in which an unstable proto-neutron star forms. Collapse to a black hole then leads to another pronounced spike in the waveform, followed by ringdown to a Kerr black hole. The ensuing accretion flow does not lead to any further radiation of appreciable size. The distinctive signature of the gravitational waves observed in these simulations would enable a LIGO detection to distinguish between core collapse leading to bounce and supernova and one leading to black-hole formation.

Living Rev. Relativity 15, (2012), 2
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