Group | Ref. | NS EOS | Mass ratio | notes | |

KT | [287] | 1 | 0.09 – 0.15 | Co/Ir | |

– | [288] | 0.89 – 1 | 0.1 – 0.17 | ||

– | [285] | 0.85 – 1 | 0.1 – 0.12 | ||

– | [286] | SLy, FPS+Hot | 0.92 – 1 | 0.1 – 0.13 | |

– | [282] | SLy, APR+Hot | 0.64 – 1 | 0.11 – 0.13 | |

– | [332] | 0.85 – 1 | 0.14 – 0.16 | BHB | |

– | [144] | APR+Hot | 0.8 – 1 | 0.14 – 0.18 | |

– | [145] | APR, SLy, FPS+Hot | 0.8 – 1.0 | 0.16 – 0.2 | |

– | [265] | Shen | 1 | 0.14 – 0.16 | -leak |

– | [134] | PP+hot | 1 | 0.12 – 0.17 | |

– | [264] | Shen, Hyp | 1.0 | 0.14 – 0.16 | -leak |

HAD | [7] | 1.0 | 0.08 | GH, non-QE | |

– | [6] | 1.0 | 0.08 | GH, non-QE, MHD | |

Whisky | [17] | 1.0 | 0.14 – 0.18 | ||

– | [18] | 1.0 | 0.20 | ||

– | [116] | 1.0 | 0.14 – 0.18 | MHD | |

– | [117] | 1.0 | 0.14 – 0.18 | MHD | |

– | [240] | 0.70 – 1.0 | 0.09 – 0.17 | ||

– | [14, 15] | 1.0 | 0.12 – 0.14 | ||

– | [241] | 1.0 | 0.18 | MHD | |

UIUC | [172] | 0.85 – 1 | 0.14 – 0.18 | MHD | |

Jena | [308, 41] | 1.0 | 0.14 | ||

– | [122] | 1.0 | 1.4 | Eccen. | |

Later works, in particular a paper by Shibata, Taniguchi, and UryΕ« [285], introduced several new techniques to perform dynamical calculations that most codes at present still include in nearly the same or lightly modified form. These included the use of a high-resolution shock-capturing scheme for the hydrodynamics, as well as a Gamma-driver shift condition closely resembling the moving puncture gauge conditions that later proved instrumental in allowing for long-term BH evolution calculations. In the series of papers that followed their original calculations, the KT group established a number of results about NS-NS mergers that form the basis for much of our thinking about their hydrodynamic evolution:

- By varying the EOS model for the NS as well as the mass ratio, it was possible to constrain the
binary parameters separating cases that form a HMNS rather than producing prompt collapse
to a BH, and it was quickly determined that the total system mass as a proportion of the
maximum allowed mass for an isolated NS is the key parameter, with only weak dependence
on the binary mass ratio.
For polytropic EOS models, the critical compactness values leading to prompt collapse for equal-mass binary mergers were found to be for and for [288]. As a rough rule, collapse occurred for polytropic EOS when the total system rest-mass was at least , where is the maximum mass of an isolated non-rotating NS for the given EOS. For physically motivated EOS models [286], the critical mass was significantly smaller; indeed, the critical NS mass was found to be for the SLy EOS [83] (i.e., collapse for with ) and for the FPS EOS [222] (collapse for with ). This was not a complete surprise, since for the physically motivated EOS the NS radius is nearly independent of the mass across much of the parameter space, limiting the ability of the HMNS to expand in response to the extra mass absorbed during the merger.

