Post on 19-Aug-2020
transcript
Chinese Astronomy and Astrophysics 42 (2018) 487–526
CHINESE
ASTRONOMY
AND ASTROPHYSICS
Gravitational Waves and Gravitational-waveSources† �
ZHAO Wen1� ZHANG Xing1 LIU Xiao-jin1 ZHANG Yang1 WANG
Yun-yong2 ZHANG Fan2 ZHAO Yu-hang2 GUO Yue-fan2 CHEN
Yi-kang2 AI Shun-ke2 ZHU Zong-hong2 WANG Xiao-ge3,4 LEBIGOT
Eric3 DU Zhi-hui3 CAO Jun-wei3 QIAN Jin5 YIN Cong5 WANG
Jian-bo5 BLAIR David6 JU Li6 ZHAO Chun-nong6 WEN Lin-qing6
1Department of Astronomy, University of Science and Technology of China, Hefei 230026, China2Department of Astronomy, Beijing Normal University, Beijing 100875, China
3Tsinghua University, Beijing 100084, China4Michigan State University, East Lansing, MI 48821, USA
5Chinese Academy of Metrology, Beijing 100013, China6University of Western Australia, WA 6009, Australia
Abstract The recent discovery of gravitational-wave burst GW150914 marksthe coming of a new era of gravitational-wave astronomy, which provides a newwindow to study the physics of strong gravitational field, extremely massive stars,extremely high energy processes, and extremely early universe. In this article,we introduce the basic characters of gravitational waves in the Einstein’s generalrelativity, their observational effects and main generation mechanisms, includ-ing the rotation of neutron stars, evolution of binary systems, and spontaneousgeneration in the inflation universe. Different sources produce the gravitationalwaves at quite different frequencies, which can be detected by different methods.In the lowest frequency range (f < 10−15 Hz), the detection is mainly depen-dent of the observation of B-mode polarization of cosmic microwave background
† Supported by National Natural Science Foundation (11633001, 11603020, 11653002, 11322324,
11173021, 11275187, 11675165, 11653002, 11421303), 973 Projects (2012CB821804, 2014CB845806), Strate-
gic Priority Research Program of the Chinese Academy of Sciences (XDB09000000, XDB23010200), Research
Fund of Beijing Normal University, and Fundamental Research Fund of Central Colleges.
Received 2016–10–24; revised version 2016–12–19� A translation of Progress in Astronomy Vol. 35, No. 3, pp. 316–344, 2017� wzhao7@ustc.edu.cn
/ /$
0275-1062/18/$-see front matter © 2018 Elsevier B.V. All rights reserved.doi:10.1016/j.chinastron.2018.10.010
488 LI Long-biao et al. / Chinese Astronomy and Astrophysics 42 (2018) 487–526
radiation. In the middle frequency range (10−9 < f < 10−6 Hz), the gravitation-al waves are detected by analyzing the timing residuals of millisecond pulsars.And in the high frequency range (10−4 < f < 104 Hz), they can be detectedby the space-based and ground-based laser interferometers. In particular, wefocus on the main features, detection methods, detection status, and the futureprospects for several important sources, including the continuous sources (e.g.,the spinning neutron stars, and stable binary systems), the burst sources (e.g.,the supernovae, and the merge of binary system), and the stochastic backgroundsgenerated by the astrophysical and cosmological process. In addition, we forecastthe potential breakthroughs in gravitational-wave astronomy in the near future,and the Chinese projects which might involve in these discoveries.
Key words gravitational wave—neutron star—compact binary system—supernova—inflation
1. INTRODUCTION
Since the Einstein’s general relativity was established one hundred years ago, it has achieved
a great progress in both theory and observational test, and it is still the most successful
gravitational theory up to now. As its basis, the equivalence principle has been verified
by the Eotvos experiment and special relativity. The predictions of its theoretical details
in the post-Newtonian level have been well tested by physical experiments and astronom-
ical observations, including the gravitational light deflection, Shapiro time delay, and the
perihelion precession of Mercury, etc.[1]. Furthermore, the cosmological standard model
of Big Bang based on the general relativity and cosmological principle, i.e., the so-called
inflation+ΛCDM model, has achieved a great success in recent years, and its basic predic-
tions have been proven by a great amount of cosmological observations (including the cosmic
microwave background radiation and large-scale structures of the universe, etc.)[2]. There-
fore, the general relativity has become the fundamental element in the frame of modern
astronomy and physics. Moreover, the most fundamental concepts of time and space etc.
mentioned by the general relativity are always the basis and frontier of physical science.
Meanwhile, as one of the three elements in the construction of this theory by Einstein,
the wave nature of gravitational field has also achieved a great progress in both theory and
observation. The mankind’s first indirect discovery of the trace of gravitational waves was
realized by observing the orbital variation of neutron stars. Since the famous binary star
PSR B1913+16 was discovered in 1974, through continuous observations of several ten years
on the main physical quantities, including the decay rate of the orbital period etc., it has
been verified that this binary star emits gravitational waves, which take off the energy and
angular momentum, and cause the decay of binary orbit, exhibiting as the reduction of
the orbital period P . The theoretically predicted radiation flux of quadrupole moment as
calculated by the general relativity has attained an accuracy of 0.3%, in comparison with
the observational data up to 2010[3]. Even so, this detection is only an indirect detection of
LI Long-biao et al. / Chinese Astronomy and Astrophysics 42 (2018) 487–526 489
gravitational waves, and the direct detection has not obtained a breakthrough until recent
years. The cooperative group of LIGO and Virgo announced on 2016 February 11 that the
signal of gravitational wave explosion GW150914 produced by the merge of double black
holes has been first detected by them[4]. Before long, this group announced on 2016 June 5
that the second example of gravitational wave explosion GW151226 caused by the merge of
double stars has been detected[5]. On 2017 May 31, in an internal media conference held by
the scientific cooperation group of LIGO and Virgo, the third example of gravitational wave
event GW170104 was announced to be detected by the advanced LIGO detector. Similar to
the previous two events, it is also produced by the merge of two black holes rotating around
each other. These achievements marked the establishment of the new field of gravitational
wave astronomy.
Similar to the electromagnetic waves, there are also various sources of gravitational
waves, the frequencies and amplitudes of gravitational waves produced by different sources
differ in a wide range. Hence, people use different methods to detect the gravitational waves
in different wavebands. At present there are mainly three kinds of detection methods in
the world (see Fig.1). The first kind is the laser interferometer gravitational wave detector,
including the ground-based AdvLIGO and AdvVirgo, and space-based eLISA etc., this kind
of detectors are mainly sensitive to the gravitational waves at high frequencies (10−4 < f <
104 Hz). There are many sources for this kind of gravitational waves, such as the merge
of double neutron stars, the merge of double black holes, the supernova explosion, rotating
neutron star, revolving binary white dwarf star, and the merge of double super-massive black
holes, etc. The second kind is the pulsar timing array, the signal of gravitational waves is
extracted by monitoring and analyzing the timing residuals of millisecond pulsars, and in the
world the running ones at present include the PPTA in Australia, the EPTA in Europe, the
NANOGrav in North America, and the IPTA combined by the three. This kind of detectors
are mainly sensitive to the isolated gravitational wave signal and stochastic gravitational
wave background in the medium frequency band (10−9 < f < 10−7 Hz), and the known
gravitational-wave sources include mainly three kinds, they are respectively the gravitational
radiation from the super-massive double black holes in the universe, the gravitational radi-
ation from the cosmic string, and the primordial gravitational wave. The third kind is the
cosmic microwave background radiation, the signal of extremely low-frequency gravitational
waves (10−18 < f < 10−15 Hz) can be extracted by analyzing the B-mode polarization in
the cosmic microwave background radiation, which are mainly the primordial gravitational
waves produced in the inflation period of the universe. At present, the best instruments in
the world are the telescopes of cosmic microwave background radiation of BICEP2 and Keck
Array. Though the later two methods have not yet detected a definite signal of gravitational
waves sofar, the people believe that the detecting sensitivity has already been very close to
the requirement of theoretical prediction, hence it is predictable that these two detection
methods will make a breakthrough in the future several years.
490 LI Long-biao et al. / Chinese Astronomy and Astrophysics 42 (2018) 487–526
Fig. 1 The detection methods and frequencies of gravitational waves emitted by various gravitational-wave
sources (BH: black hole, GW: gravitational wave)
According to the general relativity, gravitational waves can be produced in many astro-
physical and cosmological processes. A drastic astrophysical process can emit gravitational
waves, such as the revolving double compact stars, the accretion of black hole, the explosion
of supernova, the oscillation of neutron star etc., all of them can emit gravitational waves,
to carry away the energy, momentum, and angular momentum, and this can reversely affect
the evolutionary process of these celestial bodies. In the extremely early period of expanding
universe, such as the inflation process, the primordial disturbance of spacetime metric in-
cludes not only the scalar density fluctuation and vectorial rotation disturbance, but also the
tensor-type part, i.e., gravitational waves. With the expansion of the universe, the vectorial
part decays, the scalar disturbances constitute the seeds of cosmic large-scale structures,
while the tensor-type disturbances are remained to be the primordial residual gravitation-
al waves. Due to the weakness of gravitational action, the universe is almost transparent
for gravitational waves, in respect to electromagnetic waves, gravitational waves bring with
more clean information of astrophysics and cosmology. Therefore, the detection and study
on gravitational waves have an extremely important significance for astrophysics and cos-
mology. In this paper we plan to introduce the several kinds of major gravitational-wave
sources in the universe, and the main properties of the gravitational waves emitted by them,
the detection methods, and the present status of detection, including particularly the con-
tinuous gravitational-wave sources (rotating neutron star and stable binary star system), the
explosive gravitational-wave sources (supernova explosion and binary star merger), and the
stochastic gravitational-wave background (the astrophysical gravitational-wave background
and primordial gravitational waves).
In this paper, G represents the Newtonian gravitational constant, in order to make the
formulae be simple, but do not lose their physical meaning, we use the natural unit system,
LI Long-biao et al. / Chinese Astronomy and Astrophysics 42 (2018) 487–526 491
2. GRAVITATIONAL WAVES IN GENERAL RELATIVITY AND THEIR
OBSERVATIONAL EFFECT
2.1 Propagation of Gravitational Waves
In the theory of general relativity, the four-dimensional spacetime metric is expressed by
gμν , i.e., dτ2 = −gμνdx
μdxν . In the curved spacetime, a free particle travels along the
geodesic line, i.e., its motion trajectory satisfies the equation of geodesics[6]:
d2xμ
dτ 2+ Γμ
αβ
dxα
dτ
dxβ
dτ= 0 , (1)
here, Γμαβ is the Christoffel connection. The curvature of spacetime is determined by the
energy-momentum tensor of matter Tμν , and both of them are connected by the Einstein
field equation:
Rμν − 1
2gμνR = 8πGTμν , (2)
here, Rμν is the Ricci tensor, while R is the Ricci scalar, and both of them are the function of
the metric gμν . Here, we do not consider the term of cosmological constant. The symmetric
metric tensor seems to have 10 free components, but due to the existence of the Bianchi
identical relation, only 6 components are exactly independent, and the other 4 components
are the gauge degrees of freedom. In the theory of general relativity, we commonly use the
harmonic gauge to restrict the 4 gauge degrees of freedom:
(√−ggαβ),β = 0 , (3)
here, g is the determinant of the metric gαβ , and gαβ is the inverse matrix of the metric
matrix.
We can commonly rewrite the above field equation into the Landau-Lifshitz form, and
define the tensor density (gothic inverse metric)[7] as
gαβ ≡ √−ggαβ , Hαμβν ≡ gαβgμν − gανgβμ . (4)
Hence, the Einstein field equation can be rewrite into the following form:
Hαμβν,μν = 16πG(−g)(T αβ + tαβLL) , (5)
here, tαβLL is the Landau-Lifshitz pseudo-tensor, which is defined as
492 LI Long-biao et al. / Chinese Astronomy and Astrophysics 42 (2018) 487–526
(−g)tαβLL ≡ 1
16πG
{gαβ,λ gλμ
,μ − gαλ,λ gβμ
,μ +1
2gαβgλμg
λν,ρ gμρ
,ν −
gαλgμνgβν,ρ gμρ
,λ − gβλgμνgαν,ρ gμρ
,λ + gλμgνρgαλ
,ν gβμ,ρ +
1
8(2gαλgβμ − gαβgλμ)(2gνρgστ − gρσgντ )g
ντ,λ g
ρσ,μ
}. (6)
We can further define hαβ ≡ ηαβ − gαβ , thus the condition of harmonic gauge can be
changed as:
hαβ,β = 0 , (7)
while, the field equation can be written as:
ημνhαβ,μν = −16πG(T αβ + tαβLL + tαβH ) , (8)
here, (−g)tαβH = (hαν,μ hβμ
,ν − hμνhαβ,μν)/16πG.
