5.5 GW signal modeling

Measuring the GW signal from a dynamical merger calculation is a rather difficult task. One must determine, using a method unaffected by gauge effects, the perturbations at asymptotically large distances from a source by extrapolating various quantities measured at large but finite distances from the merger itself.

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 [193] for a thorough review of the theory). Temporarily reintroducing physical constants, the strains of the two polarizations for signals emitted in the z-direction are

h = G--(¨I − ¨I ), + rc4 xx yy 2G-¨ h× = rc4Ixy,
where r is the distance from the source to the observer and ¨ Iij it the traceless quadrupole moment of the system. The energy and angular momentum loss rates of the system due to GW emission are given, respectively, by
( ) dE G ... ... − -dt = 5c5I ij I ij, ( )GW dLk- 2G- ¨ ... − dt = 5c5𝜖ijkIilI lj. GW
While only approximate, the quadrupole formulae do yield equations that are extremely straightforward to implement in both grid and particle-based codes using standard integration techniques.

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 dE ∕df from the GW time-series through Fourier transforming into the frequency domain [330Jump To The Next Citation Point]. 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,

( ) dE- πGm1m2--- −1∕3 ωorb df = 3 (πG (m1 + m2 )fGW ) ; fGW ≡ 2forb = π , (32 ) GW
because of finite-size effects, and then link the deviation to the properties of the NS [98Jump To The Next Citation Point], as we show in Figure 9View Image.
View Image

Figure 9: Approximate energy spectrum dEGW ∕df derived from QE sequences of equal-mass NS-NS binaries with isolated ADM masses MNS = 1.35M ⊙ and a Γ = 2 EOS, but varying compactnesses (denoted M ∕R here), originally described in [303]. The diagonal lines show the energy spectrum corresponding to a point-mass binary, as well as values with 90%, 75%, and 50% of the power at a given frequency. Asterisks indicate the onset of mass-shedding, beyond which QE results are no longer valid. Image reproduced by permission from Figure 2 of [98Jump To The Next Citation Point], copyright by APS.

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 [239Jump To The Next Citation Point] and Zerilli [335Jump To The Next Citation Point] and written down in a gauge-invariant way by Moncrief [195Jump To The Next Citation Point] involves analyzing perturbations of the metric away from a Schwarzschild background. The second uses the Newman–Penrose formalism [202Jump To The Next Citation Point] to calculate the Weyl scalar ψ4, a contraction of the Weyl curvature tensor, to represent the outgoing wave content on a specially constructed null tetrad that may be calculated approximately [60Jump To The Next Citation Point]. 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 s = − 2 spin-weighted spherical harmonic mode. For circular binaries, the l = 2, m = 2 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 l = 2, m = 0 mode to the l = 2, m = 2 mode observed for high-ellipticity close orbits in [122Jump To The Next Citation Point]. A thorough summary of both methods and their implementation may be found in [257].

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