- The mass ratio was found to play a critical role in the evolution of the remnant/disk configuration, since unequal-mass cases are better characterized as disruptions of the smaller secondary followed by its accretion onto the primary, rather than a true merger between the two NSs. Disk masses from full GR calculations [285] are generically smaller than those predicted from non-GR calculations. For polytropic EOS, disks contain approximately 4% of the total system mass for mass ratios , varying roughly for a fixed total mass, with the disk mass decreasing for heavier binaries (and thus larger compactnesses) given the stronger gravity of the central remnant. Using the stiffer APR EOS [3], the dependence on the mass ratio was seen to be much steeper for a physical EOS than for polytropes, scaling like , where [282].
- With respect to GW emission, it was determined in [288] that in low-mass cases in which a
HMNS was formed, stiffer polytropic EOS models were able to support the development of
a bar-mode instability, leading to transient spiral arm formation from the remnant and an
extended period of strong GW emission, in the characteristic modulated form that results
from differentially rotating ellipsoids (see, e.g., [160, 274, 259, 16, 183, 72]). Unequal-mass
cases typically yielded one high-frequency peak at roughly corresponding to
non-axisymmetric oscillations, and equal-mass cases yielded multiple peaks including those
associated with quasi-radial oscillations as well [285]. For physical EOS models [286], mass
loss into a disk is reduced relative to the polytropic case given the higher compactness of the
central region, and GW oscillation peaks, while very strong, occur at correspondingly higher
frequencies. By contrast, prompt formation of a BH led to a ringdown signal with rapidly
decreasing amplitude becoming negligible within a few dynamical times.
The GW signals were evaluated under the gauge-dependent assumption of transverse tracelessness, and energy and angular momentum loss rates into each spherical harmonic mode were computed using the gauge-invariant Zerilli–Moncrief formalism [239, 335, 195] in much the same way that is used by some groups in numerical relativity today (many BH-BH and hydrodynamics simulations report GW signals derived from the alternate Weyl scalar formulation [202, 60], or use both methods).

- In [286], it was concluded that mergers of NSs with comparable masses made poor SGRB
progenitor candidates, assuming prompt emission (because of the lack of energy available for
neutrino annihilation), but that the energy budget in the HMNS case is orders of magnitude
larger. Remarkably, this discussion from 2005 predates the first identifications of SGRBs with
older populations, which greatly improved our theoretical understanding of compact object
mergers as their likely progenitors. In the first work that followed the initial localizations of
SGRBs, mergers of binaries with relatively small mass ratios, , were seen to form
sufficiently hot and massive disk to power a SGRB, albeit a relatively brief, low-luminosity one.
It was suggested that the more likely SGRB progenitor is indeed a HMNS, since dissipative
effects within the remnant can boost temperatures up to 10
^{11}K.Further approximate relativistic investigations of NS-NS mergers, along with BH-NS mergers, as potential SGRB sources quickly swept through the community after the initial localizations of SGRBs, with several groups using a wide variety of methods all concluding that mergers were plausible progenitors, but finding it extremely difficult to constrain the scenario in quantitative ways given the extremely complicated microphysics ultimately responsible for powering the burst (see, e.g., [207, 206] who investigated potential disk energies; [243], who modeled the fallback accretion phase onto a BH; and [266], who considered the thermodynamic and nuclear evolution of disks around newly-formed BHs produced by mergers). We will return to this topic below in light of recent GRMHD Simulations.

In the past few years, five groups have reported results from NS-NS mergers in full GR; KT, HAD, Whisky, UIUC, and Jena. Much of the work of the HAD and Whisky groups, developers respectively of the code of those names, began at Louisiana State University (HAD) and the Albert Einstein Institute in Potsdam (Whisky), though both efforts now include several other collaborating institutions. Two other groups, the SXS collaboration that originated at Caltech and Cornell, and the Princeton group, have reported BH-NS merger results and are actively studying NS-NS mergers as well, but have yet to publish their initial papers about the latter. All of the current groups use AMR-based Eulerian grid codes, with four evolving Einstein’s equations using the BSSN formalism and the HAD collaboration making use of the GHG method instead. HAD, Whisky, and UIUC have all reported results about magnetized NS-NS mergers (the KT collaboration has used a GRMHD code to study the evolution of magnetized HMNS, but not complete NS-NS mergers). The KT collaboration has considered a wide range of EOS models, including finite-temperature physical models such as the Shen EOS, and have also implemented a neutrino leakage scheme, while all other results reported to date have assumed a polytropic EOS model.

Given the similarities of the various codes used to study NS-NS mergers, it is worthwhile to ask whether they do produce consistent results. A comparison paper between the Whisky code and the KT collaboration’s SACRA codes [20] found that both codes performed well for conservative global quantities, with global extrema such as the maximum rest-mass density in agreement to within 1% and waveform amplitudes and frequencies differing by no more than 10% throughout a full simulation, and typically much less.

Several of the the groups listed above have also been leaders in the field of BH-NS simulations: the KT, HAD, and UIUC groups have all presented BH-NS merger results, as have the SXS collaboration [85, 84, 108], and Princeton group [294, 88] (see [284] for a thorough review).

We discuss the current understanding of NS-NS mergers in light of all these calculations below.

