学位論文

Development ofa High-Angular-Resolution Antenna
for Low-Frequency Gravitational-Wave
Observation(低周波重力波観測のための高角度分解能望遠鏡の開発)
東京大学大学院理学系研究科
物理学専攻
正田 亜八香平成 26 年 12 月

博士 (理学) 申請

Contents

1 Introduction

5

2 Gravitational Waves
2.1 Theory of the Gravitational Waves . . . . . . . . . . . . . . . . . . .
2.1.1 Geodesic Equation and Metric Tensor . . . . . . . . . . . . . .
2.1.2 Linearization of the Einstein Equation . . . . . . . . . . . . .
2.1.3 Monochromatic Plain-Wave Solution . . . . . . . . . . . . . .
2.1.4 Eﬀect on Free Masses . . . . . . . . . . . . . . . . . . . . . . .
2.1.5 Generation of Gravitational Waves . . . . . . . . . . . . . . .
2.2 Gravitational Wave Sources . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Compact Binary Coalescence . . . . . . . . . . . . . . . . . .
2.2.2 Rotating Stars . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3 Burst-like GWs . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.4 Stochastic Gravitational Wave Background . . . . . . . . . . .

9
9
9
9
11
12
13
14
14
16
17
17

3 TOBA
3.1 Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Response to the Gravitational Waves . . . . . . . . . . . . . . . . . .
3.2.1 Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Pattern function . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Scientific Targets . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Previous Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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22
24
25

4 Phase-II TOBA
4.1 Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Multi-Output System . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 Antenna Pattern Function . . . . . . . . . . . . . . . . . . . .
4.2.3 Detection Volume . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.4 Angular Resolution . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Design Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Suspension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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34
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Contents

Contents

4.4.1 Test Masses . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.2 Optical Bench . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.3 Intermediate Mass and Damping Mass . . . . . . . . . . . . .
4.5 Control System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.1 Control Loop . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.2 Photo Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.3 Laser Interferometers . . . . . . . . . . . . . . . . . . . . . . .
4.5.4 Coil-Magnet Actuators . . . . . . . . . . . . . . . . . . . . . .
4.6 Active Vibration Isolation Table . . . . . . . . . . . . . . . . . . . . .
4.6.1 Control System . . . . . . . . . . . . . . . . . . . . . . . . . .
4.7 Vacuum System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.8 Digital System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.9 Principle Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . .

37
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41
42
42
44
45
48
48
50
54
54
55

5 Experiments
5.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Calibration and Sensitivity . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Calibration from the voltage to the angular fluctuation . . . .
5.2.2 Calibration from the feedback signal to the out-of-loop signal .
5.2.3 Calibration to GW amplitude . . . . . . . . . . . . . . . . . .
5.2.4 Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Seismic Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.1 Passive Vibration Isolation . . . . . . . . . . . . . . . . . . . .
5.3.2 Seismic noise level . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Decoupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5 Common Mode Noise Reduction . . . . . . . . . . . . . . . . . . . . .
5.6 ADC/DAC Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.7 Sensor Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.7.1 Shot Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.7.2 Radiation Pressure Noise . . . . . . . . . . . . . . . . . . . . .
5.7.3 Intensity Noise . . . . . . . . . . . . . . . . . . . . . . . . . .
5.7.4 Phase Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.8 Performance of the AVIT . . . . . . . . . . . . . . . . . . . . . . . . .
5.8.1 Open Loop Transfer Function . . . . . . . . . . . . . . . . . .
5.8.2 Vibration Isolation Performance . . . . . . . . . . . . . . . . .
5.8.3 Sensitivity Improvement . . . . . . . . . . . . . . . . . . . . .

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Contents

Contents

6 Observation and Analysis
89
6.1 Intermediate-Mass Black Hole Binaries . . . . . . . . . . . . . . . . . 89
6.1.1 Observational Evidences of Intermediate-Mass Black Holes . . 89
6.1.2 Waveform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.1.3 Previous Results . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.2 Analysis Pipeline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.2.1 Segmentation and Calibration . . . . . . . . . . . . . . . . . . 93
6.2.2 Data Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.2.3 Matched Filtering . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.2.4 Signal Detection . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.2.5 Upper Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.3 Observation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.4 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.5.1 Data Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.5.2 Signal to Noise Ratio . . . . . . . . . . . . . . . . . . . . . . . 101
6.6 Observational Boundary . . . . . . . . . . . . . . . . . . . . . . . . . 101
7 Summary105
7.1 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 105
7.2 Discussion and Future Perspective . . . . . . . . . . . . . . . . . . . . 107
7.2.1 Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7.2.2 Multi-Output System . . . . . . . . . . . . . . . . . . . . . . . 108
7.2.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
A Decoupling matrices for Active Vibration Isolation System

