6.4 Full GR calculations

A summary of full GR calculations of NS-NS mergers is presented in Table 4. The KT collaboration was responsible for the only full GR calculations of NS-NS mergers that predate the breakthrough calculations of numerically stable binary BH evolutions [231Jump To The Next Citation Point, 22, 61], which have since transformed the field of GR hydrodynamics and MHD in addition to vacuum relativistic evolutions (Miller et al. [192] performed NS-NS inspiral calculations in full GR, but were not able to follow binaries through to merger). The first calculations of NS-NS mergers using a completely self-consistent treatment of GR were performed by Shibata and collaborators in the KT collaboration using a grid based code and the BSSN formalism [287Jump To The Next Citation Point]. CTS initial data consisting of equal-mass NSs described by a Γ = 2 polytropic EOS were constructed via SCF techniques [313], for both synchronized and irrotational configurations. The hyperbolic system was evolved on a grid, with an approximate maximal slicing condition that results in a parabolic equation for the lapse [271] and an approximate minimal distortion condition for the shift vector requiring the solution of an elliptic equation at every time step [272]. The shift vector gauge condition was found to fail when BHs were produced in the merger remnant, a well-known problem that had long bedeviled simulations involving binary and even single BH evolutions, so modifications were introduced to extend the stability of the algorithm as far as possible. Among the key results from this early work was a clear differentiation between mergers of moderately low-compactness NSs (π’ž ≳ 0.11), where the remnant collapsed promptly to a BH, and very low-compactness models, which yielded hypermassive remnants stabilized against gravitational collapse by differential rotation. Virtually all the NS matter was contained within the remnant for initially irrotational models, which served as evidence against equal-mass NSs mergers being a leading source of r-process elements in the universe through ejection. The lack of significant mass loss in equal-mass mergers, together with insignificant shock-heating of the material, also argued against the likelihood of such mergers as progenitors for SGRBs if the gamma-ray emission was assumed to be coincident with the GW burst; instead a delayed burst following the collapse of a HMNS to a BH appeared more likely.

Table 4: A summary of Full GR NS-NS merger calculations. EOS models include polytropes, piecewise polytropes (PP), as well as physically motivated models including cold SLy [83Jump To The Next Citation Point], FPS [222], and APR [3Jump To The Next Citation Point] models to which one adds an ideal-gas hot component to reflect shock heating, as well as the Shen [268, 267] finite temperature model and EOS that include Hyperonic contributions [264Jump To The Next Citation Point]. “Co/Ir” indicates that both corotating and irrotational models were considered; “BHB” indicates that BH binary mergers were also presented, including both BH-BH and BH-NS types, “ν-leak” indicates a neutrino leakage scheme was included in the calculation, “GH” indicates calculations were performed using the GHG formalism rather than BSSN, “non-QE” indicates superposition initial data were used, including cases where eccentric configurations were studied (“Eccen.”); “MHD” indicates MHD was used to evolve the system.
Group Ref. NS EOS Mass ratio π’ž notes
KT [287] Γ = 2 1 0.09 – 0.15 Co/Ir
[288Jump To The Next Citation Point] Γ = 2,2.25 0.89 – 1 0.1 – 0.17
[285Jump To The Next Citation Point] Γ = 2 0.85 – 1 0.1 – 0.12
[286Jump To The Next Citation Point] SLy, FPS+Hot 0.92 – 1 0.1 – 0.13
[282Jump To The Next Citation Point] SLy, APR+Hot 0.64 – 1 0.11 – 0.13
[332Jump To The Next Citation Point] Γ = 2 0.85 – 1 0.14 – 0.16 BHB
[144Jump To The Next Citation Point] APR+Hot 0.8 – 1 0.14 – 0.18
[145Jump To The Next Citation Point] APR, SLy, FPS+Hot 0.8 – 1.0 0.16 – 0.2
[265Jump To The Next Citation Point] Shen 1 0.14 – 0.16 ν-leak
[134Jump To The Next Citation Point] PP+hot 1 0.12 – 0.17
[264Jump To The Next Citation Point] Shen, Hyp 1.0 0.14 – 0.16 ν-leak
HAD [7Jump To The Next Citation Point] Γ = 2 1.0 0.08 GH, non-QE
[6Jump To The Next Citation Point] Γ = 2 1.0 0.08 GH, non-QE, MHD
Whisky [17] Γ = 2 1.0 0.14 – 0.18
[18] Γ = 2 1.0 0.20
[116Jump To The Next Citation Point] Γ = 2 1.0 0.14 – 0.18 MHD
[117Jump To The Next Citation Point] Γ = 2 1.0 0.14 – 0.18 MHD
[240Jump To The Next Citation Point] Γ = 2 0.70 – 1.0 0.09 – 0.17
[14Jump To The Next Citation Point, 15Jump To The Next Citation Point] Γ = 2 1.0 0.12 – 0.14
[241Jump To The Next Citation Point] Γ = 2 1.0 0.18 MHD
UIUC [172Jump To The Next Citation Point] Γ = 2 0.85 – 1 0.14 – 0.18 MHD
Jena [308, 41] Γ = 2 1.0 0.14
[122Jump To The Next Citation Point] Γ = 2 1.0 1.4 Eccen.

