3.7 The Fourier spectrum of gravitational waves

3.7.1 Zero BH-spin case

The final fate of the NS in BH-NS binaries is clearly reflected in the spectrum of gravitational waves. General qualitative features of the gravitational-wave spectrum for BH-NS binaries composed of non-spinning BH are summarized as follows. For the early stage of the inspiral phase, during which the orbital frequency is ≲ 1 kHz (R∕12 km )− 3∕2 and the PN point-particle approximation works well, the gravitational-wave spectrum is approximately reproduced by the Taylor-T4 formula. For this phase, the spectrum amplitude of &tidle; heff ≡ f h(f) decreases as −ni f where ni ≈ 1∕6 for f ≪ 1 kHz and the value of ni increases with f for f ≲ 1 kHz − 3∕2 (R∕12 km ). 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 f cut was given by the KT group and will be reviewed below.

By contrast, if tidal disruption does not occur or occurs at a close orbit near the ISCO, the spectrum amplitude for a high frequency region (f ≳ 1 kHz) is larger than that predicted by the Taylor-T4 formula (i.e., the value of ni decreases and can even become negative). In this case, an inspiral-like motion may continue even inside the ISCO for a dynamic time scale and gravitational waves with a high amplitude are emitted. (This property holds even in the presence of mass shedding.) This is reflected in the fact that f&tidle;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 f cut is higher for a given mass of BH and NS. This cutoff frequency is determined by the frequency of gravitational waves emitted when the NS is tidally disrupted for the stiff EOS or by the frequency of a QNM of the remnant BH for the soft EOS. Therefore, the cutoff frequency provides potential information for a EOS through the tidal-disruption event of the NS, in particular for the stiff EOS.

Figure 25View Image, plotted by the UIUC group [63Jump To The Next Citation Point], 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.

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Figure 25: The gravitational-wave spectrum for Q = 1 (Case E), 3 (Case A), and 5 (Case D) with a = 0 and Γ-law EOS with Γ = 2. This simulation was performed by the UIUC group. The solid curve shows the spectrum of a 2.5PN and numerical waveforms, while the dotted curve shows the contribution from the numerical waveform only. The dashed curve is the analytic fit derived in [5] from analysis of BH-BH binaries composed of a non-spinning BH with the same values of Q as the BH-NS. The heavy solid curve is the effective strain of Advanced LIGO. To set physical units, a rest mass is assumed to be MB = 1.4M ⊙ (R ≈ 13 km) for the NS and a source distance of D = 100 Mpc. This figure is taken from [63].

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 26View Image plots the spectrum as a function of the frequency for Q = 2, MNS = 1.35M ⊙, 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.

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Figure 26: Spectra of gravitational waves from BH-NS binaries for Q = 2, a = 0, and MNS = 1.35M ⊙ with various EOS. The bottom axis denotes the normalized dimensionless frequency f m (= Gf m ∕c3) 0 0 and the left axis the normalized amplitude c2f &tidle;h(f)D ∕Gm 0. The top axis denotes the physical frequency f in Hz and the right axis the effective amplitude &tidle; fh(f ) observed at a distance of 100 Mpc from the binaries. The short-dashed sloped line plotted in the upper left region denotes a planned noise curve of Advanced-LIGO [2] optimized for 1.4M ⊙ NS-NS inspiral detection (“Standard”), the long-dashed slope line denotes a noise curve optimized for burst detection (“Broadband”) and the dot-dashed slope line plotted in the lower right region denotes a planned noise curve of the Einstein Telescope (“ET”) [91Jump To The Next Citation Point, 92Jump To The Next Citation Point]. The upper transverse dashed line is the spectrum derived by the quadrupole formula and the lower one is the spectrum derived by the Taylor-T4 formula, respectively. This figure is taken from [107Jump To The Next Citation Point].

The features of gravitational-wave spectra for a = 0 are schematically summarized in Figure 27View Image. 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 f cut, the amplitude of the spectrum slightly increases with the frequency, that is a characteristic feature seen for the spectrum of BH-BH binaries (e.g., [36]). We refer to a spectrum of this type as type-II.

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Figure 27: Schematic figure for the gravitational-wave spectrum for the same masses of BH and NS with different EOS of NS for a = 0. For (i)-a, tidal disruption occurs far outside the ISCO. For (i)-b, tidal disruption occurs near the ISCO. For (ii), tidal disruption does not occur and the NS is simply swallowed by the BH. We refer to the spectra (i)-a and (i)-b as type-I and the spectrum (ii) as type-II. For a = 0 and 0.13 ≲ 𝒞 ≲ 0.2, the type-I spectrum is seen only for small values of the mass ratio with Q ≲ 3. The filled and open circles denote the cutoff frequencies associated with tidal disruption and QNM, respectively.