111

B Q measurement

113

C Circuits

115

5Abstract
We have developed a high-angular-resolution gravitational detector for the realization of the low-frequency gravitational wave (GW) astronomy, and searched for GWs
from intermediate-mass black hole binaries.
The low-frequency GW is an important targets for the GW astronomy. For
example, GWs from black hole binaries and a stochastic GW background are predicted to exist in low frequencies such as 10−4 − 1 Hz. In order to observe thelow-frequency GWs, we have proposed a ground-based detector Torsion-Bar Antenna (TOBA). TOBA has two test mass bars which rotate due to the tidal force
from GWs. Its main feature is that it has sensitivity around 1 Hz even on the groundbecause the resonant frequencies in torsional mode is low enough for the bars to behave as free-falling masses. TOBA is expected to detect low-frequency GWs witha relatively low-cost and compact setup compared to the other low-frequency GW
detectors such as space-borne detectors. Previously, the first prototype with a single
test mass bar had been developed and set the first upper limit on a stochastic GW
background at 0.2 Hz.
In this thesis, we developed the Phase-II TOBA for a test of technologies forthe sensitivity improvements and for introducing a new observational method. The
new observational method we proposed is called a multi-output system. While
only the rotation of the test masses in the horizontal plane is monitored in theconventional configuration, we pointed out that the bars rotate also in the verticalplane. Consequently, the three independent signals are derived from a single detector
by monitoring the vertical rotation and the horizontal rotation at the same time.
This method improves the event rate of the detection, and the angular resolution of
the GW sources. We successfully derived the three signals at the same time using
Phase-II TOBA, and established the multi-output observation system.
Other than the multi-output system, we introduced the vibration isolation systems in order to reduce the seismic noise. One is passive vibration isolation withthe suspension system for the noise reduction at higher frequencies, and the other isactive vibration isolation using the six-legged table for the noise reduction at lower
frequencies. As a result, the sensitivity in the horizontal rotation is improved at 1 10 Hz due to the vibration isolation.
Using the Phase-II TOBA, we performed an observational run for continuous
24 hours and searched for mergers of intermediate-mass black hole (IMBH) binaries
with the mass of 200M⊙ as a demonstration. This is the first search for the IMBH

Contents

Contents

binaries heavier than 100M⊙ . As a result, no GW signals are detected and we haveset an observational upper limit that there is no IMBH binaries within 1.2 × 10−4
pc of the Earth.
Though the sensitivity of the Phase-II TOBA is not enough for the GW detection,
our technologies developed in this thesis are validated for the establishment of thelow-frequency GW astronomy.

2

Glossary
Acronym Definition
ADC
AVIT
BBN
CBC
CMB
DAC
DECIGO
FAR
FDR
GW
IMBH
LIGO
LISA
LS
OLTF
PS
RMS
SGWB
SMBH
TOBA
TTgauge
UGF

Analog to Digital Converter
Active Vibration Isolation Table
Big Bang Nucleosynthesis
Compact Binary Coalescence
Cosmic Microwave Background
Digital to Analog Converter
DECi-hertz Interferometer Gravitational wave Observatory
False Alarm Rate
False Dismissal Rate
Gravitational Wave
Intermediate-Mass Black Hole
Laser Interferometer Gravitational wave Observatory
Laser Interferometer Space Antenna
Laser Sensor
Open Loop Transfer Function
Photo Sensor
Root Mean Square
Stochastic Gravitational Wave Background
Super Massive Black Hole
TOrsion-Bar Antenna
Transverse-Traceless gauge
Unity Gain Frequency

3

1

ChapterIntroduction

Since Galileo Galilei had pointed a telescope at moon in 1609, we have observed
the Universe using the electro-magnetic waves. Various laws of nature are revealedby the observation of the Universe. However, some phenomena cannot be observedwith electro-magnetic waves, e.g., information inside the astronomical object, thecharacteristics of black holes, the Universe before the transparent to radiation, andso on. One of the possible method to observe such phenomena is the gravitational
waves (GWs).
GWs are distortion of the space and time propagating with the speed of light.
They are predicted by Albert Einstein in 1916 as the consequence of the general
theory of relativity. They are considered to be emitted from astronomical objects
with heavy mass moving at accelerated speed, such as neutron star binaries, blackhole binaries, supernovae, and so on. The presence of GWs is proven from thedecreasing orbital period of the pulsar. Hulse and Taylor had measured the orbital
period of PSR B1913+16 for about 30 years to verify that the kinetic energy loss
in the binary system matches the energy loss due to gravitational radiation [33,49]. They won the Nobel prize in 1993 because of this observation. However, thedistortion due to GWs are too small to be detected directly.
Nowadays – about 100 years later from the Einstein’s first prediction, the GWs
are about to be detected. The GW detectors have been widely studied since Weberclaimed that GWs are detected with a pair of resonant bar detectors in 1970 [55].
Though the direct detection of GWs have not been verified by the further experiments, the GW detectors drastically improve their sensitivity due to keen study bymany researchers all over the world. The current main stream of the GW detectors
is ground-based interferometric GW detectors with the arm length of 3 - 4 km. In
US, Large Interferometer Gravitational wave Observatory (LIGO) [51] is now under upgrading. Also in Europe, the large-scaled interferometer Virgo [8] is under
upgrading. In addition, the cryogenic interferometer KAGRA is now under construction in Kamioka, Japan [47]. These detectors will have sensitivity high enoughto have the expected event rate of ∼ 10/yr for the mergers of neutron star binaries.
51 Introduction