Later works, in particular a paper by Shibata, Taniguchi, and UryΕ« [285Jump To The Next Citation Point], 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:

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 Γ = 2 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 [85Jump To The Next Citation Point, 84Jump To The Next Citation Point, 108Jump To The Next Citation Point], and Princeton group [294Jump To The Next Citation Point, 88Jump To The Next Citation Point] (see [284Jump To The Next Citation Point] for a thorough review).

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

6.4.1 HMNS and BH remnant properties

Using their newly developed SACRA code [332Jump To The Next Citation Point], 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 Γ = 2 ideal gas, the critical total binary mass for prompt collapse to a BH is M = 2.8 –2.9M tot βŠ™, 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 a ≈ 0.78 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 [134Jump To The Next Citation Point], finding that binaries with total masses Mtot ≲ 2.7M βŠ™ should yield long-lived HMNSs (> 10 ms) and substantial disk masses with Mdisk > 0.04M βŠ™ assuming that the current limit on the heaviest observed NS, M = 1.97M βŠ™ [81Jump To The Next Citation Point] is correct. In Figure 10View Image, 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 [134Jump To The Next Citation Point] for an explanation of the EOS used in each simulation.

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Figure 10: Type of final remnant corresponding to different EOS models. The vertical axis shows the total mass of two NSs. The horizontal axis shows the EOSs together with the corresponding NS radii for MNS = 1.4M βŠ™. Image reproduced by permission from Figure 3 of [134Jump To The Next Citation Point], copyright by APS.

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 [265Jump To The Next Citation Point] that HMNSs will form generically for binary masses ≲ 3.2M βŠ™, 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 ∼ 3 –10 × 1053 ergβˆ•s, 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 1.8M βŠ™, the KT collaboration [264] still finds that thermal support can stabilize HMNS with masses up to 2.7M βŠ™.

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 q = 0.7 –1.0 [240Jump To The Next Citation Point], they found that bound disks with masses of up to 0.2M βŠ™ can be formed, with the disk mass following the approximate form

Mdisk = 0.039(Mmax − Mtot ) + 1.115(1 − q)(Mmax − Mtot); Mmax = 1.139(1 + q)M ∗, (34 )
where Mmax the maximum mass of a binary system for a given EOS (Γ = 2 ideal gas for these calculations), M ∗ is the maximum mass of an isolated non-rotating NS for the EOS, and Mtot 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 11View Image.
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Figure 11: Evolution of the total rest mass Mtot of the remnant disk (outside the BH horizon) normalized to the initial value for NS-NS mergers using a Γ = 2 polytropic EOS with differing mass ratios and total masses. The order of magnitude of the mass fraction in the disk can be read off the logarithmic mass scale on the vertical axis. The curves referring to different models have been shifted in time to coincide at tcoll. Image reproduced by permisison from Figure 5 of [240Jump To The Next Citation Point], copyright by IOP.

