By the year 2010, five groups — AEI, CCCW, KT, LBPLI, and UIUC groups — have succeeded in fully general-relativistic simulations for BH-NS binaries and provided a variety of the numerical results. The results of these groups agree qualitatively with each other and have clarified the basic picture for the merger and tidal disruption processes, although quantitatively, minor differences (e.g., on the remnant disk mass) have been reported. They have provided the criteria for tidal disruption, the final mass and spin of the BH left after the merger, the remnant disk mass, gravitational waveforms, and gravitational-wave spectrum. In the following sections, we review the methods of numerical relativity employed in these groups, the basic picture of the merger and tidal disruption processes clarified to date, and the resulting gravitational waveforms and spectra, separately.
Numerical method
Initial condition
As shown in Table 1, the studies of quasi-equilibrium and its sequences for BH-NS binaries have been performed by several groups. In the numerical simulations, quasi-equilibrium is employed as the initial condition (besides a minor modification of the quasi-equilibrium state, reviewed below). The five simulation groups, AEI, CCCW, KT, LBPLI, and UIUC, employ the quasi-equilibrium states derived by different groups as summarized in Table 2. Depending on the implementations on which each group relies, the type of the initial condition is different. Table 3 classifies the type of the initial data by the following six parameters:
-
(1)
The EOS of the NS
The notation of “Γ = 2” or “Γ = 2.75” means the single polytropic EOS with an adiabatic index of Γ = 2 or Γ = 2.75.
The notations of “PwPoly” and “SH” mean the piecewise polytropic EOS and the Shen’s EOS, respectively.
-
(2)
The spatial background metric \({{\tilde \gamma}_{ij}} = {f_{ij}}\) and the extrinsic curvature K
The notation of “CF” means the conformally-flat condition \({{\tilde \gamma}_{ij}}\), and that of “M” means the maximal slicing K = 0.
The notation of “KS” indicates the Kerr-Schild metric. In this case, the extrinsic curvature K is also set to that derived from the Kerr-Schild metric.
-
(3)
The state of the fluid flow in the NS
The notation of “Co” means a corotating NS, and that of “Ir” an irrotational NS.
-
(4)
The method to treat the BH, i.e., the moving puncture or the excision
The notations of “Pu” and “Ex” mean that the moving-puncture and the excision approaches are used in the numerical simulation, respectively. “ExPu” means that the initial condition is prepared in the excision approach, but the simulation is done in the moving-puncture formulation (see below).
-
(5)
Presence or absence of the BH spin a and misalignment angle θ between the spin axis and orbital angular momentum vector.
-
(6)
The mass ratio Q
Table 2 Quasi-equilibrium study groups by which the initial data of each simulation group is supplied.
Table 3 Summary of the initial data used in simulations performed so far. Note that LBPLI’s data includes magnetic fields.
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.
(1) At the beginning of their procedure, they prepare a quasi-equilibrium data, which has zero approaching velocity υ
r
= ȧ0r = 0 and orbital angular velocity Ω0. (2) The second step is to evolve the quasi-equilibrium initial data for at least one and a half orbits. Then one records the time derivative of the measured coordinate separation between the center of the compact objects, ḋ(t) and fits ḋ(t) to a function of the form
$$\dot d = {A_0} + {A_1}t + B\sin (\omega t + \phi),$$
(105)
where the parameters A0, A1, B, ω, and ϕ are all determined by the fitting. The A0 + A1t part denotes the smooth inspiral, while the B sin(ωt + ϕ) part denotes the unwanted oscillations due to the eccentricity of the orbit. For a nearly circular Newtonian orbit, the eccentricity e of the orbit can be written as
$$e - {B \over {\omega {d_0}}},$$
(106)
where d0 = d(t = 0). This implies that reducing the orbital eccentricity is equivalent to reducing B. (3) The final step is to add the corrections to the approaching velocity ȧ0 and the orbital angular velocity Ω using the parameters, which appear in Equation (105). Such corrections should be chosen so that the eccentricity-induced initial radial velocity and radial acceleration can be removed. Namely, the initial radial velocity is changed as
$$\delta {\dot a_0} = - {{B\sin \phi} \over {{d_0}}},$$
(107)
and the initial orbital angular velocity as
$$\delta {\Omega _0} = - {{B\omega \cos \phi} \over {2{d_0}{\Omega _0}}}.$$
(108)
Using the corrected approaching radial velocity and the orbital angular velocity for the initial step (1), they iterated the procedure and obtained a better initial data. They showed that going twice through the iterative method described above, the orbital eccentricity is reduced by about an order of magnitude.
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.
The basic equations of the GH formalism are similar to those employed in the PN approximation [25]. Using the harmonic or generalized harmonic gauge condition,
$${\nabla _\mu}{\nabla ^\mu}{x^\nu} = {H^\nu},$$
(109)
where Hν is the arbitrary function, Einstein’s equation is written in a hyperbolic form of the spacetime metric g
μν
. The main modification, analogous to the introduction of F
i
or \({F_i} - {{\tilde \gamma}_{ij,j}} = 0\) in the BSSN formalism, is treating H
ν
as independent functions. These may be either the functions themselves or determined by evolution equations. This is essential for the stable and longterm evolution of the system. The other modification is to introduce the first spatial derivative of the metric components as new independent variables D
iμν
= ∂
igμν
to make the GH formulation first-order [129].
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.
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.
Hydrodynamics
For the evolution of orbiting NS and a disk surrounding BHs formed after the merger, hydrodynamics (or magnetohydrodynamics) equations have to be solved. For a hydrodynamic simulation specifically, one has to solve the continuity equation for the rest-mass density ρ, the relativistic Euler and energy equations for the four velocity ¾ and internal energy ε (or h), i.e.,
$${\nabla _\mu}(\rho {u^\mu}) = 0,$$
(110)
$${\gamma _{i\nu}}{\nabla _\mu}{T^{\mu \nu}} = 0,$$
(111)
$${n_\nu}{\nabla _\mu}{T^{\mu \nu}} = 0,$$
(112)
where Tμν is the stress-energy tensor. For the perfect fluid, it is written as
$${T^{\mu \nu}} = \rho h{u^\mu}{u^\nu} + P{g^{\mu \nu}}.$$
(113)
for a magnetohydrodynamics simulation, we have to solve Maxwell’s equation as well. In ideal magnetohydrodynamics, only the induction equation for magnetic-field components should be solved [73].