The observation frequency band of the ground-based interferometers are around
100 Hz. Therefore the their main observational targets are the neutron star binary
coalescence and supernovae. Observation of the neutron star binaries will reveal thestate equation of the neutron stars, the mechanism of the gamma-ray burst, andso on. GWs from the supernovae are expected to contain information about theircore which would help to unravel the mechanism of the explosion. While the large
scientific impacts are expected from the observational results of the ground-based
interferometers, it is necessary to observe GWs in various frequencies in order to
evolve astronomy using GWs. For example, mergers of black holes are expected to beobserved in low frequencies, such as mHz to 1 Hz. The space-borne interferometers,such as Laser Interferometer Space Antenna (LISA) [53] and DECI-herz Gravitational Observatory (DECIGO) [37], are planned to be launched for low-frequency
GW observation.
However, since the space GW antennas are large project such that three space
crafts construct large interferometers in space. They require many advanced technologies, but also they have diﬃculties in repairing, upgrading and so on. In orderto get rid of these problems, Torsion-bar Antenna (TOBA) had been proposed [10].
The TOBA is a GW detector composed of two bar-shaped test masses which rotate
diﬀerentially due to the tidal force from GWs. The main feature of TOBA is that
it is sensitive to GWs at lower frequencies such as around 1 Hz even on the ground.
Low resonant frequencies of the torsional mode realize the free-falling test masses
around 1 Hz, i.e, the sensitive test masses to GWs. One of our observational targets
is GW from a merger of the intermediate-mass black hole (IMBH) binary. IMBH isrecently found in globular clusters and considered to be a seed of super-massive black
holes (SMBHs). Observation of the IMBH coalescence is important for the theoryof the SMBH evolution, and the galaxy evolution. Another target is a stochastic
GW background. While the observation of a stochastic GW background with thecosmic micro-wave background (CMB) is discussed, it is also important to observe
the background in various frequencies. The frequency dependence of the energyof the background has the information of the history of the Universe, such as there-heating temperature.
For the low-frequency GW detection, the first prototype of the TOBA had beendeveloped previously [36]. It has a single test mass bar suspended by the pinningeﬀect of a superconductor. They performed the principle test and the first observation [36, 46]. In this thesis, we have developed the upgraded TOBA, which iscalled ‘Phase-II TOBA’ here. It is a new detector for the development of vibration
isolation technologies, and for the principle test of the multi-output system. Here,
we developed the passive and active vibration isolation system for the TOBA and6

1 Introduction

successfully reduced the seismic noise. Also, we established the multi-output system,
which derives three independent signals from a single detector. The multi-output
system improves the event rate and the angular resolution of the detector. Also, we
performed the observation for continuous 24 hours. Using these data, 200M⊙ IMBH
binaries are searched for for the first time. Consequently, the GW signals are notdetected and the have set an upper limit on the event rate of IMBH mergers.
In our thesis, the characteristics of GWs and the principle of the TOBA are
explained at first (Chapter 2, and 3). Subsequently, the concept of the Phase-II
TOBA, including the principle of the muti-output system and the detector design,is discussed in Chapter 4. The measured characteristics of the constructed detectors
are shown in Chapter 5. In Chapter 6, we performed the observational run andsearched for IMBH binaries as a part of an end-to-end test.

7

2

ChapterGravitational Waves
In this section, we briefly review the theory of GWs. We derive the GWs from bylinearizing the Einstein equation. Subsequently, we explain how GWs are emitted
from the astronomical sources, their characteristics, and how they aﬀect on freemasses.

2.1

Theory of the Gravitational Waves

2.1.1

Geodesic Equation and Metric Tensor

According to the General Theory of Relativity, the infinitesimal distance between
two points in the space time xµ and xµ + dxµ is
ds2 = gµν dxµ dxν ,(2.1)

where gµν is the metric tensor which represents the structure of the space time. The
motion of the free masses which do not receive the external forces except gravity
can be derived by the Geodesic Equation:µd2 xλ
λ dx dxν
Γ
=
.
µν
dτ 2
dτ dτ
Here, τ is the proper time and Γλµν is the Christoﬀel symbol:()
1 γα ∂gαµ ∂gαν
∂gµν
λ
+
−
.
Γµν = g
2
∂xnu
∂xµ
∂xα

(2.2)

(2.3)

Therefore, the motion of the free mass depends on gµν .

2.1.2

Linearization of the Einstein Equationgµν obeys the Einstein equation:
18πGRµν − gµν R = 4 Tµν ,
2
c
9

(2.4)

2 Gravitational Waves

2.1 Theory of the Gravitational Waves

where Rµν , R, and Tµν are the Ricci tensor, the Ricci scalar, and the Energy momentum tensor tensor, respectively. c is the speed of the light and G is the gravity
constant. Rµν and R are derived fromαRµν = Rµαν
R = g µν Rµν ,

(2.5)
(2.6)

γ
where Rµρν
is the Riemann curvature tensor

∂Γγµν
∂Γγµρ
−
+ Γγαρ Γαµν − Γγβν Γβµρ .
(2.7)
ρ
ν
∂x
∂x
When the perturbation hµν is added to the flat space, i.e. the metric tensor
is written by gµν = ηµν + hµν , where ηµν is the metric tensor of the Minkowski
space, Christoﬀel symbol, Riemann curvature tensor, Ricci curvature tensor, andRicci scalar are()
1 γα ∂hαµ ∂hαν
∂hµν
λ
Γµν ∼
η
+
−
(2.8)
2
∂xν
∂xµ
∂xα
(
)
1 γα ∂ 2 hαν∂ 2 hµν∂ 2 hµρ∂ 2 hαβγ
Rµρν ∼
η
−
+
− µ ν
(2.9)
2
∂xρ ∂xµ ∂xα ∂xρ ∂xα ∂xν
∂x ∂x
(
)
∂ 2 hλµ1∂ 2 hλν∂ 2 hµν
∂ 2h
Rµν ∼
−
+
− µ ν
(2.10)
2 ∂xλ ∂xµ ∂xλ ∂xλ ∂xλ ∂xν
∂x ∂x
γ
Rµρν
=