In the discussion of gravitational waves, we adopt the weak-field approximation, i.e.,
gμν = ημν + hμν , here, |hμν | � 1. Considered the linear approximation of hμν , we can
obtain:
hμν ≡ hμν − (1/2)ημνh , (9)
here, h = ημνhμν . Under the linear approximation the Einstein field equation can be
simplified into:
h ,αμν,α = −16πGTμν . (10)
Here we discuss the gravitational waves propagating in vacuum, i.e., Tμν = 0, thus the
propagation equation of gravitational waves is the standard wave equation of a massless
particle h ,αμν,α = 0. For a monochromatic wave, the solution is:
hμν = Cμν exp(ikσxσ) , (11)
here, Cμν expresses the amplitude of gravitational wave, kσ ≡ (ω,k) is the four-dimensional
wave-vector, which satisfies kσkσ = 0. The symmetric matrix Cμν has 10 components,
but considered the condition of harmonic gauge (see Eq.(7)), there are only 6 independent
components. Meanwhile we have noticed that under the coordinate transformation of xμ →xμ+ξμ, the Einstein field equation is invariant, if ξμ simultaneously satisfies the condition of
harmonic gauge, i.e., ξμ,α,α = 0 (its solution is ξμ = Bμ exp(ikσxσ)), then after a coordinate
transformation hμν also simultaneously satisfies the condition of harmonic gauge, hence, in
the remained 6 independent components of hμν , there are still 4 gauge degrees of freedom. In
LI Long-biao et al. / Chinese Astronomy and Astrophysics 42 (2018) 487–526 493
order to fix the degree of freedom brought by the coordinate transformation, we commonly
use the so-called transverse-traceless gauge, usually also called the TT gauge, which requires:
C0μ = 0 , Cii = 0 , kiCij = 0 . (12)
Eq.(12) has given 8 gauge constraints, thus in the 10 components of hμν , there are only 2
really independent components, i.e., the gravitational waves in the general relativity have
only two polarized components. Taking the gravitational wave propagating along the direc-
tion of z axis as an example, under the TT gauge, the two non-zero components of hμν are
respectively h11 = −h22 and h12 = h21, the former one is called the ”+” polarized compo-
nent of gravitational wave, and the latter one is called the ”×” polarized component. It is
easy to find that gravitational waves are transverse waves, which propagate with the speed
of light. And according to the transformation behavior under the rotation of coordinates,
we can easily find that the spin of gravitational waves equals 2.
For the gravitational wave under the non-TT gauge hμν , we can also use the following
transformation to convert it to be the counterpart hTTμν under the TT gauge,
hTTij = Λij,kl(k)hkl , (13)
here, k is the direction vector of the propagation of gravitational waves, the transformation
matrix is Λij,kl(k) = PikPjl − (1/2)PijPkl, here, the matrix P is defined as Pij(k) =
δij − kikj , which is a projective operator. It can be proven that the (spatial) tensor Λij,kl
satisfies the following identical relations:
Λii,kl = 0 , kiΛij,kl = 0 , (14)
and these relations just correspond to the TT gauge condition, as seen from Eq.(12).
2.2 Observational Effect of Gravitational Waves
In order to describe the observational effect of gravitational waves, we study the geodesic
deviation equation satisfied by the relative motion of two adjacent free-particles in the curved
spacetime. Here, we consider two adjacent particles, the difference of their coordinates is
Sμ ≡ δxμ, and we put one particle in the origin of coordinates, according to the geodesic
equation it is known that it keeps static at the origin. Another particle is put at the place
Sμ, and its four-dimensional speed is Uμ = (1, 0, 0, 0). The deviation quantity Sμ satisfies
the next equation[6]:
D2Sμ
Dτ2= Rμ
νρσUνUρSσ . (15)
We consider the gravitational-wave field under the TT gauge, and under the linear approx-
imation, the above equation is simplified as:
∂2Sμ
∂t2=
1
2Sσ ∂
2hμσ
∂t2. (16)
494 LI Long-biao et al. / Chinese Astronomy and Astrophysics 42 (2018) 487–526
Considering the gravitational wave propagating in the direction along the z axis, we
know that it has two independent polarized components. Firstly, we consider the obser-
vational effect of ”+” polarization, at this time h11 = −h22 = C+ exp(ikσxσ), thus the
solution of the geodesic deviation equation is:
S1(t) = (1 +C+
2eiωt)S1
0 , S2(t) = (1− C+
2eiωt)S2
0 , S3(t) = S30 , (17)
here, Si0 ≡ Si(t = t0). It seems to be the oscillations of a free particle in the x − y
plane under the action of the ”+” polarized gravitational wave propagating along the z
direction, the phases of the oscillations in the x and y directions just differ by 180◦; andthe amplitudes of the oscillations are consistent in the two directions, both of which are
determined by the amplitude of the gravitational wave. This is the fundamental feature of
the observational effect of gravitational waves, hence, for the detection of gravitational waves,
it commonly needs two detecting arms perpendicular to each other, thus the oscillation
signals of gravitational waves with such kind of features can be detected.
If the observational effect of the ”×” polarization is considered, we have h12 = h21 =
C× exp(ikσxσ), thus the solution of the geodesic deviation equation is:
S1(t) = S10 +
C×2
eiωtS20 , S2(t) = S2
0 +C×2
eiωtS10 , S3(t) = S3
0 . (18)
Here, we make a coordinate transformation x1 → x1 = (x2 + x1)/√2, x2 → x2 = (x2 −
x1)/√2, and x3 → x3 = x3, i.e., let (x1, x2, x3) rotate 45◦ anticlockwise around the x3 axis,
thus in the new coordinates, we have:
S1(t) = (1 +C×2
eiωt)S10 , S2(t) = (1− C×
2eiωt)S2
0 , S3(t) = S30 . (19)
Hence, we find that the observational effect of the ”×” polarized gravitational waves is
the same as that of the ”+” polarized gravitational waves, the difference is only that the
directions of oscillations differ by an angle of 45◦, as shown by Fig.2.
3. GRAVITATIONAL WAVE SOURCES
In this chapter we mainly discuss two mechanisms for generating gravitational waves: (1)
an isolated gravitational-wave source, such as the neutron star or black hole system, which
causes the gravitational radiation due to the mass distribution and motion of itself; (2) the
primordial gravitational radiation in the universe, and its main mechanism is the non-unique
definition of the vacuum state in a curved spacetime. It is assumed that the universe in the
primordial state is at the vacuum state of graviton, thus with the expansion of the universe,
the vacuum state naturally corresponds to the multiple-particle state of one graviton, i.e., the
LI Long-biao et al. / Chinese Astronomy and Astrophysics 42 (2018) 487–526 495
gravitons will be generated automatically with the expansion of the universe, and therefore
forms the gravitational wave background, which is commonly called the cosmic primordial
gravitational waves, or the cosmic relic gravitational waves. This is a fundamental property
of the quantum field theory in a curved spacetime.
Fig. 2 (a) The deviation of relative position of two adjacent particles when the ”+” polarized gravitational
wave passes through along the z direction; (b) The effect of the ”×” polarized gravitational wave[8]
3.1 Gravitational Radiation of an Isolated Source: Quadrupole Radiation
We first discuss the gravitational radiation from an isolated gravitational wave source. It is
well known that the gravitational force is very weak in comparison with the electromagnetic
force, thus in order to produce a stronger gravitational wave it is necessary to have a very
compact source with a very large mass. At present, such kind of known sources are merely
the compact stellar objects in the universe, including neutron stars, white dwarfs, black
holes etc., and some extremely drastic explosions of celestial bodies, such as the supernova
explosion, etc. Hence, the gravitational wave physics naturally becomes the gravitational
wave astronomy, rather than a science in laboratories. Because these celestial bodies are
very distant from us, and they are all the relatively isolated systems, the gravitational wave
sources we discussed in this chapter satisfy the following three conditions: isolated, far from
the observers, and moving with a low speed. Under these conditions, we solve Eq.(10)
satisfied by gravitational waves, and its solution is a typical retarded solution:
hμν(t,x) = 4G
∫1
|x− y|Tμν(tr,y)d3y , (20)
here, tr ≡ t − |x − y| is the retarded time, and the expression of its corresponding Fourier
transform is:
˜hμν(ω,x) = 4G
∫eiω|x−y| Tμν(ω,y)
|x− y| d3y . (21)
496 LI Long-biao et al. / Chinese Astronomy and Astrophysics 42 (2018) 487–526
If the distance of the source is assumed to be R from the observer, and the typical scale
of the source is ΔR, and ΔR � R, then Eq.(21) can be simplified as:
˜hμν(ω) = 4G
eiωR
R
∫Tμν(ω,y)d
3y . (22)
Using the condition of an isolated source, the above expression can be simplified as:
˜hij(ω) = −2
3Gω2 e
iωR
Rqij(ω) , (23)
and
hij(t) =2G
3Rqij(tr) , (24)
here, we have defined the quadrupole moment to be
qij(t) ≡ 3
∫yiyjT 00(t,y)d3y , qij(ω) ≡ 3
∫yiyj T 00(ω,y)d3y . (25)
Therefore, the gravitational radiation is the typical quadrupole radiation. The lowest re-
quirement and necessary condition for generating gravitational waves is the acceleration of
mass quadrupole moment, which is one of the basic differences between the gravitational
radiation and the electromagnetic radiation. Because the dipole moment of a system is
generally much larger than its quadrupole moment, thus the gravitational radiation is much
weaker than the electromagnetic radiation. Using Eq.(13), we can obtain the expression
of gravitational radiation under the TT gauge. Let the origin of coordinates be located at
the centroid of the radiation source, thus the magnitudes of gravitational waves in different
directions are:
h+(t, θ, φ) =G
3R[q11(cos
2 φ− sin2 φ cos2 θ) + q22(sin2 φ− cos2 φ cos2 θ)−
q33 sin2 θ − q12 sin 3φ(1 + cos2 θ) +
q13 sinφ sin 2θ + q23 cosφ sin 2θ]tr , (26)
h×(t, θ, φ) =G
3R[(q11 − q22) sin 2φ cos θ + 2q12 cos 2φ cos θ −
2q13 cosφ sin θ + 2q23 sinφ sin θ]tr . (27)
Hence, for a radiation system, if only its quadrupole moment tensor is derived, the magnitude
of its gravitational radiation can be acquired.
As like the electromagnetic waves, the reason why the gravitational waves are one kind
of matter is that they possess energy and momentum as well. In the general relativity, the
energy–momentum tensor of gravitational waves tμν is:
tμν = ημαηνβ(tαβLL + tαβH ) . (28)
LI Long-biao et al. / Chinese Astronomy and Astrophysics 42 (2018) 487–526 497
Under the weak-field approximation, only the lowest order of approximation is taken into
consideration, it can be written as the following expression:
〈tμν〉 = 1
32πG〈hαβ,μh
αβ,ν − 1
2h,μh,ν − hαβ
,βhαμ,ν − hαβ,βhαν,μ〉 , (29)
in which 〈· · · 〉 indicates the average that is taken in a range much larger than the wavelength
of gravitational waves. Under the TT gauge, the above expression can be simplified into the
following form:
〈tμν〉 = 1
32πG〈hTT
αβ,μhαβ TT,ν 〉 . (30)
According to its energy–momentum tensor, we can define the energy flow of gravitation-
al radiation, i.e., the radiation power P = −dEgw/dt. Using the conservation law satisfied
by the energy–momentum tensor tμν,ν = 0, we can obtain
P =
∫s
〈ti0〉nids , (31)
in which s is a two-dimensional surface, ni is the unit normal vector of this surface. For an
isolated, far from the observer, and slowly moving gravitational wave system as mentioned
above, considering a two-dimensional spherical surface surrounding the radiation source, the
radiation power can be written as:
P =G
45
⟨d3Qij
dt3d3Qij
dt3
⟩tr
, (32)
in which Qij ≡ qij − δijδklqkl, which is the traceless part of the quadrupole moment qij .
Commonly, the gravitational waves emitted in the astrophysical processes are very
weak, for instance, for a binary star system revolving around its centroid along a circular
orbit, when the masses of the double stars are respectively 1.4M�, the distance of the systemis 20 light years from the Earth, and the revolution period is 7.8 h, then the magnitude
of the emitted gravitational wave is h = 4.6 × 10−20, while its radiation power is only
P = 6.2× 10−23 W. Hence, we can find that even for such kind of very drastic gravitational
radiation source in the universe, its gravitational radiation is extremely weak. Therefore,
the detection of gravitational waves is commonly extremely difficult.