If one needs to follow the evolution of the lepton number densities, the following additional continuity equations for the leptons (the electron number density and/or the total lepton number density) have to be solved
$${\nabla _\mu}(\rho {Y_a}{u^\mu}) = \rho {S_a},$$
(114)
where Y
a
denotes a lepton number fraction per baryon and S
a
is the corresponding production/annihilation rates of the lepton in the baryon rest frame, which is associated with neutrino emission and absorption (e.g., [175, 176, 172, 173, 170, 146, 183, 184, 185]).
The hydrodynamic equation is schematically written in the form
$${{\partial U} \over {\partial t}} + {{\partial {F^i}} \over {\partial {x^i}}} + {S_{\rm{U}}} = 0,$$
(115)
where U represents the set of the evolved variables and Fi are the associated transport fluxes. The third term of Equation (115) denotes the source term, which is usually the coupling term between the hydrodynamic variables and geometric quantities or the first derivative of the metric. The third term is evaluated in a straightforward manner, which usually does not cause numerical instabilities in a straightforward evaluation (as long as an appropriate time step is chosen). A careful treatment is required in handling the transport term [73]. The numerical scheme for the transport term employed in all the groups is similar. The CCCW, LBPLI, and UIUC groups employ the scheme of Harten, Lax, and van Leer (HLL) [89], the AEI group employs several schemes prepared in their Whisky library [13, 14, 11, 12], and the KT group employs the scheme of Kurganov and Tadmor [105]. In handling the transport term, one further needs to determine the method of the interpolation by which the values of hydrodynamic quantities at the cell surfaces are specified. All these groups currently employ the third-order piecewise parabolic interpolation.
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 = κρΓ (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).
A better choice is to employ a piecewise polytropic EOS. This is a phenomenologic ally parameterized EOS, which approximately reproduces cold nuclear-theory-based EOS at a high density (above the nuclear density) only with a small number of polytropic constants and indices [167, 168, 150] (see also [87, 128]), i.e.,
$$P(\rho) = {\kappa _i}{\rho ^{{\Gamma _i}}}{\rm{for}}{\rho _{i - 1}} \leq \rho < {\rho _i}(1 \leq i \leq n),$$
(117)
where n is the number of the pieces used to parameterize an EOS and ρ
i
denote boundary densities for which appropriate characteristic values are assigned. Here, ρ0 = 0 and ρ
n
→ ∞. κ
i
are the polytropic constants and Γ
i
the adiabatic indices for each piece.
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
.
Recently, it has been shown that the piecewise polytropic EOS composed of one piece in the crust region and three pieces in the core region approximately reproduces most of nuclear-theory-based EOS at a high density [167]. Here, three pieces in the core region are required to reproduce a high-mass NS for which inner and outer cores could have different stiffness due to the variation of the properties of the high-density nuclear matter (ρ ≳ 2 × 1014 g/cm3). Thus, if one focuses on a NS of relatively low mass, a smaller number of pieces is acceptable; see [168, 107] for the two-pieces case. Table 4 lists the parameters employed in [168, 107], and Figure 10 shows the relation between the mass and the radius of a spherical NS for these piecewise polytropic EOS. We note that using a single polytrope for the low-density EOS is justified to the extent that the radius and deformability of the NS as well as resulting gravitational waveforms in the merger phase are insensitive to the low-density EOS.
Table 4 The parameters and key ingredients for a piecewise polytropic EOS employed in [107]. Γ2 is the adiabatic index in the core region and p is the pressure at the fiducial density ρfidu = 1014.7 g/cm3, which determines the polytropic constant κ2 of the core region and ρ1: the critical rest-mass density separating the crust and core regions. Mmax is the maximum mass of a spherical NS for a given EOS. R135 and \({{\mathcal C}_{135}}\) are the circumferential radius and the compactness of the NS with MNS = 1.35 M⊙.
In the presence of shocks during the merger phase, the assumption of zero-temperature breaks down. Thus, in addition to the piecewise polytropic part, a correction term has to be added. An often-used minimum prescription is to simply add the following term to the pressure [199, 168, 107]
$${P_{{\rm{th}}}} = ({\Gamma _{{\rm{th}}}} - 1)\rho {\varepsilon _{{\rm{th}}}},$$
(118)
where Γth is an adiabatic index for the thermal part, and εth = ε − εcold with εcold being determined by the piecewise polytropic EOS. This type of the piecewise polytropic EOS is employed by the KT group for a wide variety of parameters [107, 109].
To self-consistently take into account thermal effects, which play an important role in the evolution of a disk formed after tidal disruption occurs, the best way is probably to employ a finite-temperature EOS, which is derived by a detailed model based on nuclear physics. This type of EOS is usually described in table form, e.g., as
$$P - P(\rho ,T,{Y_e}){\rm{and}}\varepsilon =\varepsilon (\rho ,T, Y_{e}),$$
(119)
where T and Y
e
are the temperature and the electron fraction [119, 188, 189]. Here, approximately speaking, ρ is determined by the continuity equation (110), and T by the internal energy, which is essentially determined by the energy equation (112) in general relativity. Y
e
is determined by the continuity equation for the electron fraction (114). To solve this equation correctly, we have to take into account neutrino emission/absorption, by which the electron fraction is varied. Multidimensional numerical-relativity simulations have reached a level of incorporating the effects of neutrino emission/absorption only quite recently [183, 185]. Rather, most multidimensional numerical-relativity simulations have been performed without solving neutrino transfer equations and the equation for Ye [146, 53, 54, 74, 149], but an a priori assumption for Ye as a function of other variables such as ρ and T is imposed. Along this line (under the assumption that Ye is unchanged in the fluid rest frame or Ye is determined by the β-equilibrium), the CCCW group performed the first simulation taking into account the finite-temperature effect [57]. Recently, Sekiguchi has developed a leakage scheme of neutrinos in general relativistic simulation [183, 184, 185] and solves the equation for Ye taking into account neutrino emission for the first time. This technique will be applied to the simulation of BH-NS binaries in the near future.