∂ 2 hµν
∂ 2h
R ∼
−
.
∂xµ ∂xν∂xα ∂xα
Here, h ≡ hλλ .
Taking Lorentz Gauge(2.11)

µ

∂hν
= 0,
∂xµ
where the trace reverse tensor hµν is defined as1hµν ≡ hµν − ηµν h,2
we obtain the linearized Einstein equation∂ 2 hµν
16πG= 2hµν = − 4 Tµν .
α∂x ∂xαc
In the vacuum, i.e. Tµν = 0,2hµν = 0,(2.12)(2.13)(2.14)(2.15), i.e., the wave equation is derived. hµν propagating in the space time is the GWs.
10

2 Gravitational Waves

2.1.3

2.1 Theory of the Gravitational Waves

Monochromatic Plain-Wave Solution

The monochromatic plain-wave solution of 2.15 is given asαhµν = Aµν eikα x .

(2.16)

Substituting 2.16 to 2.12 and 2.15,
Aµν k µ = 0kµ k µ = 0(2.17)(2.18)are derived.
Here, we take the transverse-traceless gauge (TT gauge),
Aαα = 0,Aµν U ν = 0,(2.19)(2.20)

where U ν is an arbitrary time vector.
Setting U ν = δ0ν , which is a unit vector of time in the Minkowski space,Aµν δ0ν = Aµ0 = 0.

(2.21)

Also, Considering the GWs with the angular vibration ω propagating along thez-axis,Aµ3 = 0.
(2.22)
Then, Aµν is derived as0 00 0 h+ h×Aµν =  0 h× −h+
0 0
0


0
0 
.
0 
0

(2.23)

Here, h+ and h× represent two polarizations of the GWs.
Consequently, we derive the generalized solution of GWs. Defining polarization
tensorse+µν (n̂) ≡ ûµ ûν − v̂ µ v̂ µ×eµν (n̂) ≡ ûµ v̂ ν + v̂ µ ûν ,11(2.24)(2.25)

2 Gravitational Waves

2.1 Theory of the Gravitational Waves

where n̂ is a unit vector parallel to the direction in which GWs propagate, and û
and v̂ are unit vectors orthogonal to n̂ and each other, general solution of GWs isderived as∑−iω(t−n̂ẋ/c)hµν =hA (n̂)eA.
(2.26)µν (n̂)e
A=+,×

Here x is a position vector of an observation point. The GWs observed at x in t is
the superposition of GWs coming from all directions with all frequencies:∫∑ ∫ +∞−iω(t−n̂ẋ/c)hij (t, x) =dω d2 n̂hA (f, n̂)eA.
(2.27)
ij (n)e
A=+,×

2.1.4

−∞

Eﬀect on Free Masses

Here, we consider about the eﬀect of the GWs on the free-falling masses in the small
region around the coordinate x, which is the laboratory system called the proper
detector frame.
The Geodetic Equation at x + ξ, a point apart from x by inifinitestimal distanceξ isd2 (xµ + ξ µ )d(xν + ξ ν ) d(xρ + ξ ρ )
µ
=
−Γ
(x
+
ξ)
.
(2.28)
νρ
dτ 2
dτ
dτ
Taking only the first order of ξ,∂Γµνρd2 ξ µdxµ dξ ρdxµ dxρ
µ
σ
=
−2Γ
(x)
−
ξ
(x)
.
νρ
dτ 2
dτ dτ
dxσ
dτ dτ

(2.29)

Since the metric tensor can be approximated as gµν ∼ ηµν , and the velocity of thei0observer is dx≪ dx= c, the geodetic equation in the proper detector frame isdτdτ
derived from 2.7, 2.9, and 2.29 as follows:µd2 ξ µ2 ∂Γ00 σ
ξ
= −c
dτ 2
∂xσ
µ
= −c2 R0σ0ξσ
1 ∂ 2 hµσ σ
=
ξ .
2 ∂t2

(2.30)It means that the tidal force from GWs
Fµ =

m i j
ḧ ξ
2 j

is applied to the free mass with the mass of m.
12(2.31)

2 Gravitational Waves

2.1 Theory of the Gravitational Waves

Solving 2.30 with the condition that the answer does not get infinity at t → ∞,we get1
δξ i = hij ξ j .
(2.32)
2
When GW with angular frequency of ω is incoming from x axis,( x ))( x )(
1 h+ h×δξξ=eiω(t−z/c)(2.33)yδξξy2 h× −h+
( x )
( y )
1
1
ξ
ξ
iω(t−z/c)eeiω(t−z/c)
h+
=
+ h×
(2.34)yx−ξξ22
is obtained. The first term and the second term represents the two polarization of
GWs, + mode and × mode. The motion of the free masses caused by GWs witheach polarization is pictured in Fig.2.1.
0

π/2

3π/2

2π

h+

hx

Figure 2.1: A picture of the two modes of GWs.

