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
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
decreases as
where
for
and the value of
increases with
for
. 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
), the amplitude of the gravitational-wave spectra damps
above a “cutoff” frequency
. The cutoff frequency is equal to a frequency in the middle
of the inspiral phase with
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
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 () is larger than that predicted by the Taylor-T4 formula
(i.e., the value of
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
becomes a slowly varying function of
for 1 kHz
, where
.
A steep damping of the spectra for is universally observed, and for softer EOS with smaller
NS radius, the frequency of
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
, in which the NS is tidally disrupted far outside the ISCO. In this
case, the spectrum damps at
at which the tidal disruption occurs. The bottom panel (case D)
plots the spectrum for
, in which the NS is not tidally disrupted. In this case, the steep
damping of the spectrum at
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 (
; see the dashed curve). In the middle panel
(case A), the cutoff frequency, at which the steep damping of
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
,
, and with a variety of EOS
for
. 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 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
and with a relatively small value of
, 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
. 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
, 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
Among these seven free parameters, they focus on because it depends most strongly on the
compactness
and the NS EOS. Figure 28
plots
, obtained in this fitting procedure, as a
function of
for
. Also the typical QNM frequencies,
, of the remnant BH for
and
are plotted by the two horizontal lines, which show that the values of
for compact models
(
) with
agree approximately with
and indicates that
for these models are
irrelevant to tidal disruption. For
,
depends on the EOS only for
. By contrast,
for
depends strongly on NS compactness,
, irrespective of
for a wide range of
.
An interesting finding in [107] is that the following relation approximately holds for the identical value
of ,
Figure 26 illustrates that
is rather high,
2 kHz, for a variety of EOS, and thus, the
dependence of
on the EOS for
appears only for a high frequency. The reason for
this is that for
, tidal disruption can occur only for a small mass ratio (and thus for a
small total mass) with a typical NS mass of
; see Equation (8
). The effective
amplitude of gravitational waves at
is
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].
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
,
and
, and
with HB EOS (cf. Table 4; left) and for
,
,
and
with various EOS (right). The left panel of Figure 29
shows that the
spectrum shapes for
,
, and
are qualitatively different; for
, 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
, the
cutoff frequency (
) 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
is neither type-I nor type-II. In this case, there are two typical frequencies. One is at
, above which the spectrum amplitude sinks, and the other is at
, 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 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
. 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 for
. 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
In the 1.5 PN approximation, 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 on the EOS is clearly seen even for a
high value of
. The right panel of Figure 29
shows the spectrum for
,
, and
with four different EOS. For all the EOS, tidal disruption occurs, and the value of
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
.
With the increase of (for a canonical mass of a NS
), the value of
decreases
(cf. Equation (7
)), and the effective amplitude at
increases as the total mass increases. These
are also favorable properties for gravitational-wave detection. The right panel of Figure 29
indeed illustrates
that
is smaller than 2 kHz irrespective of the EOS, and also, that the effective amplitude at
is as large as the noise curve of the advanced detector for a hypothetical distance of
= 100 Mpc. For an even higher spin, say
, tidal disruption is likely to occur for a higher BH
mass with
. 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
=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.
http://www.livingreviews.org/lrr-2011-6 |
Living Rev. Relativity 14, (2011), 6
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