6.1 Quasi-equilibrium and semi-analytic methods vs fully dynamical results

Before reviewing fully dynamical calculations of NS-NS mergers, it is worthwhile to ask how much information can already be deduced from QE calculations, which may be performed at much smaller computational cost, as well as from semi-analytic PN treatments and related approximate techniques. Clearly, the details of the merger and ringdown phases fall outside the QE regime, so only dynamical calculations can yield reliable information about the stability of remnants, properties of ejecta, or other processes that arise during the merger itself or in its aftermath. Thus, the primary point of comparison is the GW signal just prior to merger, which is also easier to detect (for first and second generation interferometers).

The strength of QE calculations lies in their ability to model self-consistently finite-size effects not captured in PN treatments (which always assume two orbiting point masses). The increased tidal interaction between the objects typically results in a more rapid phase advance of the binary orbit, which is important for constructing template waveforms that cover the entire NS-NS inspiral, merger, and ringdown. While QE sequences potentially offer a wealth of information about well-separated binaries and can help fix the phase evolution of the inspiraling binary, they do have two weaknesses arising as the binary approaches the stability limit. First, most QE methods, including the CTS formalism described in Section 4.2.1, are time-symmetric, and assume that the NS possess a symmetry plane perpendicular to the direction of motion (i.e., a front-back symmetry whose axis is perpendicular to the orbital angular momentum and the binary separation vector). In reality, tidal lags develop prior to final plunge, with the innermost edge of each NS rotating forward and the outer edge backwards. This effect has been captured in analytic and semi-analytic approaches (see, e.g., [159] for an early example), and is clearly seen in dynamical calculations (see Figure 3View Image), but is not captured in CTS-based schemes (tidal lags also develop in BH-NS merger calculations when the BH has a non-zero spin, since this breaks the front-back symmetry; see [300] for an example).

A second weakness of QE methods is the treatment of the ISCO, particularly its importance as a characteristic point along an evolutionary sequence that, in theory, could encode information about the NS EOS. Simple estimates of the infall trajectory derived solely from QE sequences predict a very sudden and rapid infall near the ISCO, i.e., the point where the binding energy reaches a minimum along the sequence (see, e.g., the argument in [98]). However, this is clearly an oversimplification. In reality, binaries transition more gradually to the merger phase, and the inward plunge may occur significantly before reaching the formal ISCO; this in turns leads to more rapidly growing deviations from the QE approximation. Looking at the GW energy spectrum, one typically sees minor deviations from the point-mass predictions at frequencies below those characterizing the ISCO, but substantially more power at frequencies above it. Equivalently, the cutoff frequency for GW emission f cut, where the spectrum starts deviating strongly from the point-mass prediction, is usually higher than the QE frequency near the ISCO, fISCO, while simple QE estimates assume these two frequencies to coincide.

To date, most attempts to generate waveforms extended back to arbitrarily early starting points involve numerically matching PN signals, typically generated using the Taylor T4 approach [53], onto the early stages of numerically generated waveforms, with some form of maximum overlap method used to provide the most physical transition from one to the other. These approaches may be improved by adding tidal effects to the evolution, typically parameterized by the tidal Love numbers that describe how tidal gravity fields induce quadrupole deformations [105]. Tidal effects can be placed into a relativistic framework [46, 74], which may be included within the effective one-body (EOB) formalism to produce high-accuracy waveforms [75]. In the EOB approach [58], resummation methods are used to include higher-order PN effects, though some otherwise unfixed parameters need to be set by comparing to numerical simulations.

Work is in its early stages to compare directly the GW spectra inferred from QE sequences of NS-NS binaries with those generated in numerical relativity simulations, but this comparison has been discussed at some length with regard to BH-NS mergers. Noting that NS-NS mergers generally correspond more closely to the BH-NS cases in which an ISCO is reached prior to the onset of tidal disruption, the KT collaboration  [283Jump To The Next Citation Point, 276Jump To The Next Citation Point] concluded that the cutoff frequency marking significant deviations from PN point-mass behavior is roughly 30% higher than that marking emission near the classical ISCO for BH-NS systems (fcut ≃ 1.3fISCO).

A more detailed study has now been performed comparing EOB methods to numerical evolutions. By comparing to long-term simulations of NS-NS mergers, Baiotti et al. [15Jump To The Next Citation Point] find that EOB models may be tuned, via careful choices of their unfixed parameters, to reproduce the GW phases and amplitudes seen in NR evolutions up until the onset of merger. They further suggest that the EOB approach seems to cover a wider range of phase space than the Taylor T4 approach, presumable because of a more consistent representation of tidal effects, and offers the best route forward for construction of more accurate NS-NS inspiral templates.


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