Coalescence of Black Hole-Neutron Star Binaries

2.1.1 Formulation in the excision approach

The set of Equations (28)–(31) has four functions that can be chosen freely; \({\partial _t}{{\tilde \gamma}^{ij}}\), ∂ _t K, \({{\tilde \gamma}_{ij}}\), and K. For computing quasi-equilibrium states, one usually assumes the presence of a helical Killing vector, ξ ^μ , and the absence of gravitational waves in the wave zone. Under these assumptions, it is natural to choose the time direction so as to satisfy ξ ^μ = (∂/∂t) ^μ (in the comoving frame), and to set \({\partial _t}{{\tilde \gamma}^{ij}}\) and ∂ _t K = 0. For the choice of K and \({{\tilde \gamma}_{ij}}\), two ways have been proposed. The first one is to choose a maximal slicing K = 0 and to adopt a flat metric \({{\tilde \gamma}_{ij}} = {f_{ij}}\), for simplicity. The other choice is to use the Kerr-Schild metric for \({{\tilde \gamma}_{ij}}\) and K in the vicinity of the BH or in the whole computational space. One of the advantages of choosing maximal slicing and the flat metric background is that the source term becomes simple and falls off rapidly for r → ∞ enough to obtain accurate results. The disadvantage is that it is not possible to construct the Kerr BH even with a distant orbit, and moreover, the set of Equations (28)–(31) may have non-unique solutions [159, 17, 224]. In particular, the spin of a BH has two values for the same spin parameter Ω _r (see Equation (49) for the definition) [132]. The lower branch of the spin as a function of Ω _r , which should be physically reasonable because it approaches to the Schwarzschild BH as Ω _r → 0, can reach only a ≈ 0.85, much less than the maximum spin of a Kerr BH. The advantage of using the Kerr-Schild metric for \({{\tilde \gamma}_{ij}}\) and K is that one can calculate a spinning BH with nearly maximum spin, a ≃ 1. The disadvantage is that the source term becomes complicated and falls off slowly for r → ∞. Because of this situation, it is not easy to derive the results as accurately as those with the conformal three metric, if one adopts this background metric in the whole computational space [208]. To recover the accuracy, a modification of the Kerr-Schild metric seems to be necessary; the metric is chosen to be nearly the Kerr-Schild one in the vicinity of the BH, whereas away from the BH, the metric should approach exponentially a conformally-flat metric and a maximal slicing [75, 132].

2.1.2 Formulation in the puncture approach

A “puncture” method [32] was proposed by Brandt and Brügmann to describe multiple BH with arbitrary linear momenta and spin angular momenta, extending the original work by Brill and Lindquist [33]. A “moving-puncture approach” [39, 15] was revealed to be quite useful in dynamic simulations. Here, we describe the puncture approach for quasi-equilibrium in the context of BH-NS binaries, which was originally developed by Shibata and Uryū [202, 203] and subsequently modified by Kyutoku et al. [106]. The puncture approach employs a mixture of the conformal thin-sandwich decomposition and the conformal transverse-traceless decomposition of Einstein’s equations. The trace part of the extrinsic curvature is set to zero (maximal slicing) and the three metric is assumed to be conformally flat in the work so far.

In contrast to the case in which the excision approach is adopted, the inner boundary conditions do not have to be imposed in the puncture approach. This could be a drawback in this approach, because one cannot impose physical boundary conditions (e.g., Killing horizon boundary conditions) for the BH. However, this could also be an advantage, because we do not have to impose a special condition for the geometric variables, and as a result, flexibility for adjusting a quasi-equilibrium state to a desired state is preserved.

2.1.3 Hydrostatic equations

The hydrostatic equations governing quasi-equilibrium states are the Euler and continuity equations. The matter in the NS interior needs to satisfy those equations. There are several versions for deriving the hydrostatic equations [26, 9, 192, 215, 80, 204]. In this section, we review the version in [204, 219].

Finally, we comment on the prescription for determining the constant in the right-hand side of Equation (77). For this task, the central value of the quantities in the left-hand side is usually used. Here the center of the NS is defined as the location of the maximum baryon rest-mass density. Equation (77) includes one more constant, which should be determined for each quasi-equilibrium figure; the orbital angular velocity as found from Equations (24), (69), (74), and (82). The method for calculating it will be explained in the next Section 2.1.4.

2.1.4 Orbital angular velocity

The first integral of the Euler equation (77) includes information of the orbital angular velocity, Ω, through Equations. (24), (69), (74), and (82). Ω should be determined from the condition that a binary system is in a quasi-equilibrium circular orbit. In the following, we describe two typical methods referring to the rotation axis of the binary system as the Z-axis and to the axis connecting the BH’s and NS’s centers as the X-axis.

In the other method, Ω is determined by requiring the enthalpy at two points on the NS’s surface along the X-axis to be equal to h = 1 on the surface. At these two points, the pressure is absent. Namely, the sum of the gravitational force from the BH, self-gravitational force from the NS, and the centrifugal force associated with the orbital motion is balanced. These two conditions may be regarded as the conditions that determine C_E and Ω, and thus, Ω can be determined for a given set of gravitational field variables. The work by Taniguchi et al. [208, 209, 210, 214] confirmed that in both methods, an accurate numerical result can be computed with a reasonable number of iterations, and that the results by these two methods coincide within the convergence level of the enthalpy. Therefore, both methods work well.

2.1.5 Center of mass of a binary system

In the puncture approach, the situation is totally different from the above, because Condition (85) has already been used to calculate the linear momentum of the BH; see Equation (58) in Section 2.1.2. In this framework, there is no known natural, physical condition for determining the center of mass of the system. Until now, three methods have been employed to determine the center of mass. In the first method, the dipole part of ψ at spatial infinity is required to be zero [202, 203]. However, it was found that in this condition, the angular momentum derived for a close orbit of Ωm₀ ≥ 0.03 is ∼ 2% smaller than that derived by the 3PN approximation for Q = 3. Because the 3PN approximation should be an excellent approximation of general relativity for a fairly distant orbit, as such Ωm₀ ≈ 0.03, the obtained initial data deviates from the true quasi-circular state, and hence, the initial orbit would be eccentric.

In the second method, the azimuthal component of the shift vector β ^ϕ at the location of the puncture is required to be equal to −Ω; a corotating gauge condition at the location of the puncture is imposed [197]. This method gives a slightly better result than that of the first method. However, the angular momentum derived for a close orbit of Ωm₀ ≥ 0.03 is also ∼ 2% smaller than that derived by the 3PN relation for a larger mass ratio Q ≥ 2. The disagreement is larger for the larger mass ratio. Such initial conditions are likely to deviate from the true quasi-circular state and hence the orbital eccentricity is large as well.

In the last method, the center of mass is determined in a phenomenological manner: One imposes a condition that the total angular momentum of the binary system for a given value of Ωm₀ agrees with that derived by the 3PN approximation [106]. This condition can be achieved by appropriately choosing the position of the center of mass. With this method, the drawback in the previous two methods, i.e., the angular momentum becoming smaller than the expected value, is overcome. Recent numerical simulations by the KT group have been performed employing initial conditions obtained by this method, and showed that the binary orbit is not very eccentric with these initial conditions (cf. Section 3.1.1).

2.1.6 Equations of state

Even though the EOS used for constructing sequences is only the polytrope, a lot of quasi-equilibria with several EOS have been derived as initial data for the merger simulations. We will summarize those initial data in Section 3.1.1.

2.1.7 Physical quantities

The several key quantities that are necessary in the quantitative analysis of quasi-equilibrium sequences are summarized in this section; the irreducible mass and spin of BH, the baryon rest mass of NS, the ADM mass, the Komar mass, and the total angular momentum of the system. It is reasonable to consider that the irreducible mass and spin of the BH and the baryon rest mass of the NS are conserved during the inspiral of BH-NS binaries. In addition, temperature and entropy of the NS may be assumed to be approximately zero because the thermal effects of not-young NS are negligible to their structure, i.e., it is reasonable to use a fixed cold EOS throughout the sequence. For such a sequence, the ADM mass, the Komar mass, and the total angular momentum of the system vary with the decrease of the orbital separation. These global quantities characterize the quasi-equilibrium sequence.