2.1.5

Generation of Gravitational Waves

Considering about the GWs coming from a small source at a distant place, theamplitude of GWs is derived as below:
(
→
− →)
→
x − x′ | −∫ Tij t − |−
, x′
c
4G
hij (t, x) = 4
d3 x′ .
(2.35)−→′−→c|x − x |13

2 Gravitational Waves

2.2 Gravitational Wave Sources

In case of the GW source (the region where Tij ̸= 0) is small enough compared withthe wavelength of GWs, and the observer is far enough from the GW source, theequation above is approximated byhij (x) =2G
Q̈ij (t′ ),c4 r(2.36)−→−
where r ≡ |→x − x′ |, t′ ≡ t − r/c, and Qij is a quadrupole moment defined by
(
)
∫
−
→′ ′
1
′i ′j
′ ′
(2.37)Qij (t) = ρ( x , t ) xy xj − δij x x d3 x′ .
3Here, ρ is the density of the GW source.

2.2

Gravitational Wave Sources

GWs are considered to be emitted from various phenomena, e.g., binary coalescence,rotation of asymmetric stars, supernovae, and cosmic strings. In this section, we
will briefly review the astronomical and cosmological sources of GWs and the recent
observational results.

2.2.1

Compact Binary Coalescence

The binary systems made of compact stars, such as black holes and neutron stars,
are promising sources of gravitational waves. PSR B1913+16, called Hulse-Taylor
binary pulsar, is the neutron star binary system is the one of the strong evidenceof the existence of GWs. Hulse and Taylor measured its parameters with highprecision, such as the reduction of its orbital period, to prove their motion obeys
the General Theory of Relativity very well [33, 49].
Define the mass of the binary stars as m1 and m2 , the orbital radius as R, theamplitude of GW is derived as follows:()5/3 ()2/3
4 GMc
πfgw (t)1 + cos2 θh+ (t) =rc2
c2(2.38)∫
cos( 2πfgw (t)dt · tret + ϕ),
4h× (t) =
r

(

GMc
c2

)5/3 (

πfgw (t)c)2/3∫cos θ sin(14

2πfgw (t)dt · tret + ϕ),(2.39)

2 Gravitational Waves

2.2 Gravitational Wave Sources

where θ is the angle between the normal to the orbit and the line-of sight, and(m1 m2 )3/5,
(m1 + m2 )1/5
G(m1 + m2 )fgw (t) =.
πR(t)3(2.40)

Mc =(2.41)

Here, Mc is called the chirp mass.
Since the radiated energy due to GWs is the sum of the kinetic energy of thepotential energy, fgw (t) is derived as1
fgw (t) =π(

51
256 tcoal − t)3/8 (

GMc
c3

)
,

(2.42)

where tcoal is the time to coalesce. This waveform is called chirp, as shown in Fig.2.2.

Figure 2.2: The chirp signal of GWs from 1000 - 1000 M⊙ binary system. t = 0 is
the time to coalesce.
However, the equations above are valid only when the background space-tune
can be assumed to be flat. When the rotating stars are close and held together
15

2 Gravitational Waves

2.2 Gravitational Wave Sources

by gravitational force, i.e., the two stars are near to merge, the assumption thatthe velocity of stars and the space-time curvature are independent is not valid. Inorder to deal with such relativistic systems, a post-Newtonian (PN) formalism isintroduced. Please refer [39] for the details of the formalism of PN waveforms.
GWs from binary systems, especially double neutron stars or neutron-star-blackhole binaries are searched by various GW detectors. For example, LIGO and Virgo
have recently searched GWs associated with γ-ray bursts, which is believed to be
emitted from mergers of compact astronomical objects, and have set a lower bounds
on the distance for various GW emission models [3]. As a future perspectives, advanced LIGO and Virgo are predicted to detect GWs from such compact binary
coalescences (CBCs) about 40 times per year [6]. Other than neutron star binaries, more massive objects, i.e., black hole binaries are searched by ground-based
detectors, doppler tracking, and pulsar timing.

2.2.2

Rotating Stars

Single rotating compact stars such as neutron stars also emit GWs when the shapeof the star is asymmetric. Considering a star rotating with the angular velocity ωrot ,a emitted GW is a monochromatic signal written as2t 1 + cos2 θ
4π 2 G Ifgw
ϵcos(2πfgw t),
c4
r
2
2
t
4π 2 G Ifgw=
ϵ cos θ sin(2πfgw t),
4
c
r

h+ =

(2.43)

h×(2.44)(2.45)

where r, I, and ϵ are the distance to the neutron star, moment of inertia aroundrotating axis, and ellipticity of the star respectively. Inserting numerical reference
values, the equation gives [39])()()2( ϵ )(
I
10kpc
fgw
−27
h ∼ 3.6 × 10.
(2.46)
10−5
r
60Hz
1038 kg m2
The Crab pulsar is one of the promising objects which rotate with the frequencyof 60.2 Hz. Such monochromatic GW signals have been searched for using the resonant GW detectors [64]. Later, the interferometric detectors searched for continuous
GW signals from the Crab pulsar and set an upper limit about a factor of two below
the spin-down limit, which is the amplitude of GWs calculated with the assumptionthat all the rotational kinetic energy loss by spin-down is converted to GWs [52].
16

2 Gravitational Waves

2.2.3

2.2 Gravitational Wave Sources

Burst-like GWs

Here, we briefly introduce GW sources which emit burst-like GWs. One of the representative burst-like GW sources is supernova. Supernovae emit GWs when their corecollapse. Though their waveforms are unknown yet, they are numerically simulated
based on several models of explosion [41]. Other than supernovae, pulsar glitches1 ,gamma-ray bursts, and soft gamma repeaters2 are candidates of GW sources.
LIGO and Virgo have searched for burst-like signals without any assumptions on
waveform and set an upper limit on the event rate as 1.3 event per year [4]. In the
future, advanced LIGO is expected to be able to detect the GWs from supernovae
occurred anywhere in our galaxy [57].