6.4.2 Magnetized NS-NS mergers

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 [6Jump To The Next Citation Point]. Beginning from spherical NSs with extremely strong poloidal magnetic fields (9.6 × 1015 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 12View Image), 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.

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Figure 12: Fluid density isocontours and magnetic field distribution (in a plane slightly above the equator) immediately after first contact for a magnetized merger simulation. The cavities at both trailing edges are attributed to magnetic pressure inducing buoyancy. Image reproduced by permission from Figure 1 of [6Jump To The Next Citation Point], copyright by APS.

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 [172Jump To The Next Citation Point]. 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 q = 0.85 and Γ = 2. 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 1012 to 1017 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 fGW ≳ 2 kHz. 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 [117Jump To The Next Citation Point] 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 [241Jump To The Next Citation Point]. They find that the large, turbulent magnetic fields (B ∼ 1012 G) present in the initial binary configuration are boosted exponentially in time up to a poloidal field of strength 1015 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 13View Image, is perhaps the most definitive result indicating that NS-NS mergers should produce SGRBs for some plausible range of initial parameters.

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Figure 13: Evolution of the density in a NS-NS merger, with magnetic field lines superposed. The first panel shows the binary shortly after contact, while the second shows the short-lived HMNS remnant shortly before it collapses. In the latter two panels, a BH has already formed, and the disk around it winds up the magnetic field to a poloidal geometry of extremely large strength, ∼ 1015 G, with an half-opening angle of 30°, consistent with theoretical SGRB models. Image reproduced by permission from Figure 1 of [241Jump To The Next Citation Point], copyright by AAS.

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.

6.4.3 GW emission

In [145Jump To The Next Citation Point], the KT collaboration found a nearly linear relationship between the GW spectrum cutoff frequency fcut 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 fcut 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 14View Image and 15View Image, we show h (t) and &tidle;h (f), respectively, for four of the runs calculated by the KT collaboration and described in [134Jump To The Next Citation Point]. 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 fGW = 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 (D ≲ 20 Mpc) and the high-frequency peak of sufficiently high quality [265Jump To The Next Citation Point].

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Figure 14: Dimensionless GW strain Dh βˆ•m 0, where D is the distance to the source and m 0 the total mass of the binary, versus time for four different NS-NS merger calculations. The different merger types become apparent in the post-merger GW signal, clearly indicating how BH formation rapidly drives the GW signal down to negligible amplitudes. Image reproduced by permission from Figures 5 and 6 of [134Jump To The Next Citation Point], copyright by APS.
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Figure 15: Effective strain at a distance of 100 Mpc shown as a function of the GW frequency (solid red curve) for the same four merger calculations depicted in Figure 14View Image. Post-merger quasi-periodic oscillations are seen as broad peaks in the GW spectrum at frequencies f GW = 2 – 4 kHz. The blue curve shows the Taylor T4 result, which represents a particular method of deducing the signal from a 3PN evolution. The thick green dashed curve and orange dot-dashed curves depict the sensitivities of the second-generation Advanced LIGO and LCGT (Large Scale Cryogenic Gravitational Wave Telescope) detectors, respectively, while the maroon dashed curve shows the sensitivity of a hypothetical third-generation Einstein Telescope. Image reproduced by permission from Figures 5 and 6 of [134Jump To The Next Citation Point], copyright by APS.

Using new multi-orbit simulations of NS-NS mergers, Baiotti et al. [14, 15Jump To The Next Citation Point] 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 16View Image.

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Figure 16: Comparison between numerical waveforms, shown as a solid black line, and semi-analytic NNLO EOB waveforms, shown as a red dashed line (top panel). The top panels show the real parts of the EOB and numerical relativity waveforms, and the middle panels display the corresponding phase differences between waveforms generated with the two methods. There is excellent agreement between with the numerical waveform almost up to the time of the merger as shown by the match of the orbital frequencies (bottom panel). Image reproduced by permission from Figure 14 of [15Jump To The Next Citation Point], copyright by APS.

6.4.4 Binary eccentricity

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 Δt = 100M ≈ 280M tot βŠ™ 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.

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