2.1.8 A mass-shedding indicator

2.3.1 Binding energy and total angular momentum

Figure 2 shows the binding energy (E_b/m₀) and the total angular momentum (\({{\bar M}_B} = 0.15({\mathcal C} = 0.145)\)) as functions of the orbital angular velocity (Ωm₀) for a NS with baryon rest mass \({\mathcal C}\) and for mass ratio Q = 3. All the quantities shown are normalized to be dimensionless. The solid curves with the filled circles denote the numerical results, and the dashed curves, the results in the 3PN approximation [25]. The numerical sequences terminate at an orbit of cusp formation (i.e., at an orbit of the mass-shedding limit) before the ISCO is encountered, i.e., before a turning point (minimum) of the binding energy appears.

Open image in new window

From the qualitative argument described in Section 1.1, the binary separation d_ms at the onset of mass shedding of a NS can be approximately derived as (see Equation (5))

$${{{d_{{\rm{ms}}}}} \over {{M_{{\rm{BH}}}}}} \simeq {1 \over {{Q^{2/3}}\mathcal{C}}}.$$

(100)

If d_ms is greater by a sufficient amount than the ISCO separation d_ISCO, we may expect the NS to start shedding mass and to be tidally disrupted before being swallowed by the BH. Relation (100) suggests that the mass-shedding separation decreases with increasing mass ratio, Q, and compactness of the NS, \((J/m_0^2)\). It is natural to expect to encounter minima in the binding energy and total angular momentum for binaries with sufficiently large mass ratio and NS compactness. Figure 3 shows the binding energy (E_b/m₀) and the total angular momentum \({{\bar M}_{\rm{B}}} = {(}0.15{)}({\mathcal C} = {(}0.145{)})\) as functions of the orbital angular velocity (Ωm₀) for a NS with baryon rest mass \({\mathcal C} = 0.120\) and for Q = 5. The NS compactness is the same as, but the mass ratio is larger than, that shown in Figure 2. In this model, the binary encounters an ISCO before the onset of mass shedding, i.e., we see minima in the binding energy and angular momentum just before the end of the sequence.

Open image in new window

Figure 3 shows that the turning points in the binding energy and the total angular momentum appear simultaneously to within numerical accuracy. This fact is more clearly seen in Figure 4 in which the binding energy versus total angular momentum for sequences of mass ratio Q = 5 but with different NS compactness is plotted. A simultaneous turning point in the binding energy and total angular momentum leads to a cusp in these curves. As suggested by Equation (100), turning points are not found for small compactness (e.g., the case of \({\mathcal C} = 0.145\)), since the sequences terminate at mass shedding before encountering an ISCO. However, for larger compactness, these curves indeed form a cusp. Note that 3PN sequences cannot identify mass shedding and therefore always exhibit turning points.

Open image in new window

2.4.1 Mass-shedding limit

First, we summarize the results for the binary separation and the orbital angular velocity at which mass shedding from the NS surface occurs. In Newtonian gravity and semi-relativistic approaches, simple equations may be introduced to fit the effective radius of a Roche lobe [225, 94, 151, 60]. In [202, 203] a fitting equation is introduced for binaries composed of a non-spinning BH and a corotating NS in general relativity. In this section, we review how to derive a fitting equation from data in [210] for a non-spinning BH and an irrotational NS.

To derive a fitting formula, we need to determine the orbital angular velocity at the mass-shedding limit. However, it is not possible to construct cusp-like configurations by the numerical code used in [210], which is based on a spectral method and accompanied by the Gibbs phenomena. This is also the case for a configuration with smaller values of χ ≤ 0.5, even though a cusp-like configuration does not appear (here χ is a mass-shedding indicator defined by Equation (99)). Thus, the data points for such close orbits have to be determined by extrapolation. For this purpose, Taniguchi et al. [210] tabulated χ as a function of the orbital angular velocity and extrapolated the sequence to χ = 0 by using fitting polynomial functions to find the orbits at the onset of mass shedding. Figure 5 shows an example of such extrapolations for sequences of NS compactness \({\mathcal C}\) with mass ratios Q = 1, 2, 3, and 5 [210]. From the extrapolated results toward χ = 0, the orbital angular velocity at the mass-shedding limit is approximately determined for each set of \({\mathcal C}\) and Q.

Open image in new window

To derive a fitting formula for the orbital angular velocities at the mass-shedding limit for all the values of \({\mathcal C}\) and Q, the qualitative Newtonian expression of Equation (7) is useful. By fitting sequence data to this expression, Taniguchi et al. [210] determined the value of C_Ω for Γ = 2 polytropic EOS as 0.270, i.e.,

$${\overline \Omega _{{\rm{ms}}}} = 0.270{{{C^{3/2}}} \over {{{\bar M}_{{\rm{NS}}}}}}{(1 + {Q^{- 1}})^{1/2}},$$

(101)

or equivalently

$${\Omega _{{\rm{ms}}}}{m_0} = 0.270\mathcal{C}^{3/2}(1+Q){(1 + {Q^{- 1}})^{1/2}},$$

(102)

Figure 6 shows the results of the fitting for the mass-shedding limit. The solid curve denotes Equation (101) and the points are the numerical results. The agreement is not perfect, but fairly good for Q ≥ 2.

Open image in new window

It may be interesting to note that the value of C_Ω = 0.270 is the same as that found for quasi-equilibrium sequences of NS-NS binaries in general relativity in [214] and of BH-NS binaries in general relativity where the NS is corotating in [202, 203]. The value of C_Ω = 0.270 could be widely used for an estimation of the orbital angular velocity at the mass-shedding limit of NS in a relativistic binary system with Γ = 2.

2.4.2 Innermost stable circular orbit

One of the most important pieces of information for relativistic close binaries is the binary separation (or the orbital angular velocity) at which the minimum of the binding energy appears, corresponding to the ISCO. The minimum point is located by fitting three nearby points of the sequence to a second-order polynomial, because the numerical data is discrete and does not necessarily give the exact minimum.

A simple empirical fitting that predicts the angular velocity Ω_ISCO at the ISCO for an arbitrary companion orbiting a BH may be derived in the manner of [210]. In their approach, one searches for an expression with three free parameters that express ΩISCO ^{as a} function of the mass ratio Q and the compactness \({\mathcal C} = 0.5\) of the companion. Then the three parameters are determined by matching to three known values of Ω_ISCO, namely, (1) that of a test particle orbiting a Schwarzschild BH, Ω_ISCOm₀ = 6^−3/2 (for Q = ∞), (2) that of an equal-mass BH-BH system as computed in [40], Ω_ISCOm₀ = 0.1227 (for Q = 1 and \({\mathcal C} = 0.1452\)), and finally (3) that of a BH-NS configuration as computed in [210], Ω_ISCOm₀ = 0.0854 (for Q = 5 and \({\mathcal C}\)). A further requirement arises from the fact that for a test particle (with Q = ∞), the expression should be independent of the companion’s compactness. A good fit to the numerical data is found in [210] with the expression

$${\Omega _{{\rm{ISCO}}}}{m_0} = 0.0680\left[ {1 - {{0.444} \over {{Q^{0.25}}}}\left({1 - 3.54{\mathcal{C}^{1/3}}} \right)} \right],$$

(103)

as demonstrated in Figure 7. The exponents of Q and \({\mathcal C} = 0.145\) in Equation (103) are empirically determined by requiring that the fitted curves lie near the data points for all the models. The agreement is not perfect, but sufficient for finding the orbital angular velocity at the ISCO within ∼ 10% error.