2.2.4

Stochastic Gravitational Wave BackgroundA stochastic GW background (SGWB) is a cosmologically interesting target which
has the information about the history of the Universe. It is a superposition of
the GWs originated in the cosmological phenomena such as the fluctuations of thespace-time in the early Universe and the cosmic string, and the GWs from unresolved
astronomical objects. The amplitude of a SGWB of a cosmological origin is larger
at lower frequency when Ωgw (f ), which is the energy per unit logarithmic intervalof frequency, does not have frequency dependency, while the waveform of a SGWB
is considered to be Gaussian, and unpredictable.
A SGWB has been searched using various detectors in various frequencies. Various telescopes, e.g., Planck, WMAP, and BICEP, are eagerly searching for a SGWB
using the B-mode polarization of CMB. These telescopes can observe a SGWB at
extremely low frequencies such as 10−18 − 10−16 Hz, whose wavelength is order ofthe size of the Universe. At very low frequencies, such as 10−10 − 10−8 Hz, Pulsar
Timing Array have set upper limits on a SGWB [40]. Above those frequencies, anupper limit has been set using the Doppler Tracking measurement [12]. At higher
frequencies, the interferometers, LIGO and Virgo, have set an upper limit around
100 Hz by the simultaneous observation [7]. Other than the large-scaled interferometers, 75-cm baseline synchronous recycling interferometers have set an upper
limit at high frequency of 100 MHz [9]. Also the cryogenic resonant bar detectors, EXPLORE and NAUTILUS set upper limits by simultaneous observation [13].
Around 1 Hz, a single, and a pair of TOBA had set the upper limit around 0.2 Hz
1Pulsar glitch is a sudden increase in the rotational frequency of a pulsar.
Soft gamma repeaters are astronomical objects which emit gamma-rays and X-rays irregularly.
They are inferred to be magnetars or neutron stars
217

2 Gravitational Waves

2.2 Gravitational Wave Sources[36, 46]. Though other ground-based detectors do not have sensitivity around thesefrequencies, several upper limits have been placed recently, using Global Positioning
System (GPS) [11], the seismic waves of the moon [22], that of the Earth [23], andthe Earth as the resonant detector [24]. Note that the search methods using seismic
sensors [22, 23, 24] are diﬃcult to improve the sensitivity since the motion of theirtest masses, which are the Earth and moon, cannot be suppressed.

18

3

Chapter
TOBA
In this chapter, we will explain the merit and demerit of TOBA and its principle.
Also, we will refer to the example configuration of the large-scaled TOBA and aroad map to the ultimate detector.

3.1

Concept

The conventional GW detectors are sensitive to GWs only above about 10 Hz because of the seismic noise. In order to observe low-frequency GWs, space-borne GW
detectors such as LISA and DECIGO are planned to be launched. However, thespace detectors have diﬃculties in their cost, repair, and upgrade while they havepotential for highly sensitive GW detectors because of its long baseline, escape fromthe seismic noise and the eﬀect of the gravity, and so on.
The TOBA is now under development as a novel low-frequency GW detector. Its
main feature is to have sensitivity to GWs at low frequencies such as 1 Hz even on
the ground. The rotational motion of the test mass is monitored in the TOBA while
the pendulum motion is observed in the interferometers. The test masses behave as
free masses even at low frequencies, i.e., it is sensitive to GWs, since the resonant
frequency of the rotational mode is as low as a few mHz. Also, the configurationof TOBA is relatively simple, compact and low-cost, while it is diﬃcult to achieve
as good sensitivity as interferometers since the length of the baseline is limited.
Therefore, TOBA is expected to observe low-frequency GWs before space-borne
GW detectors are launched and to accelerate the evolution of the GW astronomy
by feeding back its scientific results to the design of space GW detectors.
19

3 TOBA

3.2 Response to the Gravitational Wavesz*:V K[
WHVWPDVVHVyx

Figure 3.1: The conceptual picture of the TOBA.

3.2

Response to the Gravitational Waves

3.2.1

Equation of Motion

The Fig.3.1 is the schematic view of the TOBA. In this section, we will explain how
the test masses of the TOBA react according to the GWs. We assume that the twobars are set along x axis and y axis respectively, and suspended at their center sothat they rotate in x − y plane.
The equation of the motion in the rotational mode isI θ̈(t) + γ θ̇(t) + κθ(t) = Fgw ,(3.1)

where θ, I, γ, and κ are angular fluctuation around z axis, the moment of inertia
around z axis, the damping constant and the spring constant respectively.
Fgw is derived by calculating the diﬀerential of energy stored in the test mass.
The tidal force from GWs on the infinitesimal volume dV of the bar isρF i dV = ḧij ξ j dV
(3.2)
2
from 2.31, where ρ and ξ are the density of the test mass and the position of theinfinitesimal volume dV . Since the test mass stores the energy
∫ ∫
U = −
F 2 dV dξi
(3.3)
∫
ḧij
ρξ i ξ j dV,
(3.4)
=
4
20