3.2 Continuous Gravitational Wave Sources
If a gravitational radiation system can continuously keep to emit a rather stable signal
of gravitational waves (including the amplitude and frequency of gravitational waves) in
a relatively long duration (in comparison with the observing time), we commonly called
it as a continuous gravitational wave source. This kind of gravitational wave sources are
frequently some rotating systems, and their rotation periods are relatively stable, which
directly determine the frequencies of gravitational waves emitted by them. We first check
if we can realize such kind source of gravitational waves in laboratories on the Earth, and
498 LI Long-biao et al. / Chinese Astronomy and Astrophysics 42 (2018) 487–526
make the amplitude of gravitational waves relatively large. Let’s consider the radiation
source consisted of a kind of dumbbell structure, if the weight of each dumbbell is 1000 kg,
and the dumbbell arm is 1 m long, we make the dumbbell structure rotate with a frequency
of 1000 Hz around its centroid. Through calculations it is ready to find that this system
can be considered as a standard quadrupole radiation system, which is an ideal source of
gravitational waves, and the frequency of gravitational waves emitted by it equals 2000 Hz,
if the observer is doing the observation at a distance of 300 m from the source, it it ready
to obtain that the amplitude of gravitational wave is h ≈10−39, which is much lower than
the sensitivity of present gravitational wave detectors, undetectable at least in the present
stage. Hence, the detectable sources of gravitational waves in the present stage all come
from the celestial body systems. In this section, we mainly introduce two kinds of common
wave sources: rotating neutron star system and stable double star system.
3.2.1 Rotating Neutron Star
Neutron star is the remnant of supernova explosion, with a mass equivalent to the solar
mass and a radius of about 10 km, it attains equilibrium relying on the neutron degeneracy
pressure and gravity, it is a kind of most compact celestial body in the universe. The
Neutron star rotates around its rotational axis with a high speed, and when the emitted
electromagnetic waves scan the Earth, we can receive a regularly pulsed signal, thus the
neutron star commonly appears as a pulsar. When the neutron star is asymmetric in respect
to the rotational axis, the quadrupole moment varies with the time, which can generate a
rather strong gravitational radiation. For a given neutron star, the frequency of the strongest
gravitational wave is two times the rotation frequency of the neutron star, for the ground-
based laser interferometer gravitational-wave observatory (such as LIGO, Virgo etc.), a kind
of detectable gravitational waves come from the fast-rotating neutron stars, their rotational
periods are usually in the order of magnitude of millisecond. In general, this kind of neutron
stars include two types: one is the young neutron stars (including the Crab pulsar, Vela
pulsar etc.), which have not yet been rotationally decelerated, and another one is the old
millisecond pulsars, which are commonly produced from the double star systems due to the
rotational acceleration caused by accreting the matter of their companions.
For the axially asymmetric neutron stars with a rapid rotation, according to the formula
of quadrupole radiation, we can calculate the amplitude of the emitted gravitational wave[9]:
h ≈ 4.2× 10−26( ε
10−6
)(I3
1038kg ·m2
)(1kpc
r
)(f
100Hz
)2
, (33)
here, r is the distance from the neutron star to the Earth; and I3 is the component of inertia
tensor along the direction of rotational axis, and for a common model of neutron stars,
the typical value is about 1038kg · m2; f is the frequency of gravitational wave, which is
double of the rotation frequency of the neutron star, in the sensitivity range of LIGO etc., it
commonly has an order of magnitude of several hundred Hz; ε = (I1−I2)/I3 is the ellipticity
LI Long-biao et al. / Chinese Astronomy and Astrophysics 42 (2018) 487–526 499
of the neutron star along the direction perpendicular to the rotational axis. There is not yet
a consensus of opinion about the formation mechanism of the neutron star’s ellipticity so
far, but some theories suggested that the maximum value can reach an order of magnitude
of 10−5[10,11]. Because gravitational waves possess energy and momentum, neutron stars
may lose their rotational energy and angular momentum due to the gravitational radiation,
which causes the rotation to be decelerated, this leads to the variation of the frequency of
emitted gravitational waves, and its variation rate is[9]:
fgw = −32π4
5GI3f
5ε2 , (34)
i.e., if the rotational deceleration of neutron stars is dominated by the gravitational radiation,
then the variation rate of rotation frequency should be directly proportional to the fifth power
of its frequency. But it is found from the practical observations that the braking index of
pulsars is commonly around 2∼3, which means that the gravitational radiation is not the
main reason to cause the rotational deceleration of neutron stars.
By analyzing the sixth series of scientific data of LIGO (S6) and the second and fourth
series of scientific data of VIRGO (VSR2 and VSR4), in respect to the 7 main young pulsars
known already, and other 172 pulsars with a rotation frequency over 10 Hz (including young
pulsars and millisecond pulsars), people did not find any obvious signal of gravitational
waves, but made some meaningful restrictions on these pulsars[12] (see Fig.3). For instance,
among the 7 mainly concerned young pulsars, for J0534+2200 (Crab), its rotation frequency
is 29.72 Hz, and the variation rate of frequency is -3.7× 10−10 Hz/s, the distance from the
Earth is 2.0 kpc, the amplitude of the emitted gravitational wave is h < 1.6×10−25 (with
a confidence of 95%), and the upper limit of corresponding ellipticity is 8.5×10−5, thus the
energy carried away by gravitational waves is smaller than 1.2% of the total energy loss rate;
for J0835-4510 (Vela), its rotation frequency is 11.19 Hz, and the variation rate of frequency
is -1.6× 10−11 Hz/s, the distance from the Earth is 0.29 kpc, the amplitude of the emitted
gravitational wave is h < 1.1×10−24 (with a confidence of 95%), and the upper limit of
corresponding ellipticity is 6.0×10−4, thus the energy carried away by gravitational waves is
smaller than 11% of the total energy loss rate; for the other several pulsars, it is also found
that the energy carried away by gravitational waves is less than 10% of the total energy loss
rate. For the millisecond pulsars J1045-4509, J1643-1224, and J2124-3358, the upper limit
of gravitational radiation also has the same order of magnitude as the total energy loss rate.
While the most strict limitation of ellipticity is taken for the millisecond pulsar J2124-3358,
its rotation frequency is 202.79 Hz, and the variation rate of frequency is -4.4× 10−16 Hz/s,
the distance from the Earth is 0.3 kpc, the amplitude of the emitted gravitational wave is
h < 4.9×10−26, and the upper limit of the corresponding ellipticity is 6.7×10−8.
500 LI Long-biao et al. / Chinese Astronomy and Astrophysics 42 (2018) 487–526
Fig. 3 The noise curves of gravitational wave detectors and the upper limits of gravitational waves for the
195 nearby neutron stars[12]
Recently, the LIGO group reported the searching results by the all-day periodically
scanning the signal of gravitational waves[13], the covered frequency range is 100∼1500 Hz,
the covered range of frequency variation is [-1.18, +1.00]×10−8 Hz/s, and the main gravita-
tional wave sources are the nearby neutron stars. Through analyzing the sixth-term LIGO
scientific data, there is no any definite signal of gravitational waves to be detected, the lowest
value of the upper limit of gravitational wave amplitude is about at 169 Hz, corresponding
to the upper limit of 9.7×10−25, while around the maximum-sensitivity frequency the upper
limit of gravitational waves is 5.5×10−24. In the highest frequency band, the most sensitive
searching of this scanning is for the neutron stars with an ellipticity of ε > 8×10−7, and the
distance from the Earth of d < 1 kpc, while in the observed results limited by this range,
there is no any neutron star existed with a rotation frequency higher than 200 Hz.
Another kind of important gravitational wave sources are the neutron stars existed
in the low-mass X-ray binary systems (LMXBs), which are accelerated by accreting the
mass and angular momentum of the companions. But the observations indicated that the
rotation frequency of almost all this kind of neutron stars is lower than 700 Hz, which is
much lower than the theoretical upper limit of 1000 Hz, then what reason stops the further
rotational acceleration? Bildsten[14] suggested that this is resulted from a balance in the
neutron star between the angular momentum loss caused by the gravitational radiation and
the increase of angular momentum caused by accretion, and the gravitational radiation may
be generated from the asymmetric structure of accretion disk or the variation of quadrupole
moment caused by an asymmetric thermal distribution of the neutron star. It is found by
the present Rossi satellite that there is an evidence about the existence of neutron stars at
least in a part of LMXBs, and the rotation frequencies of these neutron stars are distributed
LI Long-biao et al. / Chinese Astronomy and Astrophysics 42 (2018) 487–526 501
in a very narrow range of about 250∼320 Hz[15]. According to the modeling calculation the
amplitude of gravitational waves emitted by these neutron stars is:
h = 5.5× 10−27 R3/410
M1/41.4
(FX
F∗
)1/2 (300Hz
f
)1/2
, (35)
here, R10 ≡ R/10 km, M1.4 ≡ M/1.4M�, F∗=1015J·cm−2, and FX is the X-ray radiation
flux. From Eq.(35) we can find that the stronger the X-ray binary system, the stronger
the corresponding gravitational radiation, and the known strongest source is the system of
Scorpius X-1 (Sco X-1), the gravitational waves emitted from which may be observed by
AdvLIGO in the future.
3.2.2 Stable Binary Star System
Among the continuous gravitational wave sources in the low-frequency and extremely low-
frequency bands, the compact mutually revolving binary star systems are the most important
kind. For this kind of sources, only when the observational time is much shorter than the
merging time, they can be treated as continuous sources, thus in the duration of observation
the frequency of gravitational radiation is relatively stable, and its possible tiny variation is
commonly treated as the frequency variation rate fgw. For a compact binary star system,
the orbit is assumed to be circular orbit, the stellar mass is respectively m1 and m2, the
angular speed of revolution is ω, the distance between the two stars is a, and the distance
of the system from the Earth is r, according to the formula of quadrupole radiation, the
gravitational wave emitted by this system is[9]:
h+ = −h01
2(1 + cos2 ι) cos[Φ(t) + Φ0] , (36)
h× = h0 cos ι sin[Φ(t) + Φ0] , (37)
here, ι is the angle between the normal line of binary star’s revolution plane and the line
of sight, Φ0 is the initial phase of gravitational wave at the initial time t = t0, and the
evolution of phase satisfies the condition: Φ(t) = 2πf(t − t0) + πfgw(t − t0)2. It is noticed
that the frequency of gravitational wave is two times the revolution frequency of the system.
For this system, the amplitude of gravitational wave and the frequency variation caused by
the gravitational radiation are respectively:
h0 =4GM
5/3c
r(πGf)2/3 = 4.0× 10−23
(10kpc
r
)(Mc
0.52M�
)5/3 (f
10−3Hz
)2/3
, (38)
f =96
5π8/3(GMc)
5/3f11/3 . (39)
Here, M = m1+m2 is the total mass of the binary star system, μ ≡ m1m2/M is the reduced
mass, and the chirp mass is Mc ≡ μ3/5M2/5.
502 LI Long-biao et al. / Chinese Astronomy and Astrophysics 42 (2018) 487–526
At first we estimate that among this kind of gravitational wave sources, what types
of stable binary stars are included in the different detectable gravitational wave bands.
In a given observable time duration Tobs, the observable period number of gravitational
waves is Nobs = Tobsf . While for a binary star system, the period number required for an
obvious variation of the gravitational wave frequency is Nspindown = f2/f ∼ (GM · f)−5/3.
Hence, the condition if the frequency of gravitational waves has no evident variation in the
observational time is that Nspindown � Nobs. The gravitational waves in the extremely low
frequency band are mainly observed by the pulsar timing array, the sensitive gravitational-
wave frequency is about f ≈ 1μ Hz, the observational time for the pulsar timing array is
generally 10 a, thus the correspondingly observed period number is Nobs ≈ 100, in order
to satisfy the stability condition of gravitational waves, it is required that the mass of the
binary star is M ≈109M�. Therefore, in the extremely low frequency band the observable
continuous gravitational wave sources are mainly the stable super-massive double black hole
systems. While the gravitational waves in the low-frequency band are mainly detected by
the space laser interferometer gravitational-wave detector (such as LISA etc.), its sensitive
frequency band is commonly f ≈ 10 mHz, and the observational time is commonly in the
order of magnitude of year, thus we can obtain the observable period number is Nobs ≈ 106,
in order to satisfy the stability condition of gravitational waves it is required that the mass
of the binary star system is M ≈102M�, such kind of gravitational wave sources include
mainly the double white dwarf system, double neutron star system, white dwarf-neutron star
system, small double black hole system, and neutron star-small black hole binary system,
etc. For the gravitational waves at the high-frequency band, the observations are mainly
relying on the ground-based laser interferometer gravitational-wave detector (such as LIGO,
Virgo etc.), the sensitive frequency is about 100 Hz, and for the observational time of 1 a,
we have Nobs ≈109, which requires that the mass of double stars is much smaller than 1M�,but such a source does not exist, thus at the high-frequency band, it is generally suggested
that there is no any stable binary star system to be the source of gravitational waves.