Adaptive mesh refinement
There are two important length scales in the problem of compact binary coalescence. One is the scale associated with the size of compact objects, which is ∼ GMBH/c2 for BH and RNS ∼ 5–8GMNS/c2 for NS. The other is the wavelength of gravitational waves, λ. For a binary in a circular orbit of angular velocity Ω, the wavelength for the dominant quadrupole mode (l = |m| = 2 mode) is
$$\lambda = {{\pi c} \over \Omega} \approx 105{{G{m_0}} \over {{c^2}}}{\left({{{G{c^{- 3}}\Omega {m_0}} \over {0.03}}} \right)^{- 1}},$$
(120)
where m0 = MBH + MNS. A longterm numerical-relativity simulation is typically performed with initial angular velocity Gc−3Ωm0 ≈ 0.02 for NS-NS binaries, and ∼ 0.03 for BH-NS binaries. Thus, λ is larger than Gm0/c2 by a factor of ≳ 100.
To accurately compute the evolution of an orbiting BH and NS resolving the structure of each object, the grid spacing should be smaller than Δ0 ∼ GMBH/20c2 and ∼ RNS/40, respectively. In addition, the outer boundary along each axis of the computation domain should be larger than the wavelength of gravitational waves to accurately take into account gravitational radiation reaction and to accurately extract gravitational waves in the wave zone. Then, if one adopts a uniform grid with grid spacing Δ0, the total grid number in one direction should be larger than
$${\begin{array}{*{20}c} {{\lambda \over {{\Delta _0}}} \approx \max \left[ {2100(1 + {Q^{- 1}})\left({{{G{c^{- 3}}\Omega {m_0}} \over {0.03}}} \right)^{- 1}} \right.}, \\ {\left. {700(1 + Q){{\left({{{{c^2}{R_{{\rm{NS}}}}/G{M_{{\rm{NS}}}}} \over 6}} \right)}^{- 1}}{{\left({{{G{c^{- 3}}\Omega {m_0}} \over {0.03}}} \right)}^{- 1}}} \right] .} \\ \end{array}}$$
(121)
In three spatial dimensions, the required grid number is more than (2λ/Δ0)3. Thus, more than 10003 in the grid number is necessary. For such a simulation, a quite high computational cost is required even in the present computational resources. It should also be pointed out that in such studies, a survey for a large parameter space composed of the mass and the spin of the BH, and the mass and EOS of the NS is necessary (see Section 3.2). For this purpose, it is very important to make every effort to save computational costs for each run.
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.
Current parameter space surveyed
There are many free parameters that specify a BH-NS binary; the mass and spin of the BH, and the mass of the NS. Furthermore, the EOS of the NS is still unknown, and thus, it should also be regarded as one of the free parameters. Here, the internal motion of the NS is believed to be close to an irrotational velocity field [24, 104] because the viscosity of the NS matter is too small to realize a corotational velocity field, and, in addition, the typical spin angular momentum of the NS (Prot =0.1–1 s) is slow enough that we can safely assume that the NS spin is zero (because an orbital period of ≲ 10 ms for m0 ≲ 10 M⊙ is much shorter than Prot). However, we have to accept a wide possible range for other parameters and EOS of NS. In particular, the mass and spin of a BH in a realistic binary system are totally unknown, and hence, a survey for a wide parameter space is required to fully understand the nature of BH-NS coalescence and to derive a complete catalog of possible gravitational waveforms.
In the first phase, all groups employed the Γ-law EOS in the form P = (Γ − 1)ρε with the special value of Γ = 2, for which the initial condition is prepared by using the polytropic EOS P = κρΓ. In this EOS, physical parameters are non-dimensional quantities such as mass ratio, Q = MBH/MNS, and compactness of the NS, \({\mathcal C}\), because κ can be freely chosen. Since 2009, several more plausible EOS have been employed by the KT and CCCW groups.
The early work of the KT group was done with the Γ-law EOS for a = 0 and for a wide range of Q and \(0.145 \leq {\mathcal C} \leq 0.178\); 1.5 ≤ Q ≤ 5 and \((Q,a,{\mathcal C})\). Since 2009, the KT group has employed a piecewise polytropic EOS [167, 150] with a wide variety of EOS parameters (see Table 4). Simulations have been systematically performed employing this EOS for a wide range of (\(0.12 \lesssim {\mathcal C} \lesssim 0.19\)) [107, 109]; 2 ≤ Q ≤ 5, −0.5 ≤ a ≤ 0.75, and \((Q,a,{\mathcal C})\). (Here, the negative value of the spin implies that the BH spin and orbital angular momentum vector are anti-parallel.)
The simulations of the UIUC group were performed employing the Γ-law EOS with Γ = 2. The UIUC group has chosen in total nine parameter sets for (\({\mathcal C} = 0.088\)) as follows; Q = 1, 2, 3, and 5, a = 0, −0.5, and 0.75, and \((Q,a,{\mathcal C})\) and 0.145. Simulations were performed for the relatively small value of NS compactness.
The CCCW group performed simulations employing two types of EOS; Γ-law EOS with Γ = 2 and 2.75, and Shen’s EOS, which is a tabulated EOS derived in a relativistic mean field theory [188, 189]. They focused on special parameter sets of (\({\mathcal C} \approx 0.145\)) as Q = 1 and 3, a = 0 and 0.5, and \((Q,a,{\mathcal C}) = (5,0.5,0.1)\) and 0.174. In their latest work, they focused on the case Q = 3 and a = 0.5, paying particular attention to the dependence of the merger process on the EOS and on the misalignment angle of the BH spin and orbital angular momentum axes.
The LBPLI group has performed one simulation to date, using the Γ-law EOS with Γ = 2, and with (\({\mathcal C} = 0.1 - 0.17\)) = (5, 0.5, 0.1) [41]. In their first work, the compactness was chosen to be small and not very realistic. In this simulation, magnetic fields in the ideal magnetohydrodynamics MHD approximation were incorporated, but they did not play an important role. The AEI group has performed simulations using Γ-law EOS with Γ = 2, and with Q = 5, a = 0, and \({\mathcal C}\) [154].