Using their newly developed SACRA code [332], the KT group [144], found that when a hybrid EOS is used to model the NS, in which the cold part is described by the APR EOS and the thermal component as a ideal gas, the critical total binary mass for prompt collapse to a BH is , independent of the initial binary mass ratio, a result consistent with previous explorations of other polytropic and physically motivated NS EOS models (see above). In all cases, the BH was formed with a spin parameter depending very weakly on the total system mass and mass ratio.

They further classified the critical masses for a number of other physical EOS in [134], finding that binaries with total masses should yield long-lived HMNSs ( 10 ms) and substantial disk masses with assuming that the current limit on the heaviest observed NS, [81] is correct. In Figure 10, we show the final fate of the merger remnant as a function of the total pre-merger mass of the binary. “Type I” indicates a prompt collapse of the merger remnant to a BH, “Type II” a short-lived HMNS, which lasts for less than 5 ms after the merger until its collapse, and “Type III” a long-lived HMNS which survives for at least 5 ms. See [134] for an explanation of the EOS used in each simulation.

While all of the above results incorporated shock heating, the addition of both finite-temperature effects in the EOS and neutrino emission modifies the numerically determined critical masses separating HMNS formation from prompt collapse. Adding in a neutrino leakage scheme for a NS-NS merger performed using the relatively stiff finite-temperature Shen EOS, the KT collaboration reports in [265] that HMNSs will form generically for binary masses , not because they are centrifugally-supported but rather because they are pressure-supported, with a remnant temperature in the range 30 – 70 MeV. Since they are not supported by differential rotation, these HMNSs were predicted to be stable until neutrino cooling, with luminosities of , can remove the pressure support. Even for cases where the physical effects of hyperons were included, which effectively soften the EOS and reduce the maximum allowed mass for an isolated NS to , the KT collaboration [264] still finds that thermal support can stabilize HMNS with masses up to .

Using a Carpet/Cactus-based hydrodynamics code called Whisky [19] that works within the BSSN formalism (a version of which has been publicly released as GRHydro within the Einstein Toolkit [90]), the Whisky collaboration has analyzed the dependence of disk masses on binary parameters in some detail. For mass ratios [240], they found that bound disks with masses of up to can be formed, with the disk mass following the approximate form

where the maximum mass of a binary system for a given EOS ( ideal gas for these calculations), is the maximum mass of an isolated non-rotating NS for the EOS, and the mass of the binary, with all masses here defined as baryonic. The evolution of the total rest mass present in the computational domain for a number of simulations is shown in Figure 11.Using the HAD code described in [8] that evolves the GHG system on an AMR-based grid with CENO
reconstruction techniques, Anderson et al. [7] performed the first study of magnetic effects in full GR
NS-NS mergers [6]. Beginning from spherical NSs with extremely strong poloidal magnetic fields
(9.6 × 10^{15} G, as is found in magnetars), their merger simulations showed that magnetic repulsion can
delay merger by 1 – 2 orbits and lead to the formation of magnetically buoyant cavities at the trailing end
of each NS as contact is made (see Figure 12), although the latter may be affected by the
non-equilibrium initial data. Both effects would have been greatly reduced if more realistic
magnetic fields strengths had been considered. Magnetic fields in the HMNS remnant, which can
be amplified through dynamo effects regardless of their initial strengths, helped to distribute
angular momentum outward via the magnetorotational instability (MRI), leading to a less
differentially rotating velocity profile and a more axisymmetric remnant. The GW emission in
the magnetized case was seen to occur at lower characteristic frequencies and amplitudes as a
result.

The UIUC group was among the first to produce fully self-consistent GRMHD results [87]. Using a newly developed Cactus-based code, they performed the first studies of unequal-mass magnetized NS-NS mergers [172]. Using poloidal, magnetar-level initial magnetic fields, Liu et al. found that magnetic effects are essentially negligible prior to merger, but can increase the mass in a disk around a newly formed BH moderately, from 1.3% to 1.8% of the total system mass for mass ratios of and . They point out that MHD effects can efficiently channel outflows away from the system’s center after collapse [295], and may be important for the late-stage evolution of the system.