For the gravitational wave sources of stable binary star systems at the low-frequency
band, what seen most commonly are the double white dwarf systems, which are also the
main targets of the detectors like LISA etc. There is a number of such kind of sources in
the universe, and the number of detectable sources may reach the order of magnitude of
thousands for LISA, here we take the strongest source RX J0806.3+1527 as an example to
estimate the magnitude of its gravitational radiation. This binary star system includes two
white dwarf stars, in which the mass of primary star is 0.55M�, and the mass of companion
star is one half of the primary star, the revolution frequency is 3.11 mHz, the variation
rate of frequency is 3.57×10−16 Hz/s, the inclination with respect to the line of sight is
38◦[33], and the distance of the system from the Earth is between 0.5 and 5 pc. We can find
that even the distance of the source is 5 pc from the Earth, the amplitude of the emitted
gravitational wave can reach h0 ≈6.4×10−23, which is completely in the detectable range
LI Long-biao et al. / Chinese Astronomy and Astrophysics 42 (2018) 487–526 503
of LISA. In addition, for the double neutron star and double black hole systems far from
merging, when the revolution period reaches the order of magnitude of minute, they are
also the ideal continuous gravitational wave sources for the detectors of LISA etc. Through
observing this kind of sources, it is possible to make simultaneously the prediction of their
merging phase for the ground-based interferometers like LIGO etc.[16], hence, these sources
are almost the only sources of gravitational waves which can be observed simultaneously by
LISA and LIGO, and this is very important for the studies of gravitational-wave physics.
For the space gravitational wave detectors like LISA etc., another kind of important
continuous gravitational wave sources are the binary star systems with a large-and-medium
mass ratio, such as the compact binary star systems composed of a neutron star, white
dwarf, or small black hole revolving around a massive (or super-massive) black hole. For
this kind of systems, a small-mass celestial body is located in the strong gravitational field
near the horizon of a large-mass celestial body, and it also needs to spend many revolution
periods to reach the final merge. For instance, if a black hole with a mass of 10 times of
solar mass revolves around a black hole with a mass of 106 times of solar mass, even it has
approached the horizon of the large black hole, before the small black hole drops into the
large black hole, it still needs to revolve for 105 revolution periods, thus its gravitational
radiation is stable in a very long time, and it can be considered as a continuous gravitational
wave source. In respect to the LISA, if only such a system is not far from the Earth, in the
range of 1∼10 Gpc, it is completely possible to be detected, and the occurrence rate of such
an event is rather high, which is estimated to reach an order of magnitude of several tens
to several hundreds during the observational time of one year[17]. It is worth notice that in
such a double star system, the motion of small-mass celestial body can be considered as the
geodesic motion of a point particle in the strong gravitational field, thus it provides a very
clean and ideal place for studying the geometrical properties of the central black hole, which
is also one of the reasons why such kind of systems have attracted a wide attention.
In the standard hierarchical theory of galactic formation, it is generally believed that
the super-massive black hole located in the galactic center is generated by the dynamical
friction during the galactic merge, thus in this theory the revolution and merge of super-
massive double black hole system commonly exist. When it is in the merging stage, the
frequency of emitted gravitational waves is higher, which belongs to the detectable range of
space gravitational-wave detectors, and in this case the gravitational radiation appears as
an explosive source. However, when it is in the revolving stage, the revolution frequency
varies rather slowly, and it appears as a continuous gravitational wave source. Especially,
when the frequency band of gravitational radiation reaches nearly 10−8 Hz, it is just in the
detectable frequency range of pulsar timing arrays. For the super-massive double black hole
system, the amplitude of the emitted gravitational wave can be estimated by the following
expression:
504 LI Long-biao et al. / Chinese Astronomy and Astrophysics 42 (2018) 487–526
h0 = 2.76× 10−14
(10Mpc
dL
)(Mc
109M�
)5/3 (f
10−8Hz
)2/3
, (40)
here, dL is the luminosity distance of the wave source, Mc is the chirp mass of the binary
star system, and f is the frequency of gravitational wave, which is two times the revolution
frequency of the binary star. When such kind of binary star systems are sufficiently close to
the Earth, they will reach the sensitive range of the pulsar timing array. But in fact, before
the pulsar timing array is running, Jenet et al.[18] already made a meaningful constraint
on the gravitational radiation from a suspected super-massive double black hole system
3C66B by analyzing 7 a of the signal residual of the millisecond pulsar PSR B1855+09. At
that time, people thought that the 3C66B system is a double black hole system with the
revolution period of 1.05 a, the total mass of black holes of 5.4×1010M�, the mass ratio of
black holes of 0.1, and the redshift of z = 0.02. As estimated by the above expression, if
the motion of the binary star is along a circular orbit, then the emitted gravitational waves
should be found by analyzing the signal residual of the millisecond pulsar PSR B1855+09.
But the practical data analysis did not find such a signal of gravitational waves, hence this
result is a challenge for the super-massive binary black hole model of 3C66B.
In recent years, the three main pulsar timing array teams in the world have all analyzed
the signal of gravitational waves that possibly existed in their observed data, and respectively
reported their results of constraint. By analyzing 5 a of the signal residuals of 17 millisecond
pulsars, the NANOGrav working group gave the upper limit of gravitational waves at the
frequency of 10−8 to be h0 <3.0×10−14 (with a confidence of 95%)[19]. While the PPTA
team obtained the upper limit of gravitational waves at this frequency to be h0 <1.7×10−14
by analyzing the data of 20 pulsars[20]. The EPTA team analyzed the data of 6 pulsars, and
the obtained result is h0 <1.1×10−14[21]. In a recent work[22], the authors carefully analyzed
the observed data of the three teams, and reconsidered two kinds of possible sources: one
is the binary black hole system with the mass of black hole determined already by other
dynamical measures, another one is the black hole system in the 116 massive early-type
galaxies with a distance from the Earth smaller than 108 Mpc obtained by the MASSIVE
survey. It is found by the analysis that for the galaxies with a mass of central black hole
over 5×109M�, including NGC4889, NGC4486 (M87), and NGC4649 (M60), if they are
super-massive double black hole systems, then the mass ratio should be smaller than 1:10.
In a word, no matter whether the prediction of galactic formation theory or the known
suspected super-massive double black hole systems, it is difficult to detect such an isolated
system by the present three pulsar timing arrays, but this provides also a good opportunity
for the future telescopes like FAST, SKY, etc.
3.3 Explosive Gravitational Wave Sources
For a kind of gravitational wave sources, if the explosion timescale of gravitational waves
is much shorter than the observing timescale, we can called them the explosive gravita-
LI Long-biao et al. / Chinese Astronomy and Astrophysics 42 (2018) 487–526 505
tional wave sources. This kind of gravitational wave events are commonly generated in a
drastic explosive event, such as the explosion of gravitational waves caused by the merge of
double stars (double white dwarfs, double neutron stars, double black holes etc.), the gravi-
tational radiation caused by the supernova explosion, the gravitational radiation caused by
the collision of cosmic strings, and the gravitational radiation accompanied by the glitch
phenomenon in pulsars etc. In this paper, we emphasize the introduction of two types of
explosive sources: supernova explosion and binary star merge.
3.3.1 Supernova Explosion
We first consider the gravitational radiation during the collapse of a celestial body. We know
that when a main-sequence star evolves to the later period, its bulge part will collapse to
be a compact star. If the main-sequence star’s mass is lower than 8 times of solar mass, its
bulge will collapse to be a white dwarf, which reaches an equilibrium relying on the electron
degeneracy pressure to resist the gravitational force. If the white dwarf is located in a
binary star system, it can accrete the mass of its companion star to increase the mass and
temperature of itself, and when its mass increases and exceeds the Chandrasekhar limit, an
Ia-type supernova explosion will be generated. In addition, when the main-sequence star’s
mass is higher than 8 times of solar mass, a II-type (or Ib and Ic-type) supernova explosion
will be generated in the later period of evolution, while its bulge part will collapse directly
to be a neutron star or a black hole. It is commonly suggested that if a black hole forms
finally, the Gamma-ray radiation will be accompanied during the collapse, which is just the
observed Gamma-ray burst with a long timescale.
During the supernova explosion, commonly there is a strong gravitational wave emission
accompanied. But, because the physical process of celestial body collapse is very compli-
cated, the numerical calculation is very difficult, which involves the complicated numerical
relativity, neutrino effect, hydrodynamic process, microphysical process, magnetic field, and
some other effects, it is still a difficult theoretical problem up to now. Hence, the accurate
prediction of the strong gravitational radiation in this process has a very large uncertainty.
But on the other side, if such kind of gravitational wave signals can be first detected in
observations, the physical processes in the stellar collapse may be deduced inversely, and
these processes are completely non-transparent for the electromagnetic radiation. For in-
stance, Hayama et al.[23] found that by observing the gravitational waves with a circular
polarization emitted from these processes, the rotational situation of the bulge part during
a supernova explosion can be accurately deduced inversely.
It is generally suggested that strong gravitational waves will be generated in the process
of celestial collapse. For a typical process of supernova explosion, it is found by numerical
simulations that if only the matter distribution in the bulge part deviates from the complete
spherical symmetry (this may be caused by following reasons: the nonuniform collapse
of supernova due to the primordial disturbances of density and temperature; an unstable
pressure due to the high-speed rotation of bulge region; a very large convection generated
506 LI Long-biao et al. / Chinese Astronomy and Astrophysics 42 (2018) 487–526
by the hydrodynamic instability, and therefore to affect the primordial explosion of stellar
object; a very large exciting energy during the formation of remained neutron star, etc.),
there will be a great part of energy emitted outward, in which about 99% of energy is
converted into the energy of energetic neutrinos, about 0.01% is converted into the energy
of photons, and about 1% is converted into gravitational waves, i.e., the energy of 10−7 ∼10−5M� is converted into gravitational waves and emitted outward, the frequency of these
gravitational waves is commonly about 200 ∼ 1000 Hz[9], and just in the sensitive range of
the ground-based laser interferometers. To consider the supernova explosion in the Galaxy,
we may use the following formula to estimate the gravitational wave amplitude[15]:
h ≈ 6× 10−21
(E
10−7M�)
)1/2 (1ms
T
)1/2 (1kHz
f
)(10kpc
r
), (41)
here, E is the total energy that has been converted into gravitational waves, and its typical
value is 10−7M�; T is the time duration of explosion, and it commonly has an order of
magnitude of millisecond; f is the frequency of gravitational waves; r is the distance of the
supernova from the Earth, and its order of magnitude is commonly 10 kpc for a source in the
Galaxy. This amplitude can be detected by the present detectors of gravitational waves like
LIGO, VIRGO etc. with a very high confidence. However, it is necessary to emphasize that
the possibility of supernova explosion is very low in the Galaxy: for the II-type supernova,
it is commonly suggested that its explosion rate is 0.01 to 0.1 time per year in the Galaxy;
even in the Virgo super-cluster, the explosion rate is only one time every 30 years. Hence,
in the sensitive range of the detectors of LIGO and the 2nd-generation AdvLIGO etc., the
probability that such kind of explosive sources can be observed is actually very low, and in
practice the feasible observations are only expected by the ground-based gravitational wave
detectors of the third generation represented by the Einstein telescope.
Even so, in References [24, 25], by analyzing the data of the LIGO and other detectors,
the authors still made some meaningful constraints on some nearby supernova explosions.
Especially, in Reference [25], by analyzing the data of LIGO, Virgo, and GEO600 in 2011,
the authors searched the possible sign of gravitational waves due to supernova explosions.
In the analysis, the selection of observed sources should satisfy three conditions: (1) the
nearby supernova explosion events within 15 Mpc; (2) the time of supernova explosion has
already been measured rather accurately; (3) during the explosion, at least over two gravi-
tational wave detectors work normally for the data collection. There are only two supernova
explosion events in accordance with these conditions, they are respectively SNe 2007gr and
SNe 2011dh. The study found that there is no suspected candidate of gravitational waves,
hence, the energy converted into gravitational waves by these two supernova explosions is
not greater than 0.1 M� at the low frequency band, while it is not greater than 10 M� at
the high frequency band above 1 kHz. Previously, the estimated intensity of gravitational
waves during the supernova explosion was too large, and the new results suggest that even
for the 2nd-generation gravitational wave detectors, like AdvLIGO, AdvVirgo, and KRGRA
LI Long-biao et al. / Chinese Astronomy and Astrophysics 42 (2018) 487–526 507
etc., it is only possible to observe the most close supernovae (r ≤1∼100 kpc), i.e., only the
supernova events within the Galaxy, small and large Magellanic clouds can be observed, and
there are only 2∼3 such events occurred per 100 years. Therefore, in a short period it is
very difficult to detect directly the gravitational waves from supernovae.
