4.1 Overview

While dynamical calculations are required to understand the GW and EM emission from BH-NS and NS-NS mergers, some of the main qualitative features of the signals may be derived directly from QE sequences. From the variation of total system energy with binary angular velocity along a given sequence, it is possible to construct an approximate GW energy spectrum dE ∕df GW immediately from QE results, essentially by performing a numerical derivative (see Figure 6View Image). Doing so for a number of different sequences makes it possible to identify key frequencies where tidal effects may become measurable and to identify these with binary parameters such as the system mass ratio and NS radius. Similarly, since QE sequences should indicate whether a binary begins to shed mass prior to passage through the ISCO (see Figure 7View Image), one may be able to classify observed signals into mass-shedding and non-mass-shedding events, and to use the critical point dividing those cases to help constrain the NS EOS. Single-parameter estimates have been derived for NS-NS binaries using QE sequences [98Jump To The Next Citation Point] (and for BH-NS binaries using QE [301] and dynamical calculations [283Jump To The Next Citation Point]). NS-NS binaries typically approach instability at frequencies fGW ≳ 1 kHz, where laser shot noise is severely degrading the sensitivity of an interferometer detector. To observe ISCO-related effects with higher signal-to-noise, it may be necessary to operate GW observatories using narrow-band signal recycling modes, in which the sensitivity in a narrow range of frequencies is enhanced at the cost of lower sensitivity to broadband signals [56].
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Figure 6: Dimensionless binding energy E ∕M b 0 vs. dimensionless orbital frequency M Ω 0, where M0 is the total ADM (Arnowitt–Deser–Misner) mass of the two components at infinite separation, for two QE NS-NS sequences that assume a piecewise polytropic NS EOS. The equal-mass case assumes MNS = 1.35M ⊙ for both NSs, while the unequal-mass case assumes M1 = 1.15M ⊙ and M2 = 1.55M ⊙. The thick curves are the numerical results, while the thin curves show the results from the 3PN approximation. The lack of any minimum suggests that instability for these configurations occurs at the onset of mass shedding, and not through a secular orbital instability. Image reproduced by permission from Figure 16 of [305Jump To The Next Citation Point], copyright by AAS.
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Figure 7: Mass-shedding indicator ( ∂(lnh)) (∂(lnh)) χ ≡ ∂r eq∕ ∂r pole vs. orbital frequency M0 Ω, where h is the fluid enthalpy and the derivative is measured at the NS surface in the equatorial plane toward the companion and toward the pole in the direction of the angular momentum vector, for a series of QE NS-NS sequences assuming equal-mass components. Here, χ = 1 corresponds to a spherical NS, while χ = 0 indicates the onset of mass shedding. More massive NSs are more compact, and thus able to reach smaller separations and higher angular frequencies before mass shedding gets underway. Image reproduced by permission from Figure 19 of [305Jump To The Next Citation Point], copyright by AAS.

It is important to note that, while the potential parameter space for NS EOS models is still very large, a much smaller set may serve to classify models for comparison with the first generation of GW detections. Indeed, by breaking up the EOS into piecewise polytropic segments, one may use as few as four parameters to roughly approximate all known EOS models, including standard nuclear models as well as models with kaon or other condensates [237Jump To The Next Citation Point]. To illustrate this, we show in Figure 8View Image four different QE models for NS-NS configurations with different EOS, taken from [305Jump To The Next Citation Point]; all have M1 = 1.15M ⊙ and M2 = 1.55M ⊙, and they correspond to the closest separation for which the QE code still finds a convergent result.

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Figure 8: Isodensity contours for QE models of NS-NS binaries. In each case, the two NSs have masses M1 = 1.15M ⊙ (left) and M2 = 1.55M ⊙ (right), and the center-of-mass separation is as small as the QE numerical methods allow while able to find a convergent result. The models assume different EOS, resulting in different central concentrations and tidal deformations. Image reproduced by permission from Figures 9 – 12 of [305Jump To The Next Citation Point], copyright by AAS.

The inspiral of NS-NS binaries may yield complementary information about the NS structure beyond what can be gleaned from QE studies of tidal disruption. NSs have a wide variety of oscillation modes, including f-modes, g-modes, and r-modes, any of which may be excited by resonances with the orbital frequency as the latter sweeps upward. Should a particular oscillation mode be excited resonantly, it can then serve briefly as an energy sink for the system, potentially changing the phase evolution of the binary. For example, in a rapidly spinning NS, excitation of the m = 1 r-mode can be significant, yielding a change of over 100 radians for the pre-merger GW signal phase in the case of a millisecond spin period [161]. For NS-NS mergers in the field, this would require one of the NSs to be a young pulsar that has not yet spun down significantly, which is unlikely because of the difficulty in obtaining such an extremely small binary separation after the second supernova. Other modes, such as the l = 2 f-mode, may be excited in less extreme circumstances, also yielding information about NS structure parameters [105Jump To The Next Citation Point].

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