To quantitatively analyze the cutoff frequency and to strictly study its dependence on the EOS, the KT group [194, 107Jump To The Next Citation Point, 109Jump To The Next Citation Point] fits all the spectra by a function with seven free parameters

&tidle;h (f) = &tidle;h (f )e −(f∕fins)σins fit 3PN + Am0--e−(f∕fcut)σcut[1 − e−(f∕fins2)σins2], (126 ) Df
where &tidle; h3PN(f) is the Fourier spectrum calculated by the Taylor-T4 formula and fins, fins2, fcut, σins, σins2, σcut, and A are free parameters. The first and second terms on the right-hand side of Equation (126View Equation) 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
∑ { 1∕3}2 [fi&tidle;h(fi) − fi&tidle;hfit(fi)]fi , (127 ) i
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 𝒞 and the NS EOS. Figure 28View Image plots fcutm0, obtained in this fitting procedure, as a function of 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 (𝒞 ≳ 0.16) 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 𝒞 ≲ 0.16. By contrast, f m cut 0 for Q = 2 depends strongly on NS compactness, 𝒞, irrespective of M NS for a wide range of 𝒞 ≲ 0.19.

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Figure 28: −3 Gc fcutm0 as a function of 𝒞 in logarithmic scales. The solid line is obtained by a linear fitting of the data for Q = 2 and Γ 2 = 3. The short-dashed and long-dashed lines show approximate frequencies of the QNM of the remnant BH for Q = 2 and Q = 3, respectively. This figure is taken from [107Jump To The Next Citation Point].

An interesting finding in [107] is that the following relation approximately holds for the identical value of Q,

ln(Gc− 3f m ) = (3.87 ± 0.12) ln 𝒞 + (4.03 ± 0.22 ). (128 ) cut 0
Namely, fcutm0 is approximately proportional to 𝒞3.9. This is a note-worthy point because the power of 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 (7View Equation)). (For binaries composed of spinning BH, this power is smaller than 3.9, but it is still much larger than 1.5 [109Jump To The Next Citation Point].) 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 26View Image 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 –1.4M ⊙; see Equation (8View Equation). The effective amplitude of gravitational waves at f = 2 kHz is ∼ 2 × 10− 22 for a hypothetical distance to the source of 100 Mpc. The amplitude is smaller than the noise level of advanced gravitational-wave detectors, but such a signal will be detectable to next-generation detectors such as Einstein Telescope [91, 92].

3.7.2 Non-zero BH-spin case

The gravitational-wave spectrum is qualitatively and quantitatively modified by the BH spin. In the following, we focus only on the case in which the BH spin and orbital angular momentum vectors align, because gravitational waves for the misaligned case have not yet been studied in detail. Figure 29View Image shows the same relation as in Figure 26View Image but for Q = 3, MNS = 1.35M ⊙ 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 = 1.35M ⊙ with various EOS (right). The left panel of Figure 29View Image 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 27View Image. 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 27View Image. 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 18View Image). In the right panel of Figure 30View Image, we summarize three types of gravitational-wave spectrum.

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Figure 29: Left: The same as Figure 26View Image but for Q = 3, MNS = 1.35M ⊙, and a = 0.75,0.5,0, and − 0.5 with a moderately stiff EOS (EOS HB). Right: The same as the left panel but for Q = 4, a = 0.75, and MNS = 1.35M ⊙ with various EOS. The thin dashed curves show the noise curves for LCGT, advanced LIGO, broadband-designed advanced LIGO, and Einstein telescope (from upper to lower). The figures are taken from [109Jump To The Next Citation Point].

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 30View Image.

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Figure 30: Left: Schematic figure of the type-I gravitational-wave spectrum for a high-frequency side with different values of the BH spin but for the same masses of BH and NS, and NS EOS. With increase of the BH spin, the cutoff frequency decreases and the amplitude below the cutoff frequency increases. Right: Three types of the gravitational-wave spectrum. Type-I (i) and type-II (ii) are the same as those shown in Figure 27View Image. Type-III (iii) is shown for the case in which the BH spin is high and the mass ratio (and thus the area of the BH horizon) is large. The filled and open circles denote the cutoff frequencies associated with tidal disruption and with a QNM, respectively.

A more quantitative explanation follows. From the relation of the luminosity of gravitational waves, the power spectrum of gravitational waves is written as

dE- ∝ [f&tidle;h(f)]2. (129 ) df
In the 1.5 PN approximation, dE ∕df in the inspiral phase is written as
[ ] dE c5 Q −3 5∕3 −3 20Q2 + 15Q df- = 3G-(πf-)2(1 +-Q)2(Gc πm0f ) 1 + (Gc πm0f )Sˆ⋅ ˆL-3-(1-+-Q-)2- , (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. ˆS and ˆL 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 (ˆ ˆ S ⋅L > 0) than for a = 0. The binary evolves due to gravitational-wave emission, and hence, the binary with ˆS ⋅ ˆL > 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 (129View Equation) shows that dE ∕df is proportional to the square of the effective amplitude (f &tidle;h(f)). Therefore, the effective amplitude for a given frequency with ˆ ˆ S ⋅L > 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 29View Image shows the spectrum for Q = 4, MNS = 1.35M ⊙, 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 ∼ 1.4M ⊙), the value of fcut decreases (cf. Equation (7View Equation)), 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 29View Image 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.


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