3.3.2 Merge of Compact Binary Star System
Among the various known gravitational wave sources, the compact binary star systems are
the most convinced, and mostly studied wave sources. There are various kinds of compact
binary star systems, including the double white dwarf systems, double neutron star systems,
and double black hole systems with different masses. When the distance of double stars is
rather far, and the revolution orbit is relatively stable, the frequency and amplitude of
the emitted gravitational wave are also relatively stable as mentioned above, and this kind
of sources belong to the continuous gravitational wave sources. But, when the distance
of double stars is quite short, the decay of revolution orbit is quite obvious, and close
to or located in the merging state, its timescale is much shorter than the observational
time of the gravitational wave detector, then the gravitational radiation emitted by the
binary star system is explosive, and this kind of system appears as a kind of explosive
gravitational wave source. When the mass of binary star is relatively small, such as the
binary neutron star merger, neutron star-small black hole merger, and binary black hole
(solar-mass) merger, etc., the emitted gravitational waves are the main observing objects
for the ground-based laser interferometers like LIGO, Virgo etc., and in fact, there are 14
binary neutron star systems actually detected (see Table 1)[26,27], in which PSR B1913+16,
B1534+12, J0737-3039A, J1756-2251, J1906+0746 (possibly belonging to the neutron star-
white dwarf system), and J2127+11C[28] have a merging timescale smaller than the age of
the universe. While for the super-massive binary black hole merger (with a mass greater
than 106M�), and the binary black hole merger with an extreme mass ratio, they are the
main observing objects of the space gravitational wave detectors like LISA etc., even of the
pulsar timing arrays.
For this kind of gravitational radiation caused by the merge of compact binary star,
various theoretical models have been developed to describe it very well. When the distance of
double stars is relatively large, the motion of celestial bodies has not reached the relativistic
speed, and the decay of revolution orbit due to the gravitational radiation is rather slow, the
post-Newton approximation can describe the gravitational radiation very well, this stage is
called the inspiral stage. However, in the later period of revolving stage and the merging
period, which are commonly called the merging stage, the gravitational field is very strong, at
this time the post-Newton approximation is invalid, thus the method of numerical relativity
is commonly adopted to derive the solution. After the double stars are merged to be a black
hole, it is necessary to emit the redundant degrees of freedom and to become a static black
hole by through the gravitational radiation, and this stage is commonly called the ringdown
stage, and the emitted gravitational waves can be described analytically by the quasi-normal
508 LI Long-biao et al. / Chinese Astronomy and Astrophysics 42 (2018) 487–526
mode of black hole oscillation. Hence, the template of gravitational radiation in a binary star
merging event is an effective superposition of three parts, this is very important for searching
gravitational waves. But even so, the understanding on the gravitational radiation by the
binary star merge is still very unmatured at present, for example the merging effects on the
gravitational tide, neutron star state, and the spin of binary star, etc. However, people can
reversely deduce the physics in a strong gravitational field, and verify the general relativity
etc. by the observed gravitational waves, which is also one of the main scientific objectives
for the direct detection of gravitational waves.
Table 1 Binary neutron star systems detected sofar[26]
pulsar spin revolution semi-major axis of orbital total mass mass of
period period orbital projection eccentricity of system pulsar
P/ms Pb/d x axis (lt-sec) e M/M� MP/M�
J0737−3039A 22.699 0.102 1.415 0.087 777 5(9) 2.587 08(16) 1.338 1(7)
J0737−3039B 2 773.461 — 1.516 — — —
J1518+4904 40.935 8.634 20.044 0.249 484 51(3) 2.718 3(7) —
B1534+12 37.904 0.421 3.729 0.273 677 40(4) 2.678 463(4) 1.333 0(2)
J1753−2240 95.138 13.638 18.115 0.303 582(10) — —
J1756−2251 28.462 0.320 2.756 0.180 569 4(2) 2.569 99(6) 1.341(7)
J1811−1736 104.1 18.779 34.783 0.828 02(2) 2.57(10) —
J1829+2456 41.009 1.760 7.236 0.139 14(4) 2.59(2) —
J1906+0746 144.073 0.166 1.420 0.085 299 6(6) 2.613 4(3) 1.291(11)
B1913+16 59.031 0.323 2.342 0.617 133 4(5) 2.828 4(1) 1.439 8(2)
J1930−1852 185.520 45.060 86.890 0.398 863 40(17) 2.59(4) —
J0453+1559 45.782 4.072 14.467 0.112 518 32(4) 2.734(3) 1.559(5)
J1807−2500B 4.186 9.957 28.920 0.747 033 198(40) 2.571 90(73) 1.365 5(21)
B2127+11C 30.529 0.335 2.518 0.681 395(2) 2.712 79(13) 1.358(10)
For the present 2nd-generation ground-based gravitational wave detectors (such as Ad-
vLIGO, AdvVirgo etc.), the explosive gravitational wave events caused by the merge of
binary star system of solar mass are the most possible gravitational wave sources to be
observed at first. The most important wave sources are the binary stars of stellar mass,
including the merging events of neutron stars, and of the black holes with a star-like mass,
etc. Abadie et al.[28] made the following estimations for the occurrence rates of various
kinds of binary star merging events: the occurrence rate of the binary neutron star merger
within the Galaxy is about 1∼1000 for every 106 years, and the most possible rate is 100
per 106 years, which corresponds to the occurrence rate of the event in the nearby universe
of 0.1∼10 Ma−1·Mpc−3, and the most possible rate is 1 Ma−1·Mpc−3. Hence, for the final
LI Long-biao et al. / Chinese Astronomy and Astrophysics 42 (2018) 487–526 509
sensitivity of AdvLIGO, AdvVirgo etc., the detectable event rate is about (0.4∼400) a−1,
and the most possible rate is 40 events for every year. But for the binary black hole merger
with the solar mass, and the neutron star-black hole merger, there is a very large uncer-
tainty in the estimation of occurrence rate. The neutron star-black hole merging rate is
about (6×10−4 ∼1) Ma−1·Mpc−3, while the merging rate for the binary black hole merger
of solar mass is about (1×10−4 ∼0.3) Ma−1·Mpc−3, but the gravitational waves emitted
by these systems are stronger, the gravitational wave detector can detect a farther signal,
hence, the finally estimated detection rates of AdvLIGO, AdvVirgo etc. for these events are
respectively (0.2∼300) a−1 (for the neutron star-black hole merger) and (0.4∼1000) a−1 (for
the binary black hole merger).
Up to now, in the 1st-term scientific data of AdvLIGO, there are no binary neutron
star merger and neutron star-black hole merger occurred. In the template matching, the
authors assumed that the mass of neutron star is in the range of (1∼3) M�, and the spin is
smaller than 0.05; while the mass of black hole is assumed to be (2∼99) M�, without any
limitation on its spin. It is found by considering the present sensitivity curve of AdvLIGO
that if the masses of both neutron stars are all equal to (1.35±0.13) M�, then the event of
binary neutron star merger can be possibly found in a distance smaller than 70 Mpc from
the Earth; while for the neutron star-black hole merger, if the same assumption is made for
the mass of neutron star, and simultaneously the mass of black hole is required to be larger
than 5 M�, then the merging event can be found in a distance smaller than 110 Mpc from
the Earth. Hence, according to the present actually-observed results, the given upper limit
for the event occurrence rate is respectively: smaller than 12600 Gpc−3·a−1 for the binary
neutron star merging rate, and smaller than 3600 Gpc−3·a−1 for the neutron star-black hole
merging rate, and these results are consistent with the estimation as mentioned above.
However, the AdvLIGO’s observation on the binary black hole merging event has made
a great progress, two obvious explosive events of gravitational waves and one suspected event
have been detected for the first time, this is the first time that mankind detected directly the
signal of gravitational waves, and it makes a tremendous influence on the whole astronomy
and physics. Here, the gravitational wave event of GW150914 was detected simultaneously
by two detectors of AdvLIGO located in Hanford and Livingston in the universal standard
time of 2015 September 14 (see Fig.4). The signal-noise ratio has reached 23.7 by the
template matching[4] (see Table 2), which is the strongest gravitational wave signal detected
in the first 16-day running data of the detector AdvLIGO. In the frequency range of (35∼250)
GHz, the peak amplitude of this signal reaches h = 1.0 × 10−21, and the peak power of
the gravitational radiation attains 3.6×1049 J·s−1, which is equivalent to an effective energy
of 200 M� emitted per second. It is found by analysis that the gravitational waves are
generated by the merge of two revolving black holes, the luminosity distance of the wave
source is 410 Mpc from the Earth, equivalent to the cosmological redshift of z = 0.09, before
merging the masses of two black holes were respectively m1 = 36 M� and m2 = 29 M�,
510 LI Long-biao et al. / Chinese Astronomy and Astrophysics 42 (2018) 487–526
after merging, the mass of black holes became m = 62 M�, thus there is totaly a mass of 3
M� being converted into the gravitational waves and taken out. It is found by data analysis
that the spin of the larger black hole is smaller than 0.7, thus the possibility that it is an
extreme Kerr black hole is excluded; while the constraint on the spin of the smaller black
hole is very weak, but after merging the spin of this black hole is very evident, which reaches
0.67, thus it is a typical Kerr black hole. As there are only two gravitational wave detectors
at present, so the ability to locate the wave source is very worse, and the uncertainty of the
source position reaches 600 square degrees. In spite of that, this still is the first directly
detected signal of gravitational waves by mankind, and also the firstly detected merging
event of double black holes, these results show that the binary black hole system of stellar
mass really exists in the universe. Meanwhile, by analyzing the signals of gravitational waves
at different frequencies, an important constraint for the mass of graviton has been proposed:
mg < 1.2×10−22 eV, which corresponds to the Compton wavelength of λg > 1013 m.
Fig. 4 The signals of GW150914 detected by the two detectors of AdvLIGO located in Hanford (H1, Panel
a) and Livingston (L1, Panel b), and the comparison with the theoretically calculated results[4]
Up to now, the secondly strongest explosive event of gravitational waves GW151226
detected by AdvLIGO occurred in the universal standard time of 2016 December 26[5],
which was generated by the merge of two black holes with stellar masses. The duration of
the event was 1 s in the sensitive frequency band of AdvLIGO, and the signal-noise ratio to
detect this event by the template matching was 13, which exceeds the confidence of 5 σ. In
(35∼450) Hz, the revolving periods of double stars attained 55, and the peak amplitude of
gravitational waves reached h = 3.4× 10−22, the peak power of the gravitational radiation
reached 3.3× 1049 J·s−1. It is found by data analysis that the wave source has an luminosity
distance of 440 Mpc from the Earth, i.e., the cosmological redshift is z = 0.09. In this wave
source, the initial masses of two black holes are respectively m1 = 14.2 M� and m2 = 7.5
M�, after merging, the mass of black holes is m = 20.8 M�, thus there is about a mass
LI Long-biao et al. / Chinese Astronomy and Astrophysics 42 (2018) 487–526 511
of 1 M� being converted into the gravitational waves and taken out. In the meantime, it
is found by simulations that there is at least one black hole with a significant spin, which
exceeds 0.2, and after merging, the spin of the black hole equals 0.74, thus it is a typical
Kerr black hole. The practical location of wave source is still uncertain for AdvLIGO, and
the uncertainty of the source position reaches 1400 square degrees.
Table 2 Three gravitational wave events of binary black hole merge detected by
AdvLIGO[31]
Event GW150914 GW151226 LVT151012
Signal-to-noise ratio: ρ 23.7 13.0 9.7
False alarm rate: FAR/a−1 < 6.0× 10−7 < 6.0× 10−7 < 0.37
p-value 7.7× 10−8 7.5× 10−8 0.045
Significant > 5.3σ > 5.3σ > 1.7σ
Primary mass: msource1 /M� 36.2+5.2
−3.8 14.2+8.3−3.7 23+18
−6
Secondary mass: msource2 /M� 29.1+3.7
−4.4 7.5+2.3−2.3 13+4
−5
Chirp mass: Msource/M� 28.1+1.8−1.5 8.9+0.3
−0.3 15.1+1.4−1.1
Total mass: Msource/M� 65.3+4.1−3.4 28.1+5.9
−1.7 37+13−4
Effective inspiral spin: χeff −0.06+0.14−0.14 0.21+0.20
−0.10 0.0+0.3−0.2
Finial mass: Msourcef /M� 62.3+3.7
−3.1 20.8+6.1−1.7 35+14
−4
Final spin: af 0.68+0.05−0.06 0.74+0.06
−0.06 0.66+0.09−0.10
Radiated energy: Erad/(M�c2) 3.0+0.5−0.4 1.0+0.1
−0.2 1.5+0.3−0.4
Peak luminosity: �peak/(1049J · s−1
)3.6+0.5
−0.4 3.3+0.8−1.6 3.1+0.8
−1.8
Luminosity distance: DL/Mpc 420+150−180 440+180
−190 1000+500−500
Source redshift: z 0.09+0.03−0.04 0.09+0.03
−0.04 0.20+0.09−0.09
Sky licalization: ΔΩ/deg2 230 850 1 600
Furthermore, the thirdly strongest (suspected) explosive event of gravitational waves
LVT151012 occurred in the universal standard time of 2015 October 14[30], and its signal-
noise ratio reached 9.7, which is equivalent to the confidence of 2.1 σ, in respect to the
previous two events, the confidence of this source is relatively lower (thus AdvLIGO can not
definitely ensure that it is really an explosive event of gravitational waves). This explosive
source is still a merging event of double black holes, and its luminosity distance is 1100 Mpc
from the Earth, which corresponds to the cosmological redshift of z = 0.20. Before merging,
the masses of two black holes are respectively m1 = 23 M� and m2 = 13 M�, the black
holes after merging are still typical Kerr black holes, with a mass of 35 M� and a spin of
0.66, i.e., there is an energy of about 1.5 M� taken out by gravitational waves. For this
event, the corresponding peak power of gravitational radiation is 3.1× 1049 J·s−1, and the
position uncertainty of this event is 1600 square degrees.