To summarize, the total number of simulations is still small, although a systematic survey is required to fully understand the complete picture of the coalescence of BH-NS binaries. In particular, many simulations were performed in a not very realistic setting using a simple Γ-law EOS and small values of \({\mathcal C} = 0.145\). As mentioned in Section 1, tidal disruption is more subject to the less compact NS, and hence, it should be in particular cautioned that a simulation with unphysically-small values of compactness may lead to an incorrect conclusion that tidal disruption and subsequent disk formation are easily achieved.
Nevertheless, the simulations performed so far have clarified a basic picture for the merger process of BH-NS binaries, the properties for the remnant, gravitational waveforms, and gravitational-wave spectrum. In the following, we summarize our current understanding of these topics based on work to date.
Merger process
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, MBH = 2.7 M⊙, a = 0, MNS = 1.35 M⊙, and RNS = 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.
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, MBH = 4.05 M⊙, a = 0, MNS = 1.35 M⊙, and RNS = 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.
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.
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.
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, Mns = 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.
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.
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.
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 MNS = 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.
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].
Properties of the remnant black hole and disk
During merger, NS material falls into its companion BH, and then, the mass and spin of the BH vary. Because a large fraction of the NS material falls into the BH for most of the numerical experiments performed so far and the total energy of gravitational waves emitted (EGW) is much smaller than the total mass energy of the system, the final BH mass becomes roughly MBH + MNS − Mdisk − EGW ≲ MBH + MNS = m0. (The disk mass Mdisk and EGW are less than 10% of the initial total mass m0.) Numerical results indeed show that the final BH mass is larger than 0.9m0.
The final BH spin depends sensitively on the mass ratio and initial BH spin. This can be understood by the following simple analysis. In Newtonian gravity, the total orbital angular momentum for two point masses in a circular orbit with an angular velocity Ω is
$${J_{{\rm{orb}}}} = {{{G^{2/3}}{M_{{\rm{BH}}}}{M_{{\rm{NS}}}}} \over {{{(\Omega {m_o})}^{1/3}}}}.$$
(122)
Thus, the non-dimensional spin parameter of the system is approximately written as
$${a_f} = {{c{G^{- 1}}{J_{{\rm{orb}}}} + M_{{\rm{BH}}}^2a} \over {m_0^2}} = {{{{(G{c^{- 3}}\Omega {m_0})}^{- 1/3}}Q + a{Q^2}} \over {{{(1 + Q)}^2}}},$$
(123)
where we assume that the BH spin aligns with the orbital angular momentum vector. If the mass and angular momentum of the remnant disk, and loss by gravitational waves may be neglected, a
f
will be equal to the spin of the remnant BH. At the onset of the merger or at tidal disruption, the angular velocity becomes Gc−3Ωm0 ∼ 0.05−0.1, and thus, (Gc−3Ωm0)−1/3 is in a narrow range of ∼ 2.1–2.7. This implies that a
f
is primarily determined by Q and a.
Equation (123) gives rather qualitative estimate for the spin of the remnant BH. Nevertheless, it still gives a good approximate value of the final spin with the choice of (Gc−3Ωm0)−1/3 = 3 as large as the remnant disk mass is small. With this choice, a
f
= 0.67, 0.56, 0.48, and 0.42 for Q = 2, 3, 4, and 5 and for a = 0. These values agree with the results derived by the CCCW [74], KT [194, 107, 108], and UIUC [63] groups within the error of Δa = 0.01−0.02.
For a large BH spin with a ≳ 0.5, a disk of a large mass (≳ 0.1 M⊙) is often formed even for Q ∼ 3 − 5 (see below). In such cases, Equation (123) overestimates the final BH spin. However, this equation still captures the qualitative tendency of the final spin; e.g., for small BH spin, the final spin is determined by the value of Q and the larger values of Q results in smaller final BH spin; for larger values of Q with a large BH spin a ≳ 0.5, the final BH spin is primarily determined by the initial BH spin.
The mass and characteristic density of the remnant disk surrounding the BH depend sensitively on the mass ratio (Q), the BH spin (a), and the EOS (or the compactness) of the NS. Figures 19–21 illustrate this fact. Figure 19 displays a result of the disk mass as a function of the NS compactness for Q = 2 (left) and Q = 3 (right) for various piecewise polytropic EOS and for various values of a, reported by the KT group [109]. This shows that the disk mass decreases steeply and systematically with the increase of the compactness irrespective of a and Q.
The left panel of Figure 20 plots together the results obtained by the UIUC, CCCW, and KT groups for Γ = 2 EOS with \({\mathcal C}\) and Q = 3 (the revised result by the KT group is plotted here; results in early work by the KT group do not agree with the result shown here [228, 194]; see below for the reason). This shows that the disk mass increases steeply with BH spin (a) for given values of \({\mathcal C} \sim 0.17\) and Q. The results by these three groups agree approximately with each other for a = 0. The CCCW group showed for the first time that the disk mass decreases with the increase of the inclination angle of the BH spin, and toward the limit to 90 degree, the disk mass approaches to that of a = 0. The right panel of Figure 20 plots the results by the KT group for different compactness (using EOS HB with MNS = 1.35 M⊙; see Table 4). This shows again that the disk mass increases with the increase of the BH spin, and also that for high BH spin (e.g., a = 0.75), the disk mass is larger than 0.1 M⊙ even for Q = 5 with \(({\mathcal C} \approx 0.145)\).
Figure 21 shows disk mass as a function of NS compactness for a = 0.75 and a = 0.5 as performed by the KT group [109]. A steep decrease in disk mass with increasing compactness is found irrespective of the values of a and Q. Simulations for BH-NS binaries with a spinning BH for particular values of the compactness or mass ratio were also performed by the CCCW and UIUC groups for a = 0.5 and 0.75, respectively. The results of the CCCW group for a = 0.5 and Q = 3 with various EOS agree with those of Figure 21 within ∼ 10–20%. We note that the disk mass may be different in the different EOS even with the same compactness of the NS, because the density profile of the NS, and the resulting susceptibility to the BH tidal force is different. Thus, this disagreement is reasonable. Also, the results of the UIUC group for a = 0.75, Q = 3, and \({\mathcal C} = 0.145\) = 0.145 with the Γ-law EOS (the disk mass ∼ 0.15 MB) agree with the relation expected from Figure 21 within ∼ 20% error.
