In the early days of numerical merger simulations, most groups typically assumed Newtonian and/or quasi-Newtonian gravitation, for which there is no well-defined dynamical spacetime metric. GW signals were typically calculated using the quadrupole formalism, which technically only applies for slow-moving, non-relativistic sources (see  for a thorough review of the theory). Temporarily reintroducing physical constants, the strains of the two polarizations for signals emitted in the -direction are
Quadrupole methods were adopted for later PN and CF simulations, again because the metric was assumed either to be static or artificially constrained in such a way that made self-consistent determination of the GW signal impossible. One important development from this period was the introduction of a simple method to calculate the GW energy spectrum from the GW time-series through Fourier transforming into the frequency domain . GW signals analyzed in the frequency domain allowed for direct comparison with the LIGO noise curve, making it much easier to determine approximate distances at which various GW sources would be detectable and the potential signal-to-noise ratio that would result from a template search. To constrain the nuclear matter EOS, one can examine where a GW merger spectrum deviates in a measurable way from the quadrupole point-mass form,, as we show in Figure 9.
Full GR dynamical calculations, in which the metric is evolved according to the Einstein equations, generally use one of two approaches to calculate the GW signal from the merger, if not both. The first method, developed first by by Regge and Wheeler  and Zerilli  and written down in a gauge-invariant way by Moncrief  involves analyzing perturbations of the metric away from a Schwarzschild background. The second uses the Newman–Penrose formalism  to calculate the Weyl scalar , a contraction of the Weyl curvature tensor, to represent the outgoing wave content on a specially constructed null tetrad that may be calculated approximately . The two methods are complementary since they incorporate different metric information and require different numerical integrations to produce a GW time series. Regardless of the method used to calculate the GW signal, results are often presented by calculating the dominant spin-weighted spherical harmonic mode. For circular binaries, the , mode generally carries the most energy, followed by other harmonics; in cases where the components of the binary have nearly equal masses and the orbit is circular, the falloff is typically quite rapid, while extreme mass ratios can pump a significant amount of the total energy into other harmonics. For elliptical orbits, other modes can dominate the signal, e.g., a 3:1 ratio in power for the mode to the , mode observed for high-ellipticity close orbits in . A thorough summary of both methods and their implementation may be found in .
Living Rev. Relativity 15, (2012), 8
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