Open image in new window

2.4.3 Critical mass ratio

Combining Equations (102) and (103), we can identify the critical binary parameters that separate the final fates. The binary encounters an ISCO before reaching mass shedding or the NS reaches the mass-shedding limit. Figure 8 illustrates the final fate of BH-NS binaries for \({\bar \Omega}\). The solid curve denotes the orbital angular velocity at the mass-shedding limit, and the long-dashed one denotes it at the ISCO. As seen from Equations (102) and (103), both of these curves depend on the mass ratio Q, but in different ways, which leads to the intersection of the two curves. An inspiraling binary evolves along horizontal lines towards increasing \({{\tilde \gamma}_{ij}}\), starting at the left and moving toward the right, until reaching either the ISCO or the mass-shedding limit. After the binary reaches the ISCO for sufficiently large mass ratio, we cannot predict the fate of the NS, because it is in a dynamic plunge phase (see Section 3), but nevertheless the mass-shedding limit for unstable quasi-equilibrium sequences is included as the dotted curve in Figure 8. As shown in Figure 8, the model with mass ratio Q = 6 (dotted-dashed line) encounters the ISCO, while that of Q = 3 (dot-dot-dashed line) ends up at the mass-shedding limit. The intersection between the mass-shedding and ISCO curves marks a critical point that separates the two distinct fates of the binary inspiral.

Open image in new window

When we eliminate Ωm₀ from Equations (102) and (103), we can draw a curve of the critical mass ratio, which separates BH-NS binaries that encounter an ISCO before reaching mass shedding, and vice versa, as a function of the compactness of the NS. The equation, which gives the critical curve is expressed as

$$0.270{\mathcal{C}^{3/2}}(1 + Q){(1 + {Q^{- 1}})^{1/2}} = 0.0680\left[ {1 - {{0.444} \over {{Q^{0.25}}}}\left({1 - 3.54{\mathcal{C}^{1/3}}} \right)} \right],$$

(104)

and the curve that separates those two regions is shown in Figure 9. The solid curve denotes the critical mass ratio for each compactness. If the mass ratio of a BH-NS binary is larger than the critical one, the quasi-equilibrium sequence terminates by encountering the ISCO, while if smaller, it ends at the mass-shedding limit of the NS.

Open image in new window

3.1.1 Initial condition

When a quasi-equilibrium configuration constructed by the excision approach is used as initial data for a simulation code based on the moving-puncture approach, one more step is needed before starting the simulations because the data inside the BH excision surface does not exist, and thus, one needs to fill the BH interior by extrapolating all field values radially from the excision surface toward the BH center. In the extrapolation procedure, one has to employ a method in which any constraint violation introduced inside the BH is not allowed to affect the exterior spacetime. Two groups proposed such a procedure [34, 61], and the UIUC group used the method they developed in their simulations [61].

Strictly speaking, the quasi-equilibrium derived in the framework of Section 2 is not realistic because an approaching radial velocity driven by gravitational radiation reaction is not taken into account. If we adopt such quasi-equilibrium data as the initial condition, the trajectories obtained in the numerical simulation usually result in a slightly eccentric orbit. One prescription to suppress this error is to choose an initial condition in which the binary separation is large enough that the effect of the radial velocity is not serious. Another method to reduce the eccentricity was proposed by the CCCW group [157, 75] as described in the following.

3.1.2 Evolving metric

There are currently two formalisms for evolving the spacetime metric. One is the Baumgarte-Shapiro-Shibata-Nakamura (BSSN) formalism [140, 195, 18] together with the moving-puncture gauge condition [39, 15, 35, 136], and the other is the generalized harmonic (GH) formalism [129] (see also [77, 86, 162, 163]). The AEI, KT, and UIUC groups employ the BSSN formalism, while the CCCW and LBPLI [6] groups use the GH formalism. These two formalisms are different from the standard 3+1 formalism [230], which was found to be inappropriate in numerical relativity because stable and longterm numerical simulations are not feasible due to the presence of an unphysical growth mode excited even by a tiny numerical truncation error.

The BSSN formalism, the original version of which was first proposed by Nakamura in 1987 [140], is in a sense a modified version of the 3+1 formalism [230]. The essence in this formalism is to adopt not only the three metric (γ _ij ) and the extrinsic curvature (K _ij ) but also a conformal factor of the three metric (ψ or ϕ = log(ψ) or W = ψ⁻² or χ = ψ⁻⁴), the trace part of the extrinsic curvature (K), and the first spatial derivative of the three metric (\({F_i} = {{\tilde \gamma}_{ij,j}}\) or \({{\tilde \Gamma}^i} = - {{\tilde \gamma}^{ij}}_{,j}\) where \({{\tilde \gamma}_{ij}}\) is the conformal three metric) as new independent variables, and then, to appropriately rewrite the basic equations using these new variables. With this prescription, the evolution equations for the conformal three metric, \({{\tilde \Gamma}^i}\), and the conformal traceless extrinsic curvature, Ã _ij ≡ γ^−1/3(K _ij − K _γij /3), are written in the form of simple wave equations. This prescription leads to avoiding the spurious growth of unphysical modes by numerical truncation errors and enables one to perform a stable and longterm evolution for a variety of systems. Typical basic equations are described in Appendix A.

In both formalisms, the number of constraint equations is increased as a result of defining the new variables, in addition to the Hamiltonian and momentum constraints. For the BSSN formalism, \({\rm det}({{\tilde \gamma}_{ij}}) = 1,{{\tilde \gamma}^{ij}}{{\tilde A}_{ij}} = 0\), and \({{\tilde \Gamma}^i} + {{\tilde \gamma}^{ij}}_{,j} = 0\) or \({\kappa _i}\rho _i^{{\Gamma _i}} = {\kappa _i}_{+ 1}\rho _i^{{\Gamma _{i + 1}}}\) are new constraints. For the GH formalism, D _iμν − ∂ _i g _μν = 0 is a new constraint.

3.1.3 Evolving a black hole

For handling a BH, the original BSSN formalism was not satisfactory. Re-defining a spatial con-formal factor [39], adopting more than fourth-order finite-differencing schemes, and employing an appropriate moving-puncture gauge condition [39, 15, 221, 35] are required for evolving BHs accurately and stably. In such a BSSN-puncture formulation, we choose a BH spacetime, which is free of the true singularity [32]. Although a coordinate singularity may appear in the center of the BH, this can be effectively (and, in a sense, fortunately) excised in the moving-puncture gauge condition [88]. Consequently, one can evolve the whole computational region without artificially excising inside the BH horizon (i.e., without special numerical treatments for the inside of the BH horizon).

On the other hand, in the code of the CCCW and LBPLI groups using GH formalism, the gauge, similar to the moving-puncture gauge, has not yet been developed. These groups employ the excision technique, i.e., a region inside the apparent horizon is cut out of the computation domain and replaced with an inner boundary. Pretorius has successfully simulated orbiting BH-BH binaries using the GH formalism with excision for the first time [162, 163]. Subsequently, the CCCW group has successfully simulated a longterm inspiral and merger of BH-BH binaries using a high-accuracy spectral method [181, 206], verifying that this technique is also robust for handling BH spacetimes.

3.1.4 Hydrodynamics

3.1.5 Equations of state

Employing realistic EOS for modeling NS is a key ingredient in accurately determining the final fate after the onset of the merger and in deriving realistic gravitational waveforms in the merger phase. However, the EOS of the nuclear matter beyond the normal nuclear density is still uncertain due to the lack of strong constraints obtained from experiments and observations (e.g., [115, 116, 117] and see [51] for a recent devlopment). Hence, we do not know exactly the detailed physical state of NS such as density and composition profiles as functions of radius for a given mass and the relation between the mass and the radius.