To clarify the dependence of the disk mass on relevant parameters, we consider three types of comparisons. First, we consider the case in which a, \({\mathcal C}\), and the EOS are fixed, but Q is varied. In this case, the disk mass monotonically decreases with the increase of Q for many cases. For example, for a = 0 and \({\mathcal C} = 0.145\) with Γ = 2 EOS, the disk mass is larger (smaller) than 0.01 MB for Q ≲ 4 (Q ≳ 4) [63, 74, 154]. However, we should point out the exception to this rule, because for the case in which a large-mass disk is formed, this rule may not hold. For example, the comparison between the left and right panels of Figure 19 shows that for relatively small compactness \({\mathcal C} \lesssim 0.16\), a large-mass disk is formed for high BH spins and the disk mass depends only weakly on the value of Q for given values of a and \({\mathcal C}\).
Second, we consider the case in which Q and \({\mathcal C}\) are fixed, but a is varied. The UIUC group compared the results for a = −0.5, 0, and 0.75 for Q = 3 and \({\mathcal C} = 0.145\) with the Γ-law EOS (Γ = 2), and the resulting disk mass at ∼ 10 ms after the onset of the merger is 0.008, 0.039, and 0.15 MB for a = −0.5, 0, and 0.75, respectively [63]. The CCCW group performed a similar study for a = 0, 0.5, and 0.9 for Q = 3 and \({\mathcal C} = 0.144\) with the Γ-law EOS (Γ = 2), and found that the disk mass at 10 ms after the onset of the merger is 0.034, 0.126, and 0.360 Mb for a = 0, 0.5, and 0.9, respectively [74]. Both groups found the systematic steep increase of the disk mass with the increase of the BH spin (cf. the left panel of Figure 20). This was also reconfirmed by the KT group [109] (see the right panel of Figure 19 and Figure 21).
Third, we consider the case in which a and Q are fixed, but the compactness, C, is varied. Systematic work was recently performed by the KT group [107, 108, 109], employing a piecewise polytropic EOS. Figures 19 and 21 show that the disk mass decreases monotonically with the increase of \({\mathcal C}\) for given values of a and Q. For producing a disk of mass larger than 0.01 M⊙ for \(a = 0,\,{\mathcal C}\) should be smaller than ∼ 0.18 for Q = 2 and ∼ 0.16 for Q = 3 according to their results. For a = 0.75, the condition is significantly relaxed.
Finally, the KT group [107, 108] found that the disk mass depends not only on the compactness but weakly on the density profile. For the NS with a more centrally concentrated density profile, the disk mass is smaller. The reason for this is that if the degree of the central mass concentration is smaller, tidal deformation is enhanced and it encourages earlier tidal disruption after the onset of mass shedding.
It is interesting to note that for a = 0.75 and MNS ∼ 1.4 M⊙, the mass of a disk surrounding a BH can be larger than 0.1 M⊙ for Q ≲ 4 with \({\mathcal C} = 0.18\) and for Q ≲ 5 with \({\mathcal C} = 0.17\). For Q = 2, the disk mass will be ≳ 0.1 M⊙ for any realistic NS with \({\mathcal C} \lesssim 0.20\). For a BH of higher spin, the disk mass will be even larger (cf. the left panel of Figure 20). In addition, the disk mass does not decrease steeply with the increase of Q for such a high spin. In particular, for a small value of \({\mathcal C}\), the disk mass depends very weakly on Q. All these results indicate that the disk mass is likely to be large even for a higher value of Q(≥ 5) with a high BH spin (a ≥ 0.75). The maximum density of the disk increases monotonically with the disk mass. For a disk mass larger than 0.1 M⊙, the maximum density is larger than 1012 g/cm3 for Q ≲ 4 and ∼ 1011 g/cm3 for Q = 5 [109]. Hence, a high-mass disk with a relatively small value of Q ≲ 4 is likely to be universally opaque against thermal neutrinos for the typical geometrical thickness of the disk; a neutrino-dominated accretion disk is the outcome and this is favorable for copious neutrino emission. This leads to the conclusion that the coalescence of BH-NS binaries with a high-spin BH with a ≳ 0.75 and Q ≲ 4 is a promising progenitor for forming a BH plus a massive disk system; that is the candidate for a central-engine of SGRB.
Before closing this subsection, we should note that different groups have reported different quantitative results for the disk mass in their earlier work, which has been improved upon. For example, the earlier work of the KT group presented a small disk mass [228, 194]. This is mainly due to an unsuitable choice of computational domain and partly due to a spurious numerical effect associated with insufficient resolution and an unsuitable choice of AMR grid. The first work of the UIUC group also underestimated the disk mass [62]. This is due to an unsuitable prescription for handling the atmosphere. However, these have been subsequently fixed.
Generally speaking, the quantitative disagreement is due to the numerics. First, the fluid elements in the disk have to acquire a sufficiently large specific angular momentum which is larger than that at the ISCO of the remnant BH. The material that forms a disk obtains angular momentum by a hydrodynamic angular momentum transport process from the inner part of the material. This implies that such a transport process has to be accurately computed in a numerical simulation. However, it is well known that this is one of the challenging tasks in computational astrophysics. Second, to avoid spurious loss and transport of the angular momentum, a high-resolution computation is required. However, the disk material is located in a relatively distant orbit around the central BH. In the AMR scheme, which is employed in all the groups, the resolution in this region is usually poorer than that in the central region. This might induce a spurious loss of angular momentum and resulting decrease of disk mass, even by a factor of ∼ 2. However, these issues are being resolved with the improvement of computational resources, the efficiency of the numerical code, and the skill for computation with the AMR algorithm. The left panel of Figure 20 shows as much.
Criteria for tidal disruption
As mentioned in Section 1, tidal disruption occurs after the onset of mass shedding and after a substantial fraction of the NS material is removed from the inner cusp by the BH tidal field. It is important to emphasize again that the condition of tidal disruption is in general different from that of mass shedding for BH-NS binaries and that tidal disruption could occur in a more restricted condition than that for mass shedding, as mentioned in Sections 1 and 2. To determine the condition of tidal disruption, a dynamic simulation (not a study of quasi-equilibrium or equilibrium) is necessary.
