6.3 Application of CCE to stellar collapse

CCE has also been recently applied to study the waveform from the fully three-dimensional simulation of the collapse and core bounce of a massive rotating star [246Jump To The Next Citation Point]. After nuclear energy generation has ceased, dissipative processes eventually push the core over its effective Chandrasekhar mass. Radial instability then drives the inner core to nuclear densities at which time the stiffened equation of state leads to a core bounce with tremendous acceleration. The asymmetry of this bounce due to a rotating core potentially gives rise to a detectable source of gravitational quadrupole radiation, which can be used to probe the nuclear equation of state and the mass and angular momentum of the star. Simulations were carried out for three choices of initial star parameters. The gravitational waves emitted in the core bounce phase were compared using four independent extraction 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 ψ4 extraction, where the signal must be extracted in the far field. In addition, the radiation is dominant in the (ℓ = 2,m = 0 ) spherical harmonic mode, in which the memory effect complicates the relationship between ψ4 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 ψ4 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 ψ4 extraction when compared to CCE. One reason is that the double time integration of ψ4 to produce the strain introduces low-frequency errors. Also, ψ4 extraction led to larger peak amplitudes compared to either quadrupole extraction or CCE.

Several important observations emerged from this study. (i) ψ4 extraction and CCE converge properly with extraction worldtube radius. RWZM produces spurious high-frequency effects, which no other method reproduces. (ii) Waveforms from CCE, ψ4 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 ψ4 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, ψ4 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.

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