512 LI Long-biao et al. / Chinese Astronomy and Astrophysics 42 (2018) 487–526
According to the observations of these merging events of double black holes, the merging
rate of double black holes in the universe can be well constrained. The working groups of
LLGO and Virgo considered two cases[31]: in the first case it is assumed that the distribution
of binary black hole systems in the universe and the mass of binary star satisfy p(m1,m2) ∝m−1
1 m−12 and in the second case it is assumed that the distribution of binary black hole
systems in the universe and the mass of primary star satisfy p(m1) ∝ m−2.51 , which is not
related with the mass of secondary star. At the same time, the masses of double stars are
assumed to be in the range of m1 � m2 � 5 M�, and m1 + m2 � 100 M�. Here, in the
first case, it is possible to underestimate the merging probability of the double black holes
with larger masses; while in the second case, it is possible to underestimate the contribution
of the double black holes with smaller masses. In a word, by analyzing the observed three
merging events of double black holes and synthesizing two cases, the merging rate of double
black holes is obtained to be 9 ∼ 240 Gpc−3 · a−1, in which the lower limit comes from
the estimation of the first case, while the upper limit is taken from the second case. The
occurrence rate of this kind of event is consistent with the theoretical estimation mentioned
above. Meanwhile, according to the three explosive gravitational wave sources, the mass
distribution of black holes in the merging events of double black holes may be estimated:
at first, it is assumed that the mass distribution of primary stars in the binary black hole
systems satisfies a power-law relation of p(m1) ∝ m−α1 , in which α is a free parameter to
be fitted; secondly, it is assumed that the mass of secondary star is between 5 M� and m1,
and satisfies a uniform distribution, thus it is obtained by fitting that α = 2.5+1.5−1.6, which is
consistent with the assumption in the second case as mentioned above.
For the future space detectors of gravitational waves (such as LISA etc.), the merge of
super-massive double black holes will be the main objects to be observed. No matter whether
from the theoretical models of galactic formation, or from the present observations of double
black holes, the evolution and merge of super-massive black holes in a cosmological scale are
inevitable, but the present estimation of occurrence rate of this kind of merging event has a
larger uncertainty, which is related with many very complicated physical processes. Klein et
al.[32] studied the occurrence rate of the merging event of super-massive double black holes
for the different scenarios of galactic evolution with a semi-analytical method, as well as the
detection rate for such kind of events by the future eLISA project. For the formation of
super-massive black holes, the authors considered three different models: in the first one, the
so-called light-seed model (i.e., the popIII model) is taken account, it is suggested that the
super-massive black holes are originated from the evolutionary remnants of popIII stars, and
the effect of time delay between the merge of massive black holes and the merge of galaxies is
taken account simultaneously; in the second model, the so-called heavy-seed model (i.e., the
Q3-d model) is taken account, and it is suggested that the super-massive black holes with
a mass of 105 M� formed in the early period of the universe (z ≈ 15 ∼ 20), the formation
may be caused by the galaxy collision or some other factors, and the effect of time delay
LI Long-biao et al. / Chinese Astronomy and Astrophysics 42 (2018) 487–526 513
between the merge of massive black holes and the merge of galaxies is also taken account in
the second model; while the third model (i.e., the Q3-nod model) is almost identical with
the second one, but the so-called delay effect is neglected. It is found by the semi-analytical
calculation that in the popIII model, for the most optimistic design of eLISA, in the running
time of 5 years, there will be about 660 detectable merging events of super-massive black
holes, in which the high-redshift events of z > 7 may reach 401; in the Q3-d model, there are
totally 40 events, in which there are 3.6 high-redshift events of z > 7; while in the Q3-nod
model, there are totally about 596 events, in which there are 343 high-redshift events. And
even for the most pessimistic design of eLISA, the rate of detectable event can reach also
the following level: there are totally 28 events in the popIII model, in which there is one
high-redshift event; in the Q3-d model, there are totally 12 events, in which the number of
high-redshift events is 0.3; and in the Q3-nod model there are 95 events, in which there are
6 high-redshift events.
3.4 Gravitational Wave Backgrounds
Besides the various isolated gravitational wave sources mentioned above, there are still var-
ious stochastic gravitational wave backgrounds in the universe, which may be in analogy
with the generally existed photon field of microwave background radiation in the universe.
There are various kinds of gravitational wave backgrounds, which are roughly classified into
two kinds according to their origins: one is the collective contribution of the isolated grav-
itational wave sources mentioned above, such as the contribution of gravitational radiation
from a great amount of binary neutron star and binary white dwarf systems, as well as the
gravitational wave background produced by the radiation of a great amount of super-massive
black hole binary stars; and another one is of the cosmological origin, such as the primordial
gravitational wave background caused by the quantum fluctuation pulled out of the horizon
in the cosmic inflation period, the gravitational wave background caused by the oscillation
and collision of cosmic strings, and the gravitational wave background emitted from various
early cosmic phase changes (such as the QCD phase change, weak electric phase change,
etc.). In this section, we plan to introduce this two kinds of gravitational wave backgrounds
respectively.
3.4.1 Stochastic gravitational wave background caused by astrophysical processes
In many astrophysical processes, such as the rotation of neutron stars, the evolution of
compact binary star systems, and the supernova explosion etc., the radiation of gravitational
waves may be caused. Because in the cosmic space, there are always a great amount of
such events of gravitational waves, and they have an approximately uniform and stochastic
distribution, thus it is easy to yield a stochastic gravitational wave background. For the
different types of wave sources, the properties of gravitational wave backgrounds emitted
by them are also different, in this section, we mainly introduce three types of stochastic
gravitational wave backgrounds: the radiation of the revolving binary white dwarf systems,
the merging radiation of double neutron stars and of double black holes with solar masses,
514 LI Long-biao et al. / Chinese Astronomy and Astrophysics 42 (2018) 487–526
and the merging radiation of super-massive double black holes.
3.4.1.1 Revolving system of double white dwarf stars
As mentioned in the previous section, the binary white dwarf systems are a very impor-
tant kind of gravitational wave sources, and even in the Galaxy, such kind of wave sources
are very numerous, which is predicted to reach an order of magnitude of 108. Hence, syn-
thesizing the contributions of these sources, a very important low-frequency gravitational
wave background may be resulted, which will be one of the main detecting objects of future
space gravitational wave detectors such as LISA etc. However, different from the evolutions
of neutron stars and black hole binary stars, the evolution of binary white dwarf systems is
related to a complicated process of matter exchange, thus the dynamics is very complicated.
For simplicity, we will neglect those evolutionary details and consider only the gravitational
radiation in the stably revolving stage, disregarding the complicated dynamical process of
matter exchange, etc. In order to describe the magnitude of the background gravitational
wave, we commonly define the energy density of gravitational wave as:
ΩGW(f) =f
ρc
dρGW
df, (42)
here, dρGW is the energy density of gravitational waves in the frequency range of f ∼ f+df ,
ρc ≡ 3H20/8πG is the critical density of the universe, and H0 is the Hubble constant. For
a binary white dwarf system, the emitted gravitational wave background depends on the
formation rate and evolutionary process of a binary white dwarf system in the universe. In
the calculation, we make following assumptions: (1) the loss of angular momentum of the
binary star system is mainly caused by the gravitational radiation; (2) the formation rate
of the binary star system is a constant in the evolutionary process of the universe; (3) the
distribution of the binary star system in the galaxy is consistent with the distribution in the
Galaxy. According to these assumptions, we can deduce the energy density of background
gravitational wave to be the following expression[9]:
ΩGW =1
ρc
REGW
6π〈r〉2 , (43)
here, R is the formation rate of binary star (about 0.01 a−1), EGW is the averaged energy
of gravitational radiation from an individual source, 〈r〉 is the averaged distance among the
wave sources. Substituting in reasonable modeling parameters, we can obtain the following
parameterized expression:
Ωgw(f) ≈ 4× 10−8
(R
(100 a)−1
)(M
M�
)5/3 (f
10−3 Hz
)2/3 (r
10 kpc
)−2
. (44)
Here, it is necessary to indicate that the second assumption mentioned above, i.e., the
loss of angular momentum of a binary star system is mainly caused by the gravitational
LI Long-biao et al. / Chinese Astronomy and Astrophysics 42 (2018) 487–526 515
radiation, is very possibly unreasonable, thus the practical gravitational wave background
is possibly smaller than that mentioned above.
3.4.1.2 Merging system of double compact stars with solar masses
Different from the gravitational radiation of double white dwarf stars, the stochastic
gravitational wave background produced by the merging systems of the double neutron stars
and of the double black holes with solar masses is commonly at higher frequencies, and the
peak value is commonly in the order of magnitude of 100 Hz, hence it is one of the main
detecting objects for the future ground-based gravitational wave detectors. In this aspect,
there has been a lot of work to perform the detailed estimation on the gravitational wave
background emitted by these systems. Here, we adopt the calculated result of a recent
work[34], in which the authors have considered the distribution functions of neutron stars
and black holes obtained from the up-to-date observations, and simultaneously the analytical
expression of the complete gravitational radiation (including the amplitude correction caused
by the post-Newton effect).
For this kind of systems the gravitational wave energy density can be calculated ac-
cording to the following formula:
ΩGW(f) =1
ρc
∫ zmax
zmin
N(z)
(1 + z)
(dEGW
d ln fr
)∣∣∣∣fr=f(1+z)
, (45)
here, N(z) is the number density of gravitational wave events at the redshift of z,dEGW
d ln fris the gravitational wave energy spectrum emitted by an individual source. By selecting
proper modeling parameters, this formula can be simplified as:
ΩGW(f) ≈ 9× 10−10
(r
1 Mpc−3 · Ma−1
)(〈M5/3
c 〉1M
5/3�
)(f
100 Hz
)2/3
, (46)
here, r is the averaged interval of gravitational wave events, Mc is the chirp mass of the
system. This result shows that: (1) at the frequencies lower than 100 Hz, the amplitude
of gravitational wave mainly depends on the occurrence rate of binary star merging event
and the distribution of chirp mass; (2) in this frequency range, the energy spectrum of
gravitational wave can be described very well by the power-law form ΩGW ∝ f2/3.
The gravitational wave event GW150914 detected recently by LIGO shows that the
binary black hole systems with larger masses exist broadly in the universe. The occurrence
rate of a binary black hole merging event similar to GW150914 is 16+38−13 Gpc−3 · a−1 in the
universe. According to this result, the working group of LIGO estimated the magnitude
of the stochastic gravitational wave background generated by the merging events of binary
black hole systems[35]: at the frequency f ≈ 25 Hz that most sensitive to the stochas-
tic background for AdvLIGO/AdvVirgo, the predicted amplitude of gravitational wave is
ΩGW(f = 25 Hz) = 1.1+2.7−0.9 × 10−9, which is much higher than the previous predictions.
According to this calculated result, and combining with the sensitivity curve of the gravita-
516 LI Long-biao et al. / Chinese Astronomy and Astrophysics 42 (2018) 487–526
tional wave detector, this background can be finally detected by AdvLIGO/AdvVirgo (see
Fig.5), without doubt this is an important discovery.