It is not easy to strictly determine the condition of tidal disruption, because its concept is not as clear as that of mass shedding. One way to determine the condition is to use the property of gravitational waveforms in the merger phase; if the “cutoff” frequency is smaller than the frequency of a quasi-normal mode (QNM) of the remnant BH, we may conclude that tidal disruption occurs (see Section 3.6 for details of the cutoff frequency). Another way is to use the property of the merger remnant; we may recognize that tidal disruption occurs if the mass of a remnant disk surrounding a BH is substantial, say larger than 1% of the total rest mass at t ∼ 10 ms after the onset of mass shedding. We employ both criteria here to determine whether tidal disruption occurs or not.
First, we summarize the criterion for the case in which the BH spin is zero (a = 0). In this case, the condition of tidal disruption depends strongly on the mass ratio and compactness of the NS. According to the criterion associated with a gravitational-wave spectrum (see Figure 28, as an example), the condition of tidal disruption is \({\mathcal C} \lesssim 0.19\) for Q = 2 and \({\mathcal C} \lesssim 0.16\) for Q = 3.
Figure 19, which displays the result of the disk mass for a = 0 with several piecewise polytropic EOS, also illustrates that the condition of tidal disruption depends strongly on the values of Q and \({\mathcal C}\): the approximate condition for a = 0 is \({\mathcal C} \lesssim 0.18 - 0.19\) for Q = 2 and \({\mathcal C} \lesssim 0.16\) for Q = 3. We note that for Q = 3, this conclusion is consistent with those of the CCCW [57] and UIUC [63] groups. These criteria agree approximately with those derived from the gravitational-wave spectrum. The AEI group studied the case of Q = 5 with Γ-law EOS (Γ = 2) and showed that the condition is \({\mathcal C} \lesssim 0.13 - 0.14\) for Q = 5 [154].
For a spinning BH, in particular for the case in which the spin vector aligns with the orbital angular momentum vector, the condition for tidal disruption is highly relaxed. Here, we focus only on the case in which the BH spin vector is aligned with the orbital angular momentum vector. From the disk mass as a function of compactness shown in Figure 21 derived by the KT group, we can read the criteria as follows. For Q = 2, tidal disruption occurs irrespective of the value of \({\mathcal C}(\lesssim 0.2)\) for a ≥ 0.5. However, for a counter-rotating spin (a = −0.5), the criterion for tidal disruption is significantly restricted (\(({\mathcal C} \lesssim {(}0.16{)})\)). For a = 0.75, tidal disruption occurs irrespective of the value of \({\mathcal C}(\lesssim 0.2)\) as far as Q ≲ 4.
Finally, we compare the conditions for tidal disruption and mass shedding for a = 0. Figure 22 plots threshold curves of tidal disruption, mass shedding in general relativity, and mass shedding in a tidal approximation in the plane of (\({\mathcal C}\), Q). If the value of Q or \({\mathcal C}\) is smaller than that of the threshold curves, tidal disruption or mass shedding occurs. The points with error bar approximately denote the numerical results for the criterion of tidal disruption, based on the results by the KT group for Q = 2, by the CCCW, KT, and UIUC groups for Q = 3, and by the AEI group for Q = 5. The solid and dashed curves denote the critical curves for the onset of mass shedding for Γ = 2 EOS in general relativity [209, 210], and for the incompressible fluid in a tidal approximation (see Equation (12)), respectively. This shows that, for a realistically compact NS, \(0.13 \lesssim {\mathcal C} \lesssim 0.21\), the condition for tidal disruption is more restricted than that for mass shedding. The primary reason for this is that for such large compactness with a = 0, tidal disruption occurs only for a small value of Q. For such a system, the time scale for gravitational radiation reaction is as short as the orbital period in close orbits. This indicates that at the onset of mass shedding, the radial approaching velocity induced by gravitational radiation reaction is high enough to significantly decrease (increase) the orbital radius (angular frequency) during the subsequent mass-shedding phase up to final tidal disruption. If the fraction of this decrease in the orbital radius is large enough to enforce the orbit inside the ISCO, tidal disruption is prohibited. This mechanism makes the condition of tidal disruption for BH-NS binaries more restricted than that of mass shedding.
On the other hand, if the value of \({\mathcal C}\) is small, tidal disruption may occur even for a large value of Q. With increase of Q, the ratio of the time scale of gravitational radiation reaction to the orbital period at the ISCO increases (e.g., Equation (2)). For such a high-Q case, the effect of orbital decrease by gravitational radiation reaction after the onset of mass shedding becomes relatively minor, and therefore, the critical curves of tidal disruption and mass shedding approach each other.
For a BH-NS binary with high BH spin, mass shedding may occur even for high mass ratio (say Q = 10 for a ≳ 0.9, e.g., Equation (12)). For such a case, the conditions for tidal disruption and mass shedding may approximately agree with each other. This point should be clarified through future study. If this is the case, the quasi-equilibrium study plays an important role in determining the condition of mass shedding, because this also gives the (approximate) condition of tidal disruption.
Gravitational waveforms In this section, the features of gravitational waveforms emitted by BH-NS binaries, found from numerical simulations to date, are summarized, showing the numerical results by the KT group [107, 109]. Waveforms of qualitatively similar features have been also derived by the UIUC [63] and CCCW groups [58].
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 ≫ RNS and r ≫ GMBH/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].
For a close orbit in which the finite-size effect is still negligible but general relativistic gravity between two objects plays a role, it is known that the simple PN analysis fails to provide a precise waveform. Comparisons of the waveforms derived through PN analysis and through numerical computation for BH-BH binaries [36, 30, 31, 5, 179] propose that a better waveform is phenomenologically derived using the Taylor-T4 formula. This method requires a special summation method of PN high-order terms in the equations of motion, which include gravitational radiation reaction effects in an adiabatic approximation. First, one needs to calculate the evolution of the orbital angular velocity Ω(t) through X(t) = [Gc−3Ω(t)m0]2/3 up to 3.5PN order by solving the following ordinary differential equations [36]
$${{dX} \over {dt}} = {{64\nu {X^3}} \over {5G{c^{- 3}}{m_0}}}F(X),$$
(124)
where F(X) is a polynomial of X as \(1 + \Sigma _{i = 2}^7{a_i}{X^{i/2}}\) and a
i
denote coefficients (functions of mass and spin of compact objects). ν is the ratio of the reduced mass to the total mass m0, ν = Q/(1 + Q)2. For a solution of X(t), then, the orbital phase Θ(t) is derived by integrating the following equation
$${{d\Theta} \over {dt}} = {{{X^{3/2}}} \over {G{c^{- 3}}{m_0}}}.$$
(125)
After X(t) and Θ(t) are obtained, the complex gravitational-wave amplitude h22 of (l, m) = (2, 2) mode is calculated up to 3PN order using the formula of [101].