Gravitational waveforms emitted in the tidal-disruption phase as well as properties of the remnant BH-disk system such as the mass and the typical density of the disk depend strongly on the EOS, as shown in the following. The primary reason for this is that these depend strongly on the relation between the mass and the radius of the NS, the sensitivity of a NS to the tidal force by its companion BH depends primarily on its radius, e.g., a NS of larger radii (with a stiffer EOS) will be disrupted at a larger orbital separation (or a lower orbital frequency; see equations in Section 1); if the tidal disruption of a NS occurs at a larger distance, more material will be widely spread around its companion BH, and consequently, a high-mass remnant disk is likely to be formed; also, the gravitational-wave frequency at tidal disruption, which will be one of the characteristic frequencies, is lower for a NS of larger radius.

In the study of binaries composed of NS, rather than require one to prepare a “realistic” EOS, we should consider exploring the possibility of determining the NS EOS by gravitational-wave observation, as discussed in [127, 220, 168, 70]. For this purpose, we need to prepare theoretical templates of gravitational waves employing a wide variety of possible EOS for the NS matter, and systematically perform a wide variety of simulations employing a large number of plausible EOS. Then, the next task is to determine what kinds of EOS should be employed in their theoretical work.

Thermal energy per baryon inside NS, except for newly-born ones, is believed to be much lower than the Fermi energy of the constituent particles, because a young NS is quickly cooled down by neutrino emission (e.g., [116, 99]). This implies that we can safely neglect thermal effects on the NS in the inspiral and early merger phases, and can employ a cold EOS, for which the pressure, P, the specific internal energy, ε, and other thermodynamic quantities are written as functions of the rest-mass density ρ. In particular, employing the cold EOS is appropriate for the case in which the merger does not result in the formation of a disk surrounding the remnant BH, i.e., for the case in which the NS is simply swallowed by the companion BH, because a heating process, such as shock heating, never plays a role. By contrast, to follow a longterm evolution of tidally disrupted material, which forms a disk around the remnant BH, the finite-temperature effects of the EOS and the neutrino cooling have to be taken into account. This is because the temperature of the disk material is likely to be increased significantly by shock or a longterm viscous heating, while the density of the disk is lower than that of the NS, and as a result, the Fermi energy is lowered, resulting in a situation where the roles of the pressures by degenerate electrons and thermal gas become less important and increasingly important, respectively.

Because the simulation for BH-NS binaries in general relativity is still in an early phase (by 2010), much work has been done with a rather simple (not very realistic) EOS. The often-used EOS in the early phase of this study within the numerical-relativity community is the Γ-law EOS,

$$P = (\Gamma - 1)\rho \varepsilon ,$$

(116)

with Γ = 2. In this EOS, the initial condition for the NS is determined by the polytropic EOS, P = κρ ^Γ (see Section 2). However, this EOS is known to disagree with typical nuclear-theory-based EOS for a small value of Γ ∼ 2 (e.g., [107] and Figure 10).

Open image in new window

At each boundary density, ρ = ρ _i (i = 1, ⋯, n − 1), the pressure is required to be continuous, i.e., \({\mathcal C} = {M_{{\rm{NS}}}}/{R_{{\rm{NS}}}}\). Thus, if one gives k₁, Γ _i , and ρ _i (i = 1, ⋯, n), the EOS is totally determined. For the zero-temperature EOS, the first law of thermodynamics (dε = −Pd(1/ρ) or dh = dP/ρ) holds, and thus, ε and h are also determined except for the choice of the integration constants, which are fixed by the continuity condition of ε (hence equivalently h) at each ρ _i .

3.1.6 Adaptive mesh refinement

Therefore, to efficiently perform a large number of numerical simulations with the finite-differencing method, adaptive mesh refinement (AMR), such as that proposed by Berger and Oliger [23], is an indispensable technique. In this technique, one prepares several refinement levels of Cartesian boxes (or other geometrical domains) with different grid resolutions; usually in the smaller boxes, the grid resolution is higher. At such a fine refinement level, the BH and the NS are evolved with sufficiently high resolution with grid spacing ≲ Δ₀. On the other hand, the propagation of gravitational waves, which have a wavelength much longer than the size of the compact objects, is followed in large-size boxes (in the coarser levels) with a large grid spacing, which is still much smaller than the gravitational wavelength (say, ∼ λ/10).

Mesh refinement techniques in numerical relativity have been developed by several groups (e.g., [93, 182, 35, 228]). The AEI and UIUC groups use the <monospace>Carpet</monospace> module of the <monospace>Cactus</monospace> code [182] developed primarily by Schnetter, the KT group the <monospace>SACRA</monospace> code [228] developed by Yamamoto, and the <monospace>LBPLI</monospace> group the <monospace>HAD</monospace> code [7]. Recently, two independent codes (<monospace>Whisky</monospace>, which employs <monospace>Carpet</monospace>, and <monospace>SACRA</monospace>) were compared, performing simulations of NS-NS binaries [14], and it was illustrated that the AMR scheme in <monospace>Carpet</monospace> and <monospace>SACRA</monospace> work well at the same level. The CCCW group employs a two-grid technique, which was originally proposed by the Meudon-Valencia-MPA team [52]. In this approach, geometric-field quantities are evolved with the multi-patch pseudo-spectral method and the hydrodynamic equations are solved using a standard finite-volume scheme on a second grid [58]. In solving the equations for the geometric field, the computational domain encompassing the local wave zone is prepared, while the hydrodynamic equations are solved in the region where the matter presents.

3.3.1 Zero BH spin

As mentioned in Section 1, broadly speaking, the final fates of BH-NS binaries are divided into two classes. One in which the NS is tidally disrupted before it is swallowed by the companion BH and the other is that the NS is simply swallowed by the BH. Figures 11 and 12 display snapshots of the rest-mass density profiles and the location of the apparent horizon on the equatorial plane at selected time slices for two typical cases [107]. For these results, the NS are modeled by the piecewise polytropic EOS described in Table 4. Figure 11 illustrates the process in which the NS is tidally disrupted before the binary reaches the ISCO and then a disk is formed surrounding the companion BH. For this model, M_BH = 2.7 M_⊙, a = 0, M_NS = 1.35 M_⊙, and R_NS = 15.2 km (EOS 2H); the mass ratio (Q = 2) is small and the NS radius is large. For this setting, the NS is significantly tidally deformed in close orbits, and eventually, mass shedding from an inner cusp of the NS sets in far outside the ISCO. After a substantial fluid element is removed from the inner cusp, the NS is tidally disrupted outside the ISCO. It should be emphasized that tidal disruption does not occur immediately after the onset of mass shedding in this case. Tidal disruption occurs for an orbital separation smaller than that for the onset of mass shedding.

Open image in new window

Open image in new window

After tidal disruption occurs, the material of the NS forms a one-armed spiral. As a result of angular momentum transport in the arm, a large amount of material spreads outward, and after the spiral arm is wound from the differential rotation, a disk of approximately axisymmetric configuration is formed around the BH located approximately at the center. However, because of the presence of a non-axisymmetric structure at its formation, however, the disk does not completely relax to an axisymmetric state in the rotational period ∼ 10 ms. Rather, a one-armed spiral of small amplitude is present for a long time, and helps gradually transporting angular momentum outward, resulting in a gradual mass infall into the BH. However, the mass accretion time scale is much longer than the rotational period, and hence, the disk remains quasi-stationary for;≫ 10 ms. This evolution process agrees qualitatively with that found in the study of the longterm evolution of BH-disk systems [90, 103].

The tidal disruption process as illustrated in Figure 11 is qualitatively common for the model with a low-mass BH or a large-radius NS or a high-spin BH. However, quantitative details depend on the parameters of the binary. For a small mass ratio (Q ∼ 2) with a = 0, the typical size of the disk (the region with ρ > 10¹⁰ g/cm³) is 50–100 km with maximum density ≳ 10¹² g/cm³ for a disk of mass ∼ 0.1 M_⊙ as shown in Figure 11. Thus, the disk is rather compact. The disk relaxes to a nearly axisymmetric configuration in a short time duration, approximately equal to the rotational period around the BH. We note that these properties depend on the parameter of the binary. For example, for a large mass ratio with a high BH spin (e.g., Q = 5 and a = 0.75), the typical size is also ∼ 100 km, but the maximum density is smaller than those for the smaller value of Q; the time scale until the disk relaxes to an axisymmetric configuration is relatively long. A remarkable point is that the tidal debris of relatively low density ∼ 10¹⁰ g/cm³ could be ejected to a distance of ≫ 100 km [63, 57, 41, 109], i.e., a wider but less dense disk is formed (see also Section 3.3.3).

