3.6 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 [107Jump To The Next Citation Point, 109Jump To The Next Citation Point]. Waveforms of qualitatively similar features have been also derived by the UIUC [63Jump To The Next Citation Point] and CCCW groups [58Jump To The Next Citation Point].

3.6.1 Zero BH spin

Figure 23View Image displays the typical gravitational waveforms for a = 0, which clearly reflect the features of the orbital evolution and subsequent merger processes (tidal disruption or not) as described in the following. In the early inspiral phase in which r ≫ R NS and r ≫ GM ∕c2 BH, 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].

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Figure 23: Gravitational waveforms observed along the axis perpendicular to the orbital plane for Q = 3 and a = 0 with very stiff (2H), stiff (H), moderate (HB), and soft (B) EOS. The simulation was performed by the KT group. The solid and dashed curves denote the numerical results and results derived by the Taylor-T4 formula. D is the distance from the source and m = M + M 0 BH NS. The left and right axes show the normalized amplitude (2 cDh ∕Gm0) and physical amplitude for D = 100 Mpc, respectively. This figure is taken from [107Jump To The Next Citation Point].

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 [36Jump To The Next Citation Point, 30, 31, 5Jump To The Next Citation Point, 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 [36Jump To The Next Citation Point]

3 dX--= -64νX----F (X ), (124 ) dt 5Gc− 3m0
where F(X ) is a polynomial of X as ∑7 i∕2 1 + i=2aiX and ai 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
3∕2 dΘ-= --X-----. (125 ) dt Gc −3m0
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 23View Image 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 18View Image). In such a case, a fundamental QNM of a BH is excited (see the waveform for the soft EOS in Figure 23View Image), 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 18View Image). 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 23View Image). For the case in which tidal disruption occurs, the highest frequency of gravitational waves is approximately determined by the orbital frequency at tidal disruption, not by the frequency of a QNM.

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

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

3.6.2 Nonzero BH spin

Gravitational waveforms are significantly modified in the presence of BH spin. Figure 24View Image 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.

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Figure 24: The same as Figure 23View Image but for Q = 3 and a = 0.75 (top left), 0.5 (top right), 0 (bottom left), and –0.5 (bottom right) with HB EOS. This figure is taken from [109Jump To The Next Citation Point].

Figure 24View Image 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 18View Image).

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 24View Image 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.

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