Many of the results later confirmed using relativistic QE sequences were originally derived in Newtonian and PN calculations, particularly as explicit extensions of Chandrasekhar’s body of work (see ). Chandrasekhar’s studies of incompressible fluids were first extended to compressible binaries by Lai, Rasio, and Shapiro [156, 155, 158, 157, 159], who used an energy variational method with an ellipsoidal treatment for polytropic NSs. They established, among other results, the magnitude of the rapid inspiral velocity near the dynamical stability limit , the existence of a critical polytropic index () separating binary sequences undergoing the two different terminal instabilities , the role played by the NS spin and viscosity and magnitude of finite-size effects in relation to 1PN terms [158, 157], and the development of tidal lag angles as the binary approaches merger . They also determined that for most reasonable EOS models and non-extreme mass ratios, as would pertain to NS-NS mergers, an energy minimum is inevitably reached before the onset of mass transfer through Roche lobe overflow. The general results found in those works were later confirmed by , who used a SCF technique [131, 132], finding similar locations for instability points as a function of the adiabatic index of polytropes, but a small positive offset in the radius at which instability occurred. Similar results were also found by [311, 312], but with a slight modification in the total system energy and decrease in the orbital frequency at the onset of instability.
The first PN ellipsoidal treatments were developed by Shibata and collaborators using self-consistent fields [270, 269, 279, 281, 299] and by Lombardi, Rasio, and Shapiro . Both groups found that the nonlinear gravitational effects imply a decrease in the orbital separation (increase in the orbital frequency) at the instability point for more compact NS. This result reflects a fairly universal principle in relativistic binary simulations: as gravitational formalisms incorporate more relativistic effects, moving from Newtonian gravity to 1PN and on to CF approximations and finally full GR, the strength of the gravitational interaction inevitably becomes stronger. The effects seen in fully dynamical calculations will be discussed in Section 6, below.
The first fully relativistic CTS QE data for synchronized NS-NS binaries were constructed by Baumgarte et al. [26, 25], using a grid-based elliptic solver. Their results demonstrated that the maximum allowed mass of NSs in close binaries was larger than that of isolated NSs with the same (polytropic) EOS, clearly disfavoring the “star-crushing” scenario that had been suggested by [327, 187] using a similar CTS formalism (but see also the error in these latter works addressed in , discussed in Section 6.3 below). Baumgarte et al. also identified how varying the NS radius affects the ISCO frequency, and thus might be constrained by GW observations. Using a multigrid method, Miller et al.  showed that while conformal flatness remained valid until relatively near the ISCO, the assumption of syncronized rotation broke down much earlier. Usui et al.  used the Green’s function approach with a slightly different formalism to compute relativistic sequences and determined that the CTS conditions were valid up until extremely relativistic binaries were considered.
The first relativistic models of physically realistic irrotational NS-NS binaries were constructed by the Meudon group  using the Lorene multi-domain pseudo-spectral method code. Since then, the Meudon group and collaborators have constructed a wide array of NS-NS initial data, including polytropic NS models [125, 303, 304], as well as physically motivated NS EOS models  or quark matter condensates . Irrotational models have also been constructed by Uryū and collaborators [313, 317] for use in dynamical calculations, and nuclear/quark matter configurations have been generated by Oechslin and collaborators [212, 209]. A large compilation of QE CTS sequences constructed using physically motivated EOS models including FPS (Friedman–Pandharipande) , SLy (Skyrme Lyon) , and APR  models, along with piecewise polytropes designed to model more general potential cases (see ), was published in .
The most extensive set of results calculated using the waveless/near-zone helical symmetry condition appear in , with equal-mass NS-NS binary models constructed for the FPS, Sly, and APR EOS in addition to polytropes. Results spanning all of these QE techniques are summarized in Table 2.
|Lai||||Newt.||Ellips.||N/A||0.2 – 1.0||Syn./Irr.|
|Taniguchi||||CTS||Spectral||0.9 – 1.0||Syn./Irr.|
|Taniguchi||||CTS||Spectral||0.83 – 1.0||Syn./Irr.|
|Taniguchi||||CTS||Spectral||Physical||0.7 – 1.0||Irr.|
Living Rev. Relativity 15, (2012), 8
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