Gravitational Waves and Gravitational-wave...

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� [email protected]

/ /$

0275-1062/18/$-see front matter © 2018 Elsevier B.V. All rights reserved.doi:10.1016/j.chinastron.2018.10.010

http://crossmark.crossref.org/dialog/?doi=10.1016/j.chinastron.2018.10.010&domain=pdf

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.


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,


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


(−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


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)


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


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)


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)


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


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


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.


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


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.


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


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:


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-


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


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


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


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


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�,


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


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.


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


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,


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


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-


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:


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


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


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:


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)


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]:


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


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


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


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.

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