Figure 12 illustrates the case in which the NS is not tidally disrupted before it is swallowed by the BH. For this model, M_BH = 4.05 M_⊙, a = 0, M_NS = 1.35 M_⊙, and R_NS = 11.0 km (EOS B in Table 4). In this case, the NS is tidally deformed only in a close orbit. Then, mass shedding sets in for a too close orbit to induce subsequent tidal disruption outside the ISCO. As a result, most of the NS material falls into the BH approximately simultaneously. Also, the infall occurs from a narrow region of the BH horizon. These processes help exciting a non-axisymmetric fundamental quasi-normal mode (QNM) of the remnant BH. The mass of the disk formed after the onset of the merger is negligible (much smaller than 0.01 M_⊙), because the BH simply engulfs the NS.

Generally speaking, the final fate depends on the location where mass shedding of the NS sets in. If the location is in the vicinity of or inside the ISCO, most of the NS material falls into the companion BH, and a BH with negligible surrounding material is the outcome. With the increase of the orbital separation at the onset of mass shedding, the mass of the material surrounding the BH increases. Note again that the mass shedding has to set in sufficiently outside the ISCO to induce tidal disruption, because tidal disruption occurs only after substantial mass is removed from the NS.

3.3.2 Nonzero BH spin

The effect of the BH spin significantly modifies the orbital evolution process in the late inspiral phase and merger dynamics, as first demonstrated by the UIUC group [63]. Figure 13 shows the trajectories of the BH and NS for models with Q = 3, \({\mathcal C} = 0.145\), and a = 0 (left) and a = 0.75 (right) [63]. The NS is modeled by the Γ-law EOS with Γ = 2. The initial orbital angular velocity is the same for two models. For the binary composed of a non-spinning (a = 0) BH and NS, the merger occurs after about 4 orbits, whereas for the case with a spinning BH (a = 0.75), it occurs after about 6 orbits. For the case with a spinning BH, the decreased rate of the orbital separation appears to be small. Qualitatively, these differences may be explained primarily by the presence of a spin-orbit coupling effect, which is accompanied by an additional repulsive force for a > 0 (and attractive force for a < 0), and thus, reduces the attractive force between two objects (see the equations of motion for two-body systems in the context of the PN approximation [102, 226, 100]). In the presence of this additional repulsive force, centrifugal forces should be reduced for a given orbital separation. This slows down the orbital velocity, and therefore, the luminosity of gravitational waves is decreased and orbital evolution due to gravitational radiation reaction is delayed (the lifetime of the binary becomes longer). In addition, the orbital radius at the ISCO around the BH is decreased (and the absolute value of the binding energy at the ISCO around the BH is increased) due to the spin-orbit coupling effect (e.g., [16]). This further helps to increase the lifetime of the binary because it evolves as a result of gravitational radiation reaction and hence has to emit more gravitational waves to reach the ISCO.

Open image in new window

This longer lifetime for a binary with a spinning BH enhances the possibility of tidal disruption, and the final outcome is modified. Figure 14 displays the contour curves and the location of the remnant BH for the same models as in Figure 13. For both models, the NS is tidally disrupted outside the ISCO and a disk is formed. For the spinning BH case (a = 0.75), a more extended, more massive, and denser disk is the outcome. For the non-spinning case (a = 0), the disk mass is only ≈ 4% of the total rest mass whereas for a = 0.75, it is ≈ 13% (see Section 3.4 for details of the remnant disk). This is probably due to the effect that the physical radius of the ISCO (or specific angular momentum of a particle orbiting the ISCO) around the spinning BH is smaller than that for the non-spinning BH and also that the radial approaching velocity at tidal disruption is smaller for a spinning BH because of the repulsive nature of the spin-orbit coupling effect.

Open image in new window

The CCCW group subsequently studied the effects of BH spin on the final remnant with several EOS [57]. They reached a similar conclusion about the orbital evolution and final outcome to that of the UIUC group even for the case in which BH spin is slightly smaller, a = 0.5. This conclusion was reconfirmed also by the KT group [109] for a wide variety of piecewise polytropic EOS and mass ratios. Therefore, a moderately large BH spin, a = 0.5, is substantial enough for modifying the merger process and enhancing the disk formation. The CCCW group also performed a simulation with a = 0.9 and \({\mathcal C} \lesssim 0.18\) [74] and found that tidal disruption occurs far outside the ISCO and the resulting disk mass is very high, ∼ 36% of the total rest mass (see Section 3.4).

The KT group also found that [109] for binaries composed of a high-spin BH (a = 0.75) and NS, tidal disruption may occur for a large value of mass ratio, Q ∼ 5, for a wide variety of NS EOS as far as it gives \({\mathcal C} = {(}0.144{)} - 0.173\). This implies that tidal disruption of a NS may be possible for a large BH mass over a wide area. In such case, the material of the tidally-elongated NS is swallowed from a relatively narrow region of the BH surface. As will be discussed in Section 3.6, this helps excite a non-axisymmetric fundamental QNM of the remnant BH. On the other hand, for the non-spinning BH case for which tidal disruption occurs only for a small BH, the material of a tidally-elongated NS is always swallowed by a wide region of the BH surface. This suppresses the excitation of non-axisymmetric QNM. This difference is reflected in gravitational waveforms and spectra, as predicted in [177, 178] (in which a BH perturbation study was performed).

To clarify the fact described above, Kyutoku et al. [109] generated Figures 15–17, which show the evolution of the rest-mass density profile for Q = 3, M_ns = 1.35 M_⊙, and EOS HB (cf. Table 4) with a = 0.75, 0.5, and −0.5, respectively. The evolution processes shown in Figures 15 and 17 are similar to those in Figures 11 and 12, respectively. Figure 15 shows the case in which mass shedding of the NS occurs at an orbit sufficiently far from the BH. Subsequently, the NS is extensively elongated, a one-armed spiral is formed, and then the spiral arm composed of dense material is wound around the BH. The material located in the outer region of the spiral arm forms a disk, while that in the inner region falls into the BH. The infall of the dense material proceeds from a wide region of the BH surface as seen in the fourth panel of Figure 11. By contrast, for a = −0.5 (see Figure 17), tidal disruption does not occur and more than 99.99% of the NS matter falls into the BH from a narrow region of the BH horizon and in a short time scale.

Open image in new window

Open image in new window

Open image in new window

The type of merger process for a = 0.5 shown in Figure 16 is qualitatively new. Tidal disruption occurs in a relatively close orbit (in the vicinity of the BH ISCO). The subsequent evolution process is similar to that for a = 0.75, but the infall of dense NS material to the BH occurs from a relatively narrow region (see the second to fourth panels of Figure 16). Eventually, the infall proceeds from a wide region of the BH surface, but the density of the infalling material seems to be too low to enhance a QNM oscillation of the BH significantly (see the fifth panel of Figure 16). This feature is often found for a binary of high-mass ratio and high BH spin.

To date, three types of merger process have been found. Type-I: the BH mass is low or the BH spin is high, and the NS is tidally disrupted for an orbit far from the BH ISCO; Type-II: the BH mass is not low, the BH spin is small (or a < 0), and the NS is not tidally disrupted; Type-III: the BH mass is not low, the BH spin (a > 0) is high, and the NS is tidally disrupted for an orbit close to the BH ISCO. Their merger processes are schematically described in Figure 18. These differences in the infall process are well reflected in gravitational waveforms and spectra.