3 TOBA

3.2 Response to the Gravitational Waves

the tidal force Fgw is derived as
Fgw = −

∂U1
= ḧij q ij .
∂θ
4

(3.5)

Here, θ is the angular fluctuation around z axis and q ij is the quadrupole momentof the test mass defined as)∫ (2 ij kij
i jj i
q = ρ ξ w + ξ w − δ ξ wk dV.
(3.6)
3
wi is the mode function, which in case of the rotation mode is written as
 
−y
i

x .
w =
0
Substituting 3.7 into 3.6, the quadrupole moments are derived as follows:∫q+ = q11 = −q12 = ρ2xydV,∫()
q× = q12 = q21 = − ρ x2 − y 2 dV.
Assuming that a GW with the amplitude ofh+ h× 0h(t) = h× −h+ 0
0
0
0

(3.7)

(3.8)
(3.9)

(3.10)

is incoming into the torsion pendulum, the equation of motion in frequency domain
is written as1− ω 2 I θ̃(ω) + iωγ θ̃(ω) + κ(1 + iϕ)θ̃(ω) = − ω 2 (h+ q+ + h× q× ).
2
Therefore, the frequency response of the angular fluctuation to the GW is∑θ̃(ω) =
HA (ω)h̃A ,(3.11)(3.12)

A=+,×

HA (ω) =qA
ω2,
2I ω 2 − ω02 (1 + iϕ) + iω/I
21

(3.13)

3 TOBA

3.2 Response to the Gravitational Waves√
where ω0 = 2πf0 = |κ|/I is the resonant frequency of the torsional mode.
Considering uniform rectangular test mass,q+ = 01
1
M (l2 − r2 ) ∼ M l2 ,
q× =
12
6
1
1
I =
M (r2 + l2 ) ∼ M l2 ,1212(3.14)(3.15)(3.16)

where M, l, and r are the mass, the length of the longer side and the shorter side of
the test mass, respectively. Substituting the equations above into 3.13, the transfer
functions from GWs to the angular fluctuations are
H+ (ω) = 0
1
ω2
H× (ω) =,2 ω 2 − ω02 (1 + iϕ) + iω/I

(3.17)(3.18)

which indicates that the TOBA is sensitive only to the GWs with ×-mode.
The other orthogonal test mass also rotates similarly, but in the opposite direction. Therefore, the common mode noise between the two test masses can berejected by taking the diﬀerential θ1 − θ2 .
Also, the absolute value of H× vs the frequency is plotted in Fig.3.2. It shows thatH× ∼ 1/2 is larger at higher frequencies than the resonant frequency while it getssmall below the resonant frequency. Around the resonant frequencies, the sensitivity
is better when the Q-value is better. The conventional resonance detectors searches
GWs around their resonant frequencies while their resonant frequencies are as highas 300 Hz. In case of TOBA, the observation is performed at higher frequencies thanf0 as interferometric GW detectors so that broadband observation is enabled. The
main feature of TOBA is that f0 is lower than that of interferometers. Since theinterferometers detect GWs using the pendulum motion of the test masses, whose
resonant frequency is about 1 Hz, the observation band of interferometers is around
100 Hz. On the other hand, the observation band of the TOBA is around 1 Hz evenon the ground since the resonant frequencies of torsional mode are as low as a few
MHz.

3.2.2

Pattern function

The pattern function is the detector response function of the incident angle of GWs
to the detector. Considering a GW hij (t) propagating along n = (1, Θ, Φ) in the22

3 TOBA

3.2 Response to the Gravitational Waves

Figure 3.2: The transfer function from GWs to the angular fluctuation of the test
mass. The resonant frequency f0 and Q-value, which is defined by Q = ω0 I/γ, are
set to be 0.1 Hz and 10, respectively. While the TOBA do not respond to GWs
at lower frequencies than the resonant frequency, it has a flat response at higher
frequencies than the resonant frequency.
polar coordinate, the amplitude of the GW h(t) is
h(t) =∑F A (n)hA (t),(3.19)

A=+,×

with F A (n) is the pattern function defined as
F A (n) ≡ Dij eA
ij (n).

(3.20)

ij
is called theHere, eAij (n) is the polarization tensors defined as shown in 2.24 D
detector tensor which depends on the detector geometry. In case of the TOBA set
as pictured in Fig.3.1, Dij is1
Dij = (nx i ny j + ny i nx j ) = nx i ny j ,223(3.21)

3 TOBA

3.3 Scientific Targets

where nx and ny are the unit vectors pointing toward the x and y-direction. Substituting 2.24 and 3.21 into 3.20, the pattern function is obtained as below:
1 + cos2 Θ
sin 2Φ cos 2Ψ + cos 2Φ cos Θ sin 2Ψ,2
1 + cos2 Θ
sin 2Φ sin 2Ψ + cos 2Φ cos Θ,
F × (Θ, Φ, Ψ) =2

F + (Θ, Φ, Ψ) = −(3.22)(3.23)

where Ψ is the polarization angle, which is the rotation angle of GW polarization in
the transverse plane. Please refer [40] for more details. This pattern function is assame as interferometric detectors.