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.
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.
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.
The Fourier spectrum of gravitational waves
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 fcut. The cutoff frequency is equal to a frequency in the middle of the inspiral phase with fcut ∼ 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 fcut 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 ≲ fcut, where fcut ∼2−3 kHz.
A steep damping of the spectra for f ≳ fcut is universally observed, and for softer EOS with smaller NS radius, the frequency of fcut 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 heff 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.
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, MNS = 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.
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 ≳ fcut. 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 fcut, 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.
To quantitatively analyze the cutoff frequency and to strictly study its dependence on the EOS, the KT group [194, 107, 109] fits all the spectra by a function with seven free parameters
$$\begin{array}{*{20}c} {{{\tilde h}_{{\rm{fit}}}}(f) = {{\tilde h}_{3{\rm{PN}}}}(f){e^{- {{(f/{f_{{\rm{ins}}}})}^{{\sigma _{{\rm{ins}}}}}}}}} \\ {+ {{A{m_0}} \over {Df}}{e^{- {{(f/{f_{cut}})}^{{\sigma _{{\rm{cut}}}}}}}}[1 - {e^{- {{(f/{f_{ins2}})}^{{\sigma _{{\rm{ins}}2}}}}}}],} \\ \end{array}$$
(126)
where \({{\tilde h}_{3{\rm{PN}}}}(f)\) is the Fourier spectrum calculated by the Taylor-T4 formula and f
ins
, fins2, fcut, σins, σins2, σcut, and A are free parameters. The first and second terms on the right-hand side of Equation (126) denote spectrum models for the inspiral and merger phases, respectively. These free parameters are determined by searching the minimum for a weighted norm defined by
$$\sum\limits_i {{{\left\{{[{f_i}\tilde h({f_i}) - {f_i}{{\tilde h}_{{\rm{fit}}}}({f_i})]f_i^{1/3}} \right\}}^2},}$$
(127)
where i denotes the data point for the spectrum.
Among these seven free parameters, they focus on fcut because it depends most strongly on the compactness \({\mathcal C}\) and the NS EOS. Figure 28 plots fcutm0, obtained in this fitting procedure, as a function of \({\mathcal C}\) for a = 0. Also the typical QNM frequencies, fQNM, of the remnant BH for Q = 2 and 3 are plotted by the two horizontal lines, which show that the values of fcutm0 for compact models (\({\mathcal C}\)) with Q = 3 agree approximately with fqnm and indicates that fcut for these models are irrelevant to tidal disruption. For Q = 3, fcutm0 depends on the EOS only for \(({\mathcal C} \lesssim 0.16)\). By contrast, fcutm0 for Q = 2 depends strongly on NS compactness, \({\mathcal C} \lesssim 0.16\), irrespective of MNS for a wide range of \({\mathcal C}\).
An interesting finding in [107] is that the following relation approximately holds for the identical value of Q,
$$\ln (G{c^{- 3}}{f_{{\rm{cut}}}}{m_0}) = (3.87 \pm 0.12)\ln C + (4.03 \pm 0.22).$$
(128)
Namely, fcutm0 is approximately proportional to \({{\mathcal C}^{3.9}}\). This is a note-worthy point because the power of \({\mathcal C}\) is much larger than a well-known factor 1.5, which is expected from the relation for the mass-shedding limit presented in Section 1 (cf. Equation (7)). (For binaries composed of spinning BH, this power is smaller than 3.9, but it is still much larger than 1.5 [109].) This difference indicates that the cutoff frequency is not determined simply by mass shedding. Qualitatively, this increase in the power is natural because the duration of a NS for the survival against tidal disruption after the onset of mass shedding is in general longer for a more compact NS due to a stronger central condensation of the density profile.
Figure 26 illustrates that fcut is rather high, ≳ 2 kHz, for a variety of EOS, and thus, the dependence of fcut 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].
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, MNS = 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 MNS = 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 (fcut ∼ 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.
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 ≲ fcut 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 = fcut. 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 < fcut for a > 0. These two effects are schematically described in the left panel of Figure 30.
A more quantitative explanation follows. From the relation of the luminosity of gravitational waves, the power spectrum of gravitational waves is written as
$${{dE} \over {df}}\propto{[f\tilde h(f)]^2}.$$
(129)
In the 1.5 PN approximation, dE/df in the inspiral phase is written as
$${{dE} \over {df}} = {{{c^5}} \over {3G{{(\pi f)}^2}}}{Q \over {{{(1 + Q)}^2}}}{(G{c^{- 3}}\pi {m_0}f)^{5/3}}\left[ {1 + (G{c^{- 3}}\pi {m_0}f){\bf{\hat S}} \cdot {\bf{\hat L}}{{20{Q^2} + 15Q} \over {3{{(1 + Q)}^2}}}} \right],$$
(130)
where we write only the terms associated with the lowest-order spin-orbit coupling term assuming that the NS spin is zero, and omit other terms. Ŝ and \({{\bf{\hat L}}}\) are unit vectors of the BH spin and orbital angular momentum, respectively. Thus, for a given frequency f, |dE/df| is larger in the presence of the prograde spin (\(({\bf{\hat S}} \cdot {\bf{\hat L}} > 0)\)) than for a = 0. The binary evolves due to gravitational-wave emission, and hence, the binary with \({\bf{\hat S}} \cdot {\bf{\hat L}} > 0\) has to emit more gravitational waves than for a = 0 to increase the frequency (to decrease the orbital separation). This agrees with the explanation given above. In addition, Equation (129) shows that dE/df is proportional to the square of the effective amplitude \((f\tilde h(f))\). Therefore, the effective amplitude for a given frequency with Ŝ· > 0 should be larger than that for a = 0. This agrees completely with the numerical results.