Open image in new window

In the latest work of the CCCW group [74], the effect of spin orientation, which is misaligned with that of the orbit rotation axis, was studied for the first time. They performed simulations for a = 0, 0.5, and 0.9, and Q = 3 with Γ-law EOS (Γ = 2), and found that the remnant disk mass decreases sensitively with increase of the inclination (misalignment) angle for given values of a and Q (see Figure 20). This is quite natural because the spin-orbit coupling force is proportional to S·L, where S and L denote BH spin and orbital angular momentum vectors, respectively, and the radius of the ISCO around the BH approaches that for a = 0 with increasing of the inclination angle. Thus, the BH spin effect becomes less important with the increase of the inclination angle. They also found that the inclination angle is significantly decreased after a substantial mass of the NS falls into the companion BH, implying that the angular momentum vector of the remnant disk misaligns only modestly with the BH spin vector (by ≲ 20°). This is also quite natural because the orbital angular momentum is as large as or larger than the spin angular momentum of the BH for small mass ratios like Q = 3. However, for a high value of Q for which the fraction of the BH spin angular momentum in the total angular momentum is large, this conclusion will be modified. The initial inclination angle will not be significantly modified and an inclined disk, which subsequently precesses around the spinning BH, may be the outcome.

3.3.3 Extent of remnant disk

A tail of a one-armed spiral formed at tidal disruption often extends far away from the BH, in particular for the case in which the BH spin is high. The CCCW group reported that for Q = 3, a = 0.5, and \(\sqrt {{l^3}/G{m_0}}\) (i.e., for realistic values of the compactness), a tidal tail of mass 0.01–0.1 M_⊙ goes to a distance l = 200–2000 km away from the BH before falling back to the central region [57]. They also reported that the fall-back time scale was ∼ 200 ms for l = 2000 km assuming that the material obeys geodesic motion. Here, 200 ms agrees roughly with the dynamic infall time scale \(\sqrt {{l^3}/G{m_0}}\). This indicates that the time scale of mass accretion from the disk onto the BH is much longer than the rotational period of the disk in the vicinity of the BH ∼ 10 ms. The typical duration of SGRB is 0.1–1 s [142]. Such a time scale may be explained by the time scale of the fall-back motion.

The LBPLI group also estimated the fall-back time for their simulation with Q = 5, a = 0.5, and \({\mathcal C} = 0.16 - 0.18\) [41]. In their simulation, the compactness of the NS was assumed to be smaller than that for a realistic NS, and thus, the formation of the tidal tail can be significantly enhanced. They reported that for a large fraction of the material ∼ 0.05 M_⊙, the fall-back time may be longer than 1 s. However, they followed the motion of the tidal tail only for a short time duration (16.3 ms; i.e., they did not show that the material really went far away, l ≳ 10⁴ km), and in addition, did not describe the detail of the method for estimating the fall-back time. (Note that a fall-back time longer than 1 s is realized only for an element, which reaches a distance from the BH of l ≳ 10⁴ km.) Hence, their conclusion should be confirmed by a longer-term simulation in the future. Their finding in the framework of numerical relativity that the disk can extend to a large distance ≫ 10 GM₀/c² for a high BH spin was qualitatively new.

The KT group performed simulations for M_NS = 1.35 M_⊙, Q = 2–5, a = 0.75 with several piecewise polytropic EOS [109]. They found that even for a binary with a NS of realistic compactness \({\mathcal C} \approx 0.145\), the disk can extend to ≳ 500 km (which is approximately the location of the outer boundary of the computational domain in their simulations). This conclusion agrees qualitatively with the previous results by CCCW and LBPLI. Thus, for the merger of a rapidly-spinning BH and a NS, it may be concluded that a widely-spread disk is formed, and the lifetime of the accretion disk will be fairly long ≳ 0.1 s. The KT groups also found that the density of the disk decreases with increase of Q (or the BH mass); for a high-mass BH, a widely spread but less dense disk is formed.

3.3.4 Effects of EOS

The dependence of the merger process on the NS EOS comes primarily from the fact that the NS radius depends sensitively on the EOS. For a NS with a large radius, tidal disruption (and subsequent disk formation) is more likely. This fact was clearly shown in the works by the KT group [107, 109], performed employing a variety of the piecewise polytropic EOS.

The EOS also determines the density profile of a NS. Even if the radius is the same for a given mass, the density profiles for two NS may be different if the hypothetical EOS is different. The KT group showed that a NS with a small adiabatic index for the core region, with which the density is concentrated in the central region, is less subject to tidal disruption than that with a larger adiabatic index (with relatively uniform density profile), even if the radius and mass are identical; e.g., for the piecewise polytropic EOS listed in Table 4, a NS with a smaller value of Γ₂ is less subject to tidal disruption. The reason for this is that the star with a high degree of central density concentration is less subject to tidal deformation, as reviewed in Section 1.2.

The CCCW group performed a simulation incorporating a tabulated finite-temperature EOS for the first time [57]. The advantage of this approach is that one can determine the temperature and composition, such as electron fraction in the disk formed after tidal disruption occurs. This will be useful for discussing the possibility that the remnant BH-disk system could be a central engine of an SGRB. To avoid taking into account the effects of neutrino emission, they assumed that the system is in β-equilibrium or that the electron fraction is unchanged in the fluid-moving frame. In the former and latter, they assumed that the system is in either of the following two limiting cases; the weak interaction time scale is either much shorter or much longer than the dynamic time scale, respectively. They performed simulations focusing on the case a = 0.5 and Q = 3. Irrespective of the EOS, they found that the remnant disk is neutron rich with Y _e ∼ 0.1−0.2 and the temperature is only moderately high (maximum is ∼ 10 MeV with the average ∼ 3 MeV) with the maximum density ∼ 10¹² g/cm³ and disk mass ∼ 0.1 M_⊙. Perhaps, because of the relatively low mass and density of the disk, the temperature is not as high as that found in a Newtonian simulation with detailed microphysics [95].

3.6.1 Zero BH spin

Figure 23 displays the typical gravitational waveforms for a = 0, which clearly reflect the features of the orbital evolution and subsequent merger processes (tidal disruption or not) as described in the following. In the early inspiral phase in which r ≫ R_NS and r ≫ GM_BH/c², two objects behave like point masses. In addition, general relativistic effects to the orbital motion are not extremely strong. For such a phase, the signal of gravitational waves is the chirp signal that can be well reproduced by the PN approximation for the two-body problem [25].

Open image in new window

Figure 23 shows that gravitational waveforms in the late inspiral phase before the onset of the merger (or tidal disruption) indeed agree with the result derived by the Taylor-T4 formula for the BH-NS binaries, as in the case of BH-BH binaries. This is natural because of the equivalence principle for general relativity [227]. (Note that in the first few wave cycles, the agreement is not very good. This is because the initial condition given for their simulation was not in an exact quasi-circular orbit.)

The waveforms may deviate from the prediction by the Taylor-T4 formula before the onset of the merger for a small value of Q or for a large NS radius. The reason for this is that the NS is tidally deformed by the BH, and, as a result, the pure Taylor-T4 formula, in which the tidal-deformation effect is not taken into account, is not a good formula for such a phase. The features of gravitational waveforms in the final inspiral phase are summarized later.

By contrast, for a sufficiently large value of Q or for a sufficiently compact NS, the tidal-deformation effect is negligible, and hence, the waveforms are quite similar to those for a BH-BH binary as mentioned above. For a small degree of tidal deformation and mass shedding, most of the NS material falls into the BH simultaneously (this case corresponds to type-II according to the definition of Figure 18). In such a case, a fundamental QNM of a BH is excited (see the waveform for the soft EOS in Figure 23), and the highest frequency of gravitational waves is determined by the QNM.