3.3

Scientific Targets

Sensitivity [1/Hz

1/2]

Using the principle mentioned above, the TOBA is expected to detect low-frequency
GWs, such as IMBHs and a stochastic GW background. In this section, we will
explain about the example sensitivity required for the GW detection.
The example sensitivity for the GW detection is shown in Fig.3.3. It has two
10-m scaled test mass bars. Their angular fluctuations are read using Fabry-Perotcavity in order to reduce the shot noise of the sensor. Also, the suspension thermal
noise should be reduced by cryogenic system.
10

-14

10

-16

10

-18

10

-20

10

Las
Gravity-gradientenoise
pre r Radssu i are tiono nSuisethespenrm si oShot noise
al n
no
ise
Bar th erm
al-3

Seismic noise

10

-2

10

-1

10

0

Frequency [Hz]

10

1

10

2

Figure 3.3: The example sensitivity of the ultimate TOBA. It has two bar-shaped
test mass bars with a length of 10 m. The resonant frequency of the torsional modeis 1 mHz, and the test masses are suspended by the system with loss angle ϕ of
10−7 .
Fig.3.4 is the simulated observable distance for a merger of equal-mass binary
system. It shows that the Ultimate TOBA will observe a merger of 105 M· black24

3 TOBA

3.4 Previous Results

Luminosity
Distance [Mpc]

holes at 10 Gpc with a signal to noise ratio (SNR) of 5. The observation of suchintermediate-mass black holes will reveal how the super massive black holes evolved
and how our galaxies evolved.
PRL 105, 161101 (2010)
10

4

10

3

10

2
1

R
SN 3

10 2
10

5

0
5 8 12 2

10

4

10

6

Total Mass of Black holes [M䖭]

108Figure 3.4: Simulated luminosity distance which ultimate TOBA will reach for amerger of equal-mass black holes with a each SNR.
Also, it will be possible to set an upper limit on a stochastic GW background.
(See Fig.3.6.) Performing one-year simultaneous observation with the ultimate TOBAs, the upper limit will exceed the upper limit derived from the theory of the Big
Bang Nucleosynthesis (BBN).

3.4

Previous Results

In order to reach the final sensitivity as shown in Fig.3.3, the detector should be
developed step by step.
As a first step, the first prototype has been developed [36, 35]. It has a single
test mass with the length of 20 cm. The main feature is that the test mass ismagnetically levitated by the pinning eﬀect of a superconductor, which realize thelow-damping and low-loss suspension system.
The achieved sensitivity is shown in Fig.3.5. The sensitivity is limited by magnetic coupling eﬀect caused by magnetic suspension system at low frequencies, andby seismic coupling noise at higher frequencies.
The principle test and the first observation are performed with this prototype.
The first upper limit on a stochastic GW background at 0.2 Hz has been set withthe single prototype TOBA [36]. Also, the upper limit is updated by a simultaneous
observation with two prototypes set at Tokyo and Kyoto in Japan [46]. Figure 3.6
shows the upper limits obtained using the TOBAs and other detectors.

25

GW strain-equivalent noise level [Hz-1/2]

3 TOBA

3.4 Previous Results

10

-5

10

-6

10

-7

10

-8

10

-9

-10

10

-11

10

0.01

Magnetic noise

Current noise level
Seismic noise
Sensor noise
Thermal noise
0.1

Electronics noise
1

Frequency [Hz]10

Figure 3.5: The sensitivity of the first prototype TOBA. The total sensitivity (redline) is limited by the seismic noise (the blue line) and the magnetic coupling noise(the green line).

26

3 TOBA

3.4 Previous Results

Figure 3.6: The upper limits on a stochastic GW background. TOBAs set an upper
limit around 0.1 Hz, which had been unexplored region previously. The green linewith the caption of ‘Single TOBA’ is obtained from the observation with the single
detector. The red line with the caption of ‘Two TOBAs’ is obtained from the crosscorrelation analysis with the two TOBAs set at Tokyo and Kyoto in Japan, whichenables to distinguish the noise and the stochastic GW signal. The line with caption‘future plane’ is an estimated upper limit set by the 1-year simultaneous observation
with the two ultimate TOBAs.

27

4

Chapter
Phase-II TOBA

We have developed Phase-II TOBA as a next step of the first prototype. As described in Sec.4.1, the main feature of Phase-II TOBA is the development of the
new suspension system which realize a new observation system and seismic coupling noise reduction. In this chapter, we will explain about the method and thecharacteristics of the new observation system and the design of the Phase-II TOBA

4.1

Concept

The main purpose of the phase-II TOBA is to establish some of the technologies
necessary for the detection of the low-frequency GWs using TOBA. In our setup,
we established the new suspension system with four features:• A multi-output system,• A common mode noise rejection system,• A passive vibration isolation system,• An active vibration isolation system.
The multi-output system is the novel observation method which improves the angular resolution and the event rate by deriving three independent signals from asingle detector. This system is realized by monitoring the vertical rotational modeof the test mass as well as the horizontal rotational mode. We introduce two test
masses for the common mode noise reduction, and a suspended optical bench for thepassive vibration isolation, and an active vibration isolation table (AVIT) for theseismic noise reduction at low frequencies. Though the rotational seismic vibration
is too small to observe [48], the translational motion of the test masses couples tothe rotational monitor as shown in Sec.3.4. Our suspension system have establishedthe seismic isolation system for such seismic coupling noise.
29

4 Phase-II TOBA

4.2 Multi-Output System

See Fig.4.1 for the significance of the Phase-II TOBA in the road to the ultimate
TOBA.
Phase-I

Phase-II- A single 20-cm
test mass- Room Temperature- Pinning eﬀectsuspension-Michaelsoninterferometer- