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 3), 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 , where the
spectrum starts deviating strongly from the point-mass prediction, is usually higher than the QE
frequency near the ISCO,
, 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 [283, 276
] 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 (
).
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. [15] 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.
http://www.livingreviews.org/lrr-2012-8 |
Living Rev. Relativity 15, (2012), 8
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