In [116], the Whisky group performed simulations of magnetized mergers with field strengths ranging
from 10^{12} to 10^{17} G. Agreeing with the UIUC work that magnetic field strengths would have
essentially no effect on the GW emission during inspiral, they note that magnetic effects become
significant for the HMNS, since differential rotation can amplify B-fields, with marked deviations
in the GW spectrum appearing at frequencies of . They also point out that
high-order MHD reconstruction schemes, such as third-order PPM, can produce significantly more
accurate results that second-order limiter-based schemes. A follow-up paper [117] showed that a
plausible way to detect the effect of physically realistic magnetic fields on the GW signal from a
merger was through a significant shortening of the timescale for a HMNS to collapse, though a
third-generation GW detector could perhaps observe differences in the kHz emission of the HMNS as
well.

More recently, they have used very long-term simulations to focus attention on the magnetic field
strength and geometry found after the remnant collapses to a BH [241]. They find that the large, turbulent
magnetic fields () present in the initial binary configuration are boosted exponentially in time
up to a poloidal field of strength 10^{15} G in the remnant disk, with the field lines maintaining a half-opening
angle of 30° along the BH spin axis, a configuration thought to be extremely promising for
producing a SGRB. The resulting evolution, shown in Figure 13, is perhaps the most definitive
result indicating that NS-NS mergers should produce SGRBs for some plausible range of initial
parameters.

It is worth noting that all magnetized NS-NS merger calculations that have been attempted to date have made use of unphysically large magnetic fields. This is not merely a convenience designed to enhance the role of magnetic effects during the merger, though it does have that effect. Rather, magnetic fields are boosted in HMNS remnants by the MRI, whose fastest growing unstable mode depends roughly linearly on the Alfvén speed, and thus the magnetic field strength. In order to move to physically reasonable magnetic field values, one would have to resolve the HMNS at least a factor of 100 times better in each of three dimensions, which is beyond the capability of even the largest supercomputers at present, and likely will be for some time to come.

In [145], the KT collaboration found a nearly linear relationship between the GW spectrum cutoff frequency and the NS compactness, independent of the EOS, as well as a relationship between the disk mass and the width of the kHz hump seen in the GW energy spectrum. While is a somewhat crude measure of the NS compactness, it occurs at substantially lower frequencies than any emission process associated with merger remnants, and thus is the parameter most likely to be accessible to GW observations with a second generation detector.

The qualitative form of the high-frequency components of the GW spectrum is primarily determined by the type of remnant formed. In Figures 14 and 15, we show and , respectively, for four of the runs calculated by the KT collaboration and described in [134]. Type I collapses are characterized by a rapid decrease in the GW amplitude immediately after the merger, yielding relatively low power at frequencies above the cutoff frequency. Type II and III mergers yield longer periods of GW emission after the merger, especially the latter, with the remnant oscillation modes leading to clear peaks at GW frequencies = 2 – 4 khz that should someday be detectable by third generation detectors like the Einstein Telescope, or possibly even by advanced LIGO should the source be sufficiently close () and the high-frequency peak of sufficiently high quality [265].

Using new multi-orbit simulations of NS-NS mergers, Baiotti et al. [14, 15] showed that the semi-analytic effective one-body (EOB) formalism severely underestimates high-order relativistic corrections even when lowest-order finite-size tidal effects were included. As a result, phase errors of almost a quarter of a radian can develop, although these may be virtually eliminated by introducing a second-order “next-to-next-to-leading order” (NNLO) correction term and fixing the coefficient to match numerical results. The excellent agreement between pre-merger numerical waveforms and the revised semi-analytic EOB approximant is shown in Figure 16.

The effects of binary eccentricity on NS-NS mergers was recently studied by the Jena group [122]. Such systems, which would indicate dynamical formation processes rather than the long-term evolution of primordial binaries, evolve differently in several fundamental ways from binaries that merge from circular orbits. For nearly head-on collisions, they found prompt BH formation and negligible disk mass production, with only a single GW burst at frequencies comparable to the quasi-normal mode of the newly formed BH. For a collision in which mass transfer occurred at the first passage but two orbits were required to complete the merger and form a BH, a massive disk was formed, containing 8% of the total system mass even at time after the formation of the BH. During that time, the black hole accreted an even larger amount of mass, representing over twice the mass of the remaining disk. Between the first close passage and the second, during which the two NS merged, the GW signal was seen to be quasi-periodic, and a a frequency comparable to the fundamental oscillation mode of the two NS, a result that was duplicated in a calculation for which the periastron fell outside the Roche limit and the eccentric binary survived for the full duration of the run, comprising several orbits.

Living Rev. Relativity 15, (2012), 8
http://www.livingreviews.org/lrr-2012-8 |
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