The degree of the QNM excitation depends strongly on the degree of tidal deformation and mass shedding. The primary reason for this is that a phase cancellation is concerned in the excitation; here, the phase cancellation is the amount that the gravitational waves emitted in a non-coherent manner (with different phases) interfere with each other to suppress the amplitude of gravitational waves [141, 187, 139]. With increasing degree of mass shedding, the phase cancellation effect plays an increasingly important role and the amplitude of the QNM-ringdown gravitational waves decreases. For the case in which a NS is tidally disrupted far outside the ISCO, this effect is significantly enhanced because the NS material does not simultaneously fall into the BH. Rather, a widely spread material, for which the density is much smaller than the typical NS’s density, falls into the BH from a wide region of the BH surface spending a relatively long time duration (this case corresponds to type-I according to the definition of Figure 18). Here, it is appropriate to point out why infall occurs from a wide region of the BH surface; the BH mass is small for the case in which mass shedding occurs for a = 0, and thus, the areal radius of the BH is smaller than or as small as the NS radius. All these effects are discouraging for efficiently exciting a QNM, and therefore, the amplitude of the QNM-ringdown gravitational waves is strongly suppressed for the case in which tidal disruption occurs (see the waveform for EOS 2H in Figure 23). For the case in which tidal disruption occurs, the highest frequency of gravitational waves is approximately determined by the orbital frequency at tidal disruption, not by the frequency of a QNM.

One important remark here is that this highest, characteristic frequency is not in general determined by the frequency at the onset of mass shedding. Even after the onset of mass shedding, the NS continues to be a self-gravitating star for a while and gravitational waves associated with an approximately-inspiral motion are emitted. After a substantial fraction of gravitational waves is emitted and thus, the orbital separation becomes sufficiently small, tidal disruption occurs. At such a moment, the amplitude of the gravitational waves damps steeply, and hence, the highest frequency of gravitational waves should be determined by the tidal-disruption event.

The qualitative features summarized above depend on the BH spin; for binaries composed of a BH of high spin, tidal disruption may occur for a high mass ratio, and hence, the infall process of the tidally-disrupted material into the BH may be qualitatively modified. This is well reflected in gravitational waveforms, as described in the next Section 3.6.2.

3.6.2 Nonzero BH spin

Gravitational waveforms are significantly modified in the presence of BH spin. Figure 24 plots gravitational waveforms for Q = 3 with the same stiff EOS (HB EOS) and with the same initial angular velocity (Gc⁻³Ωm₀ = 0.030) but with different values of the BH spin. This obviously shows that with increasing the BH spin, the lifetime of the binary system increases and hence the number of wave cycles increases. This is explained primarily by the spin-orbit coupling effect (see also Section 3.3), which brings a repulsive force into the BH-NS binary for the prograde spin of the BH. Due to the presence of this repulsive force, the orbital separation of the ISCO (the absolute value of the binding energy there) can be smaller (larger) than that for the non-spinning BH. This effect increases the lifetime of the binary, and furthermore, enhances the chance for tidal disruption of the NS because a circular orbit with a closer orbital separation is allowed. Second, the repulsive force reduces the orbital velocity for a given separation, because the centrifugal force may be weaker for a given separation to maintain a quasi-circular orbit. The decrease of the orbital velocity results in the decrease of the gravitational-wave luminosity, and this decelerates the orbital evolution as a result of gravitational radiation reaction, making the lifetime of the binary longer and increasing the number of cycles of gravitational waves. We note that all these effects are also clearly reflected in the gravitational-wave spectrum, as is shown in Section 3.7.

Open image in new window

Figure 24 shows that for a ≤ 0, a ringdown waveform associated with a QNM of the BH is clearly seen, whereas for a = 0.75, such a feature is absent. This reflects the fact that tidal disruption of the NS occurs for a = 0.75 far outside the ISCO, whereas it does not for a ≤ 0. For a = 0.5, tidal disruption occurs but a ringdown waveform associated with a QNM is seen. This is a new type of gravitational waveform. In this case, tidal disruption occurs near the ISCO and a large fraction of the NS material falls into the BH. The infall occurs approximately simultaneously and proceeds from a narrow region of the BH surface. This new type appears for the case in which the BH mass (or mass ratio Q) is large enough that the surface of the event horizon is wider than the extent of the infalling material, as explained in Section 3.3.2 (see Figure 18).

The inspiral waveform matches well to that of the Taylor-T4 formula for binaries composed of a spinning BH as well as for a non-spinning BH. Figure 24 also shows that matching is achieved as well as for a = 0, irrespective of the spin, except for the final phase just before the onset of the merger. As in the case of a = 0, the deviation of gravitational waveforms from the prediction by the Taylor-T4 formula is enhanced with increasing degree of tidal deformation, and with subsequent mass shedding and tidal disruption.

3.7.1 Zero BH-spin case

The final fate of the NS in BH-NS binaries is clearly reflected in the spectrum of gravitational waves. General qualitative features of the gravitational-wave spectrum for BH-NS binaries composed of non-spinning BH are summarized as follows. For the early stage of the inspiral phase, during which the orbital frequency is ≲ 1 kHz (R/12 km)^−3/2 and the PN point-particle approximation works well, the gravitational-wave spectrum is approximately reproduced by the Taylor-T4 formula. For this phase, the spectrum amplitude of \({h_{{\rm{eff}}}} \equiv f\tilde h(f)\) decreases as \({f^{- {n_i}}}\) where n _i ≈ 1/6 for f ≪ 1 kHz and the value of n _i increases with f for f ≲ 1 kHz (R/12 km)^−3/2. As the orbital separation decreases, both the non-linear effect of general relativity and the finite-sized effect of the NS come into play, and thus, the PN point-particle approximation breaks down. When tidal disruption (not mass shedding) occurs for a relatively large separation (e.g., for a NS of stiff EOS or for a small value of Q), the amplitude of the gravitational-wave spectra damps above a “cutoff” frequency f_cut. The cutoff frequency is equal to a frequency in the middle of the inspiral phase with f_cut ∼ 1−2 kHz for this case (it is lower than the frequency at the ISCO). The cutoff frequency depends on the binary parameters as well as on the EOS of the NS. A more strict definition of f_cut was given by the KT group and will be reviewed below.

By contrast, if tidal disruption does not occur or occurs at a close orbit near the ISCO, the spectrum amplitude for a high frequency region (f ≳ 1 kHz) is larger than that predicted by the Taylor-T4 formula (i.e., the value of ni decreases and can even become negative). In this case, an inspiral-like motion may continue even inside the ISCO for a dynamic time scale and gravitational waves with a high amplitude are emitted. (This property holds even in the presence of mass shedding.) This is reflected in the fact that \(f\tilde h(f)\) becomes a slowly varying function of f for 1 kHz ≲ f ≲ f_cut, where f_cut ∼2−3 kHz.

A steep damping of the spectra for f ≳ f_cut is universally observed, and for softer EOS with smaller NS radius, the frequency of f_cut is higher for a given mass of BH and NS. This cutoff frequency is determined by the frequency of gravitational waves emitted when the NS is tidally disrupted for the stiff EOS or by the frequency of a QNM of the remnant BH for the soft EOS. Therefore, the cutoff frequency provides potential information for a EOS through the tidal-disruption event of the NS, in particular for the stiff EOS.

Figure 25, plotted by the UIUC group [63], clearly illustrates the facts described above. The top panel (case E) plots the spectrum for Q = 1, in which the NS is tidally disrupted far outside the ISCO. In this case, the spectrum damps at f ∼ 1 kHz at which the tidal disruption occurs. The bottom panel (case D) plots the spectrum for Q = 5, in which the NS is not tidally disrupted. In this case, the steep damping of the spectrum at f ∼ 2 kHz is determined by the swallowing of the NS by the companion BH, and thus, the cutoff frequency is characterized by ringdown gravitational waves associated with the QNM of the remnant BH. Because the finite-sized effect of the NS is not very important in this case, the gravitational-wave spectrum is similar to that of the BH-BH binary merger with the same mass ratio (Q = 5; see the dashed curve). In the middle panel (case A), the cutoff frequency, at which the steep damping of h_eff sets in, is different from that for the BH-BH binary with the same mass ratio. This implies that tidal deformation and disruption play an important role in the merger process and in determining the gravitational waveform.