Fig. 5 a) Comparison of the stochastic gravitational wave background generated by the merge of double black
holes and the sensitivity curve of the AdvLIGO detector; b) the predicted relation between the signal-noise
ratio of the gravitational wave background and the observing time length for the detection of AdvLIGO[35]
In 2014, the cooperative group of LIGO and Virgo searched the stochastic gravitational
wave background in the recently observed data, but the signal of gravitational waves was not
discovered[36]. By assuming that the gravitational wave energy spectrum has the following
expression: ΩGW(f) = Ωα(f/fref)α, they made the constraint on the gravitational wave
amplitude with a confidence of 95%: in the frequency range of (41.5 ∼ 169.25) Hz, it is
assumed that α = 0, thus ΩGW(f) < 5.6× 10−6; in the frequency range of (170 ∼ 600) Hz,
ΩGW(f) < 1.8 × 10−4; in the frequency range of (600 ∼ 1 000) Hz, the constraint on the
gravitational waves is ΩGW < 0.14(f/900 Hz)3; and in the frequency range of (1 000 ∼1 726) Hz, ΩGW < 1.0(f/1 300 Hz)3. Furthermore, by analyzing the H1 and H2 data of
LIGO, and by using the correlation, the working group of LIGO[37] gave a more strict
constraint on the high-frequency gravitational waves of (460 ∼ 1 000) Hz, i.e., ΩGW <
0.14(f/900 Hz)3.
3.4.1.3 Merging system of super-massive double black holes
According to the hierarchical clustering theory of galaxies, a large-mass galaxy is pro-
duced by the continuous merge of small-mass galaxies. In the galactic merging process, the
central large-mass black hole attains first the hard state by through the dynamic friction,
then through some other ways (such as the interaction of gas, star scattering etc.), it steps
into the final parsec, and enters the stage dominated by the gravitational radiation to re-
lease the gravitational radiation. When there is no obvious explosive source, an incoherent
superposition of the gravitational radiation sources from different places will uniformly and
isotropically construct a stochastic gravitational wave background, and the corresponding
gravitational-wave frequency is commonly about 10−9 Hz, hence it is the main wave source
to be detected by the present pulsar timing arrays. For this kind of gravitational wave
sources, the energy density is commonly written as the following expression:
LI Long-biao et al. / Chinese Astronomy and Astrophysics 42 (2018) 487–526 517
ΩGW(f) =2π2
3H20
f2h2c(f) , (47)
here, hc is the characteristic intensity of gravitational wave, which can be commonly param-
eterized by the following power-law form: hc(f) = A(f/fa)α, in which A is the amplitude
of gravitational wave, α is the spectral index, fa = 1/a = 3.17 × 10−8 Hz. At present, the
strongest observational constraint for the gravitational waves in this frequency band comes
from the observational group of PPTA[38], when α = −2/3, the upper limit is obtained as
A < 1.0× 10−15 (see Fig.6).
Fig. 6 Comparisons between the upper limit of stochastic gravitational wave background given by PPTA
and the various theoretical predictions for the gravitational wave background generated by super-massive
double black holes[38]
We discuss below the properties of the theoretically predicted stochastic gravitational
wave background. The gravitational radiation observed on the Earth is the incoherent su-
perposition of the gravitational radiations of binary back hole mergers with different masses
at different redshifts, the characteristic intensity can be calculated by the following formula:
h2c =
∫ ∞
0
dz
∫ ∞
0
dMcd3N
dzdMcd ln frh2(fr) , (48)
in which fr = (1 + z)f is the frequency of gravitational wave after the redshift correction,
z is the redshift, N is the number of merging events, h(fr) is the radiation intensity after
518 LI Long-biao et al. / Chinese Astronomy and Astrophysics 42 (2018) 487–526
averaging in both direction and polarization for the sources with a chirp mass of Mc at the
luminosity distance of dL, and its expression is:
h(fr) =8π2/3M
5/3c√
10dL(z)f2/3r . (49)
It can be seen that the calculation of background radiation relies on the determination
of the number of black hole mergers in the redshift and mass space d3N/dzdMcd ln fr,
and its essence is to solve the merge and evolution of galaxies, the interactions between
the central black hole and surrounding stars, and between two black holes. Though at
present it is impossible to strictly deduce a detailed expression of this distribution from the
galactic formation theory, but according to the presently observed distribution function of
galactic mass, the galactic merging rate, and the mass relation between the central black
hole and host galaxy, in combination with the hierarchical clustering theory of galaxies, it
can be calculated that hc(f) really exhibits a power-law relation, and its spectral index is
α = −2/3, this is one of the most important features of the gravitational wave background.
But its amplitude estimation still has a considerable uncertainty, however most models show
an order of magnitude of A ≈ 10−15, hence, it seems that the present observations have made
already a rather strict constraint on a part of these models[38].
3.4.2 Cosmic primordial gravitational waves
Besides the gravitational wave background generated by the astrophysical processes as men-
tioned above, there is still a very important kind of stochastic gravitational wave background,
which is originated from the cosmic expansion and evolution processes, so that it is called
the cosmologically originated background gravitational waves. This kind of gravitational
wave source may also come from the different stages of cosmic evolution, including the pri-
mordial gravitational waves produced in the cosmic inflation stage, the gravitational waves
generated from the reheating process of the universe, the gravitational waves generated by
the process of phase change in the early universe, and the gravitational waves generated by
the motion and evolution processes of large-scale structures such as the cosmic string, etc.
In this section, we only pay attention to the most important part of primordial gravitational
waves, and this is the most definite kind of cosmic background gravitational wave source in
current models.
3.4.2.1 Inflationary cosmology
At present we know that before the standard thermal big bang, the universe experienced
a period of fast expansion, which is commonly called the inflation process. In this process,
the universe experienced an approximately exponential expansion. It is suggested commonly
by the model that it requires the cosmic scale factor to expand at least over 1020 times in a
very short time, while the cosmic horizon almost does not change in this period. Hence, the
fast cosmic expansion quickly push the original casually-related regions outside the horizon.
When it enters the period dominated by material objects, these regions slowly return to
LI Long-biao et al. / Chinese Astronomy and Astrophysics 42 (2018) 487–526 519
the inside of horizon, thus the various cosmological difficulties existed in the model of big
bang are naturally solved, such as the difficulty of horizon, the difficulty of uniformity, and
the difficulty of magnetic monopole etc. Moreover, the inflation has the early quantum
fluctuations pushed out of the horizon to become classical fluctuations, and therefore to
yield the original seeds of cosmic structures, thus the difficulty about the origin of cosmic
structures is also naturally answered, this is just the basic idea of the inflation theory.
There are mainly two kinds of the quantum fluctuations pushed out of the horizon in the
stage of inflation: i.e., the density fluctuation of scalar type and the gravitational wave
of tensorial type. The density fluctuation directly couples with matter, thus to provide
the initial condition for the formation of cosmic large-scale structures. Meanwhile, because
the interaction between gravitational waves and matter is very weak, in the propagating
process they propagate almost freely, and their evolutionary behaviors only depend on the
cosmic expansion behaviors in the different periods of the universe; hence by detecting the
primordial gravitational waves in different frequency bands, it is possible to deduce the
evolutions of the universe at the different stages (including the inflation stage), this is also
the most important scientific significance for detecting the primordial gravitational waves.
In the standard inflation model, the inflation process is realized through a scalar field.
The action of inflation field is[39]:
S =
∫d4x
√−gL =
∫d4x
√−g
(1
2∂μφ∂
μφ+ V (φ)
). (50)
We commonly assume that the inflation field is approximately uniform and isotropic, and
there is a very small disturbing component existed, i.e.
φ(x, t) = φ(t) + δφ(x, t) . (51)
The energy density and pressure of inflation filed are respectively:
ρφ = φ2/2 + V (φ) , pφ = φ2/2− V (φ) . (52)
When V (φ) � φ2, we have pφ −ρφ. From this we can see that the universe dominated by
a scalar field and with a potential energy much larger than kinetic energy is situated in the
de Sitter phase, i.e., we obtain an inflation process driven by the vacuum energy of scalar
field. At this moment, the kinetic equation of inflation field is:
φ+ 3Hφ+ Vφ(φ) = 0 , (53)
in which, Vφ ≡ dV/dφ. Due to the appearance of 3Hφ, we see that the cosmic expansion
impedes the rolling motion of the inflation field along the potential V (φ). In order to have
an enough long time of rolling motion, we require φ � 3Hφ, thus generally we call the
following two conditions as slowly rolling conditions:
520 LI Long-biao et al. / Chinese Astronomy and Astrophysics 42 (2018) 487–526
φ2 � V (φ) =⇒ V 2φ
V� H2 , φ � 3Hφ =⇒ Vφφ � H2 . (54)
When these two conditions are satisfied, the inflation field will roll down along the function
of potential energy, and this time interval is commonly called the slowly rolling. We can
define the slowly rolling parameters as:
ε ≡ − H
H2= 4πG
φ2
H2 1
16πG
(Vφ
V
)2
, η ≡ 1
8πG
(Vφφ
V
). (55)
Then, the corresponding slowly-rolling conditions can be expressed as:
ε � 1 , η � 1 . (56)
3.4.2.2 Primordial gravitational waves
We discuss below the primordial tensorial disturbance, i.e., the primordial spectrum of
cosmic residual gravitational waves. Under the universe of Friedmann-Robertson-Walker,
the metric of linear tensorial disturbance can be commonly written as:
ds2 = a2(τ)[−dτ2 + (δij + hij)dxidxj ] , (57)
in which, τ is the conformal time, its relation with the cosmic time is adτ = dt. hij is the
transverse traceless tensorial disturbance, and its power spectrum Pt(k) can be defined as:
〈hk,λ, h∗k′,λ〉 =
2π2
k3Pt(k)δ
3(k − k′) , (58)
here, hk represents the Fourier expansion coefficients of hij .
According to the Einstein-Hilbert action, we write the 2nd-order tensorial disturbance
as [39]:
S =1
8
∫a2[(h′ij)
2 − (∂lhij)2]dτd3x,
=1
2
∫d3k
∑λ
∫ [|v′k,λ|2 − (k2 − a′′
a|vk,λ|2)
]dτ , (59)
in which f ′ ≡ df/dτ , vk,λ ≡ ahk,λ/2. The kinetic equation of vk can be obtained by
quantization:
v′′k +
(k2 − a′′
a
)vk = 0 . (60)
In the meantime, the adiabatic condition is selected as:
vk → 1√2k
e−ikτ , (61)
LI Long-biao et al. / Chinese Astronomy and Astrophysics 42 (2018) 487–526 521
and we can obtain the Stewart-Lyth formula of tensorial disturbance:
Pt(k) 8
M2pl
H2
4π2
∣∣∣∣∣aH=k
8
M2pl
V
12π2
∣∣∣∣∣aH=k
, (62)
here, Mpl = 1/√8πG is the reduced Planck mass. This power spectrum can be commonly
parameterized to be the power-law form Pt(k) = At (k/k0)nt , and its spectral index is
nt = −2ε. In order to associate the magnitudes of scaler and tensorial disturbances, we
can define a new parameter: the tensor–scalar ratio r ≡ Pt(k0)
Ps(k0). It can be proven that
the tensor–scalar ratio predicted by the simple scalar field satisfies the consistency relation
r = 16ε = −8nt. Hence, by verifying this consistency relation, it is possible to make an
observational test for this big class of inflation models[40].
3.4.2.3 Evolution of Primordial Gravitational Waves
For the gravitational waves defined by Eq.(57), the equation of evolution is:
∂μ(√−g∂μhij) = −16πGπij , (63)
in which, the anisotropic part πij can be considered as the creating source of gravitational
waves, which may be provided by the free-particle flows (such as neutrinos etc.) in the
universe. However, the study shows that it only a little affects the gravitational waves in
the frequency band of 10−16 ∼ 10−10 Hz, thus this effect can be commonly neglected. In
the Fourier space, the evolution equation can be rewritten as:
h′′k + 2a′
ah′k + k2hk = 0 , (64)
here, ’ indicates the derivative for the conformal time τ . For a give wave number k, and at
a given time τ , we can define a transfer function tf to be:
tf (τ, k) ≡ hk(τ)/hk(τi) , (65)
in which, τi indicates the initial time of the thermal big bang. The strict expression of this
transfer function can be obtained by strictly solving Eq.(64). Here, we use its analytical
approximate expression. From the evolution equation of gravitational waves, we know that
for the gravitational wave with a given wave number k, when its wavelength is much larger
that the horizon, i.e., k � aH, the amplitude of gravitational wave keeps invariant; while
when its wavelength is much smaller than the horizon, i.e., k � aH, the amplitude of
gravitational wave is hk ∝ 1/a(τ), which decays with the expansion of the universe. In the
standard ΛCDM universe, after the inflation is ended, it is successively the period dominated
by radiation, the period dominated by matter, and the period dominated by cosmological
constant. In this model, the time derivative of the transfer function can be approximated
by the following expression[41]:
522 LI Long-biao et al. / Chinese Astronomy and Astrophysics 42 (2018) 487–526
tf (τ0, k) = −3j2(kτ0)Ωm
kτ0
√1 + 1.36
(k
keq
)+ 2.50
(k
keq
)2
, (66)
here, keq = 0.073Ωmh2Mpc−1, which indicates the wave number of the gravitational wave
that entered the cosmic horizon when the universe is at the time of radiation–matter equality,
τ0 = 1.41×104 Mpc is the conformal time of the present universe, and Ωm is the proportion
occupied by the present matter in the total cosmic energy. Of course, if the neutrinos in
the early universe are considered as free fluid, it can a little change the energy spectrum
of gravitational waves at the frequencies of 10−16 ∼ 10−10 Hz; moreover, the various phase
changes in the early universe (including the phase change of positive and negative electron
annihilation, QCD phase change etc.) can also a little change the gravitational waves at the
frequencies above 10−10 Hz.