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 fcut 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, MNS = 1.35 M⊙, and a = 0.75 with four different EOS. For all the EOS, tidal disruption occurs, and the value of fcut 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 fcut decreases (cf. Equation (7)), and the effective amplitude at f = fcut increases as the total mass increases. These are also favorable properties for gravitational-wave detection. The right panel of Figure 29 indeed illustrates that fcut 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.
Summary and issues for the near future
The inspiral and merger of BH-NS binaries are among promising sources for kilometer size laser-interferometric gravitational-waves detectors. The merger remnant is also a possible candidate for the progenitor of the central engine of SGRB. BH-NS binaries are also invaluable experimental fields for studying high-density nuclear matter through astronomical observations. To derive accurate gravitational waveforms in the late inspiral and merger phases, and to explore the compact-binary-merger hypothesis for the central engine of SGRB, the numerical simulation in full general relativity, taking into account realistic physics, is the unique approach. We review the progress and current status of numerical simulations for BH-NS binaries, and summarize the current understanding obtained from numerical results. The following is a summary as of June 2011:
-
In the final phase of BH-NS binaries, the NS is either tidally disrupted or swallowed by the companion BH. For a typical compactness of the NS, \({\mathcal C} = 0.14 - 0.20\), with zero BH spin, tidal disruption occurs outside the ISCO only for a small mass ratio of Q ≲ 3. For the case in which tidal disruption occurs, the final remnant is a BH surrounded by a disk of relatively small mass (say ≲ 0.1 M⊙).
-
The effects associated with BH spin enhance the possibilities for tidal disruption and for disk formation. Even for a moderately large spin a = 0.5, the criterion of tidal disruption for the mass ratio is relaxed to Q ≲ 5, for, e.g., \({\mathcal C} = 0.17\). In addition the effects of BH spin increase the mass of a remnant disk surrounding a BH to ∼ 0.1−0.3 M⊙ for a typical NS of mass MNS = 1.35 M⊙ with a = 0.75 and Q ∼ 5 [109]. For a favorable condition, such as \({\mathcal C} = 0.145\) and a = 0.9, the disk mass may be ≳ 0.5 M⊙ [74].
-
For a binary composed of a high-spin BH with high mass ratio, the disk, if it is formed, has a spread structure. However, the simulations so far (in which detailed microphysical effects such as neutrino wind are not taken into account) have not shown any evidence that a fraction of NS material escapes from the system.
-
The final merger process is well reflected in gravitational waveforms. Up to now, it has been found that there are at least three types of gravitational waveform. (i) For the case in which tidal disruption occurs far outside the ISCO, the amplitude of gravitational waves steeply damps in the middle of the late inspiral phase, and ringdown gravitational waves associated with a QNM of the remnant BH are not seen clearly. This type is referred to as type-I. For the case in which tidal disruption does not occur outside the ISCO, ringdown gravitational waves associated with a QNM of the remnant BH are clearly seen in the final phase of the gravitational-wave signal. This type is referred to as type-II. For the case in which tidal disruption occurs near the ISCO, there are two possibilities. One is that the amplitude of the gravitational waves steeply damps and the ringdown signal of a QNM is not seen clearly. This is the case that the mass ratio Q (and thus BH mass) is small, and the resulting type of the gravitational waveform is type-I. For a large value of Q (for a high BH mass), the disrupted material falls into the BH from a narrow region of the BH surface. In such cases, both the feature of steep damping associated with tidal disruption and ringdown gravitational waves associated with a QNM are seen. This type is referred to as type-III.
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Reflecting that there are three types of gravitational waveforms, the gravitational-wave spectrum is also classified into three types. For type-I and type-III, the gravitational-wave spectrum is characterized by the cutoff frequency associated with tidal disruption. For a given set of masses of BH and NS, and BH spin, the cutoff frequency is determined by the EOS of the NS. Thus, if the cutoff frequencies are determined for a detected signal of gravitational waves, the EOS of the NS will be constrained. The cutoff frequency is higher than ∼ 1 kHz (e.g., Equation (8)). The frequency is lower for a NS of larger radii or for a rapidly spinning BH with a large BH mass. In particular, for a binary composed of a high-spin BH, the effective amplitude at f = fcut is enhanced by the spin-orbit coupling effect. This effect is favorable for detecting a gravitational-wave signal at the cutoff frequency by advanced detectors.
There are several issues to be solved for the near future. First, more realistic modeling of NS is required because numerical studies have been performed with quite simple EOS and microphysics up till now. For more realistic modeling of BH-NS binaries (in particular for modeling formation and evolution processes of a disk surrounding a BH), more physical EOS should be taken into account; we have to take into account finite-temperature EOS, neutrino process, and magnetic fields (accurately evolving magnetic field configurations). Second, there is still a wide range of the parameter space that has not been studied. In particular, binaries of high BH spin (a > 0.9) have not been studied yet. For the case in which BH spin is close to unity, the NS may be tidally disrupted even for a high mass ratio Q ∼ 20; cf. Equation (12). This possibility has not been explored yet. For such a high-mass binary, tidal disruption occurs at a relatively-low gravitational-wave frequency ∼ 1 kHz. This is favorable for observing the tidal-disruption event by gravitational-wave detectors, and thus, this deserves intense study. Recent work by Liu et al. [130] indicated it feasible to perform a simulation with a high spin a ∼ 0.99 using a simple prescription (see also [133]). A simulation with such a high spin will be done in a few years. Third, only one study has been done for the merger process of the binaries in which the BH spin and orbital angular momentum vectors misalign. In particular, any study of gravitational waveforms has not been done for this case. This is also an issue to be explored. Finally, it is necessary to optimize simulation codes to efficiently and accurately perform a large number of longterm simulations for a wide range of parameter space. This is required for preparing template sets of gravitational waves that are used for gravitational-wave data analysis. Work along this line has recently begun in 2010 [153, 110], and in the next several years, it will be encouraged because the preparation of theoretical templates is an urgent task for advanced gravitational-wave detectors.