Open image in new window

As described above, the cutoff frequency at which the steep damping of the spectrum occurs will bring us the information for the degree of tidal deformation and where tidal disruption occurs in a close orbit just before the merger. The degree of tidal deformation and the frequency at which tidal disruption occurs depend on the EOS of the NS. This suggests that the cutoff frequency should have the information of the EOS. Motivated by this idea, the KT group performed a wide variety of simulations, changing the mass ratio, EOS, and BH spin, and systematically analyzed the resulting gravitational waveforms. Figure 26 plots the spectrum as a function of the frequency for Q = 2, M_NS = 1.35 M_⊙, and with a variety of EOS for a = 0. Irrespective of the EOS, the spectrum has the universal feature mentioned above. However, the cutoff frequency, at which the steep damping sets in, depends strongly on the EOS.

Open image in new window

The features of gravitational-wave spectra for a = 0 are schematically summarized in Figure 27. Here, three curves are plotted assuming that the masses of the BH and the NS are all the same with a = 0 and with a relatively small value of Q, but the NS EOS is different. The curves (i)-a, (i)-b, and (ii) schematically denote the gravitational-wave spectra for the stiff, moderately stiff, and soft EOS. For (i)-a and (i)-b, the damping of the spectrum is determined by tidal disruption. In this case, the spectrum is characterized simply by exponential damping for f ≳ f_cut. We refer to a spectrum of this type as type-I. For the case (ii), on the other hand, tidal disruption does not occur, and the cutoff frequency is determined by the QNM of the remnant BH. In this case, for a frequency slightly smaller than f_cut, the amplitude of the spectrum slightly increases with the frequency, that is a characteristic feature seen for the spectrum of BH-BH binaries (e.g., [36]). We refer to a spectrum of this type as type-II.

Open image in new window

Among these seven free parameters, they focus on f_cut because it depends most strongly on the compactness \({\mathcal C}\) and the NS EOS. Figure 28 plots f_cutm₀, obtained in this fitting procedure, as a function of \({\mathcal C}\) for a = 0. Also the typical QNM frequencies, f_QNM, of the remnant BH for Q = 2 and 3 are plotted by the two horizontal lines, which show that the values of f_cutm₀ for compact models (\({\mathcal C}\)) with Q = 3 agree approximately with f_qnm and indicates that f_cut for these models are irrelevant to tidal disruption. For Q = 3, f_cutm₀ depends on the EOS only for \(({\mathcal C} \lesssim 0.16)\). By contrast, f_cutm₀ for Q = 2 depends strongly on NS compactness, \({\mathcal C} \lesssim 0.16\), irrespective of M_NS for a wide range of \({\mathcal C}\).

Open image in new window

Figure 26 illustrates that f_cut is rather high, ≳ 2 kHz, for a variety of EOS, and thus, the dependence of f_cut on the EOS for a = 0 appears only for a high frequency. The reason for this is that for a = 0, tidal disruption can occur only for a small mass ratio (and thus for a small total mass) with a typical NS mass of 1.3−1.4 M_⊙; see Equation (8). The effective amplitude of gravitational waves at f = 2 kHz is ∼ 2 × 10⁻²² for a hypothetical distance to the source of 100 Mpc. The amplitude is smaller than the noise level of advanced gravitational-wave detectors, but such a signal will be detectable to next-generation detectors such as Einstein Telescope [91, 92].

3.7.2 Non-zero BH-spin case

The gravitational-wave spectrum is qualitatively and quantitatively modified by the BH spin. In the following, we focus only on the case in which the BH spin and orbital angular momentum vectors align, because gravitational waves for the misaligned case have not yet been studied in detail. Figure 29 shows the same relation as in Figure 26 but for Q = 3, M_NS = 1.35 M_⊙ and a = 0.75, 0.5, 0, and −0.5 with HB EOS (cf. Table 4; left) and for Q = 4, a = 0.75, and M_NS = 1.35 M_⊙ with various EOS (right). The left panel of Figure 29 shows that the spectrum shapes for a ≤ 0, a = 0.5, and a = 0.75 are qualitatively different; for a ≤ 0, the exponential damping above a cutoff frequency, which is determined by the fundamental QNM of the remnant BH, is seen. In this case, tidal disruption does not occur. This is the type-II spectrum according to the classification in Figure 27. On the other hand, for a = 0.75, the cutoff frequency (f_cut ∼ 1.5 kHz) is determined by the frequency at which tidal disruption occurs. This is the type-I spectrum according to the classification in Figure 27. The spectrum for a = 0.5 is neither type-I nor type-II. In this case, there are two typical frequencies. One is at f ∼ 2 kHz, above which the spectrum amplitude sinks, and the other is at f ∼ 3 kHz, above which the spectrum amplitude steeply damps. The first frequency is determined primarily by the frequency at which tidal disruption occurs, and the second one is the QNM frequency of the remnant BH. We call this new type of spectrum type-III (according to the definition of Figure 18). In the right panel of Figure 30, we summarize three types of gravitational-wave spectrum.

Open image in new window

Open image in new window

For the type-III spectrum, we refer to the first (lower) typical frequency as the cutoff frequency in the following. In this definition, with increasing BH spin, the cutoff frequency decreases and the amplitude of the gravitational-wave spectrum for f ≲ f_cut increases. These two effects are preferable for gravitational-wave detection by planned advanced laser-interferometric detectors, because their sensitivity is better for smaller frequencies around f = f_cut. These quantitative changes come again from the spin-orbit coupling effect (see also Sections 3.3 and 3.6), as explained in the following.

Due to the spin-orbit coupling effect, which brings a repulsive force into BH-NS binaries (for the prograde spin), the orbital velocity for a given separation is reduced, because the centrifugal force may be weaker for a given separation to maintain a quasi-circular orbit. Due to the decrease of the orbital velocity (a) the orbital angular velocity at a given separation is decreased and (b) the luminosity of gravitational waves is decreased. Effect (a) results in the decrease of the cutoff frequency at which tidal disruption occurs. Effect (b) decelerates the orbital evolution as a result of gravitational radiation reaction, resulting in a longer lifetime of the binary system and in increase in the number of the gravitational-wave cycle. This effect increases the amplitude of the gravitational-wave spectrum of f < f_cut for a > 0. These two effects are schematically described in the left panel of Figure 30.

For binaries composed of a high-spin BH, tidal disruption can occur outside the ISCO even for a high-mass BH for a variety of EOS. Thus, the dependence of f_cut on the EOS is clearly seen even for a high value of Q. The right panel of Figure 29 shows the spectrum for Q = 4, M_NS = 1.35 M_⊙, and a = 0.75 with four different EOS. For all the EOS, tidal disruption occurs, and the value of f_cut depends strongly on the EOS. For EOS 2H and H, the spectra are type-I, but for EOS HB and B (stiff EOS), they are type-III. Thus, for a high BH spin, a type-I or type-III spectrum is often seen even for a high value of Q.

With the increase of Q (for a canonical mass of a NS ∼ 1.4 M_⊙), the value of f_cut decreases (cf. Equation (7)), and the effective amplitude at f = f_cut increases as the total mass increases. These are also favorable properties for gravitational-wave detection. The right panel of Figure 29 indeed illustrates that f_cut is smaller than 2 kHz irrespective of the EOS, and also, that the effective amplitude at f ≲ 2 kHz is as large as the noise curve of the advanced detector for a hypothetical distance of D = 100 Mpc. For an even higher spin, say a ≳ 0.9, tidal disruption is likely to occur for a higher BH mass with Q ∼ 10. For such a case, the amplitude of gravitational waves at tidal disruption is likely to be high enough to be observable irrespective of EOS for D = 100 Mpc. This indicates that a BH-NS binary with a high BH spin will be a promising experimental field for constraining the EOS of high-density nuclear matter, when advanced gravitational-wave detectors are in operation.