The energy density ΩGW of gravitational waves can be expressed as:
OmegaGW(k) =Pt(k)
12H20
tf (τ0, k)
. In Fig.7 we show the dependence of the energy density of primordial gravitational waves on
the tensor–scalar ratio and spectral index. Because the primordial gravitational waves are a
kind of background gravitational wave source of full frequency range, thus in principle it is
possible to detect gravitational waves by using various different gravitational wave detectors.
However, from Fig.7, we can very easily find that currently, it is most possible to detect the
gravitational waves in the extremely low frequency band by observing the polarized signal
of cosmic microwave background radiation. At present, the best observational results are
taken from the BICEP2 and Keck Array telescopes, the upper limit of gravitational waves
obtained by them is r < 0.07[42]. It is predicted that in the future five years, through various
ground-base detectors it is possible to raise the upper limit to the level of r = 0.01.
4. SUMMARY AND PROSPECTION
Gravitational wave is one of the most important predictions of general relativity. In the past
one century, the theoretical study of gravitational waves has made a great progress in many
aspects, including the development of high-order post-Newtonian approximation theory, the
breakthrough of numerical relativity, and the establishment of cosmic disturbance theory,
etc. Meanwhile, people have also made a great effort in the aspect of gravitational wave
detection. The first experiment of direct detection of gravitational waves may be traced
to Weber in the 1950s, who designed the first detector of gravitational waves, i.e., Weber’s
rod, but the signals of gravitational waves have not been observed by it. In the subsequent
several ten years, there were five resonant rods completed in the world, though the sensitivity
of these resonant rods has been improved continuously, the signals of gravitational waves
were still not detected. Afterward, people first obtained the evidence for the existence of
LI Long-biao et al. / Chinese Astronomy and Astrophysics 42 (2018) 487–526 523
gravitational waves through observing the orbital decay of the pulsar PSR B1913+16, but it
was still an indirect detection. A breakthrough for the direct detection was made until 2016,
i.e., in 2016 February the LIGO Scientific Collaboration first announced that the signal of
gravitational waves GW150914 that produced by the merge of two black holes has been
detected, which indicates the establishment of a new field of gravitational wave astronomy.
Of course, it is necessary to detect gravitational waves simultaneously by multiple detectors
further in order to verify each other.
Fig. 7 The dependence of the energy density of primordial gravitational waves on the tensor–scalar ratio r
and spectral index nt, as well as the limitations of various detection methods[41]
The importance of gravitational waves relies on the one hand that it is the most impor-
tant prediction of general relativity, and its detection plays an irreplaceable role for studying
the wave nature and quantum nature of gravitational field. In the meantime, through grav-
itational waves, people can study the physics of strong gravitational field, to verify the
general relativity accurately in the strong gravitational field, as well as to distinguish the
different gravitational theories[1]. For instance, in the general relativity, gravitational waves
only have two polarized components, but in the revised gravitational theories there are at
most 6 independent polarized components, this provides an ideal way to verify the various
gravitational forces. As another example, the graviton is massless in the general relativity,
but in many revised gravitational theories the graviton is allowed to have a tiny mass, thus
the measurement on the upper limit of graviton mass can be used as well to distinguish the
different gravitational theories. On the other hand, because that gravitational waves are
mostly produced by the drastic motion of compact celestial bodies or by the evolution of the
early universe, and once a gravitational wave is produced, there is almost no any interaction
occurred during its propagation in the universe, thus it can bring with the clean signals of
524 LI Long-biao et al. / Chinese Astronomy and Astrophysics 42 (2018) 487–526
astrophysics and early universe, and provide a unique probe for human beings to study the
internal structure of neutron stars, the physics nearby the horizon of black holes, the explo-
sive process of supernovae, and the evolution of extremely early universe, etc. Furthermore,
the compact binary stars are considered as the gravitational wave sources, the luminosity
distance of the wave source can be determined accurately through the observation of the
amplitude and phase of gravitational wave, hence, in combination with the information of
redshift determined by the electromagnetic measure, this kind of gravitational wave sources
can be taken as the standard rings for exploring the history of cosmic expansion[43]. Based
on these reasons, the detection of gravitational radiation from various sources is always an
important subject of studies in the world.
In this paper, we have introduced the main kinds of gravitational wave sources existed
in the universe. (1) Two kinds of continuous gravitational wave sources, i.e., the rotating
neutron stars and stable binary star systems. In respect to the former, there is no such grav-
itational wave signal detected sofar, but according to the theoretical prediction, the ground-
based laser interferometers such as AdvLIGO etc. are possible to detect the gravitational
waves emitted from the new-born neutron stars or the neutron stars in the X-ray binary star
systems. While for the latter, including the gravitational radiations emitted from the binary
white dwarf systems, binary neutron star and black hole systems, and super-massive bina-
ry black hole systems, in which the gravitational radiations emitted from the binary white
dwarf systems are very possible to be detected by the space laser interferometers like LISA
etc., while the gravitational wave of the binary black hole system with a large mass-ratio is
one of the main targets to be detected by the space detectors like LISA etc. (2) Two kinds
of explosive gravitational wave sources, i.e., supernovae and binary star mergers. Though
the former is the candidate of a kind of strong gravitational wave sources, it is difficult to
expect to be detected really in the near future due to its low event rate. While, the latter
is the main detecting target for the present ground-based laser interferometers (AdvLIGO
etc.) and space laser interferometers (LISA etc.), and the presently observed gravitational
wave events GW150914, GW151226, and GW170104 are just the gravitational wave bursts
produced by the merge of double black holes. (3) Two kinds of main stochastic gravitational
wave backgrounds, i.e., the gravitational wave background produced by the astrophysical
processes and the primordial gravitational wave background produced in the inflation period
of the universe. In respect to the former, according to the present observational facts, the
predicted gravitational wave background produced by the compact binary stars with a solar
mass can be detected finally by the ground-based interferometers AdvLIGO and AdvVirgo
etc., while the gravitational wave background produced by the super-massive double black
holes will be observed by the pulsar timing arrays in the near future. And for the latter, it
is mainly detected by the polarized power spectrum of the cosmic microwave background,
the experiments in this aspect have developed very fast, and it is predicted that a break-
through will be made in the next 10 years. It is necessary to point out here that besides the
LI Long-biao et al. / Chinese Astronomy and Astrophysics 42 (2018) 487–526 525
gravitational wave sources discussed in this paper, there are many other gravitational wave
sources noteworthy, such as the gravitational wave produced by the R-mode instability of
neutron stars, the gravitational wave produced by the motion and collision of cosmic strings,
the gravitational wave produced by the various phase changes in the early universe, and the
quasi-normal mode of black holes etc. may also cause a rather strong gravitational wave to
be detected in the near future.
In a word, the present various detectors of gravitational waves in the world are joining
in the all kinds of observations. In the next several years we expect that some new break-
throughs will be obtained at least in the following aspects of gravitational wave detection. (1)
Through the 2nd-generation ground-based laser interferometers represented by AdvLIGO,
the gravitational waves produced by the merge of double neutron stars or neutron star-black
hole will be first observed, and it is possible to realize a double-channel detection of electro-
magnetic waves and gravitational waves for such kind of wave sources. In this aspect, the
space project of Einstein Probe in our country is predicted to participate this observation at
the X-ray band. Meanwhile, through the cooperation with the University of Western Aus-
tralia, a 3rd-generation ground-based gravitational wave observatory is prepared to build in
our country, i.e., the ground-based laser interferometer of 8 km×8 km, and it is predicted
to participate this kind of observations after it is completed. Especially, through the joint
observation of this detector with the other ground-base interferometers, the constraint on
the positions of wave sources will be improved greatly, which is very important for the s-
tudy on the gravitational wave sources. (2) Through the data accumulation of the pulsar
timing arrays like PPTA, EPTA, and NONAGrav etc., as well as the participation of the
new high-precision telescopes in our country such as FAST etc., the stochastic gravitational
wave background produced by the super-massive binary black hole systems will be detected
firstly. (3) Through the careful observations of ground-based detectors of the microwave
background radiation represented by BICEP2, the signal of cosmic primordial gravitational
waves will be first observed at the extremely low-frequency band. In this aspect, the project
of microwave background radiation in Ali of our country will make first the observation in
the northern sky area, and it is predictable to join in this discovery.
References
1 Will C. M., Theory and Experiment in Gravitational Physics, Cambridge: Cambridge University
Press, 1993, 1
2 Weinberg S., Cosmology, Oxford: Oxford University Press, 2008, 1
3 Maggiore M., Gravitational Waves Volumn 1: Theory and Experiments, Oxford: Oxford University
Press, 2008, 1
4 LIGO Scientific Collaboration and Virgo Collaboration, PRL, 2016, 116, 061102
5 LIGO Scientific Collaboration and Virgo Collaboration, PRL, 2016, 116, 241103
6 Misner C. W., Thorne K. S., Wheeler J. A., Gravitation, New York: W. H. Freeman and Company,
1973, 1
526 LI Long-biao et al. / Chinese Astronomy and Astrophysics 42 (2018) 487–526
7 Poisson E., Will C., Gravity, Cambridge: Cambridge University Press, 2014, 1
8 Carroll S. M., arXiv:gr-qc/9712019
9 Creighton J. D. E., Anderson W. G., Gravitational-Wave Physics and Astronomy, Singapore: Wiley-
VCH Verlag GmbH & Co. KGaA, 2011, 1
10 Lasky P. D., PASA, 2015, 32, 34
11 Horowitz, C. J., Kadau K., PRL, 2009, 102, 191102
12 LIGO Scientific Collaboration and Virgo Collaboration, ApJ, 2014, 785, 2
13 LIGO Scientific Collaboration and Virgo Collaboration, PRD, 2016, 94, 042002
14 Bildsten L., ApJ, 1989, 501, L89
15 Sathyaprakash B. S., Schutz B. F., Living Reviews in Relativity, 2009, 12, 2
16 Sesana A., PRL, 2016, 116, 231102
17 Barack L., Cutler C., PRD, 2004, 69, 082005
18 Jenet F. A., Lommen A., Larson S. L., Wen L., ApJ, 2004, 606, 799
19 Arzoumanian Z., et al., ApJ, 2014, 794, 2
20 Zhu X., et al., MNRAS, 2014, 444, 3709
21 Babak S., et al., MNRAS, 2015, 455, 1665
22 Schutz K., Ma C., MNRAS, 2016, 459, 1737
23 Hayama K., Kuroda T., Nakamura K., Yamada S., PRL, 2016, 116, 151102
24 Zhu X., Howell E., Blair D., MNRAS, 2010, 409, L132
25 LIGO Scientific Collaboration and Virgo Collaboration, arXiv:1605.01785
26 Martinez J. G., et al., ApJ, 2015, 812, 143
27 Lazarus P., et al., arXiv:1608.08211
28 Kuroda K., Ni W., Pan W., IJMPD, 2015, 24, 1530031
29 Abadie J., et al., Classical and Quantum Gravity, 2010, 27, 173001
30 LIGO Scientific Collaboration and Virgo Collaboration, PRD, 2016, 93, 122003
31 LIGO Scientific Collaboration and Virgo Collaboration, arXiv:1606.04856
32 Klein A., et al., PRD, 2016, 93, 024003
33 Roelofs G. H. A., Rau A., Marsh T. R., Steeghs D., Groot P. J., Nelemans G., ApJ, 2010, 711, L138
34 Zhu X., Howell E. J., Blair D. G., Zhu Z., MNRAS, 2013, 431, 882
35 LIGO Scientific Collaboration and Virgo Collaboration, arXiv:1602.03847
36 LIGO Scientific Collaboration and Virgo Collaboration, RPL, 2014, 113, 231101
37 LIGO Scientific Collaboration and Virgo Collaboration, PRD, 2015, 91, 022003
38 Shannon R. M., et al., Science, 2015, 349, 1522
39 Baumann D., arXiv:0907.5424
40 Zhao W., Huang Q. Classical and Quantum Gravity, 2011, 28, 235003
41 Liu X. J., Zhao W., Zhang Y., Zhu Z., PRD, 2016, 93, 024031
42 BICEP2 and Keck Array Collaboration, PRL, 2016, 116, 031302
43 Schutz B., Nature, 1986, 323, 310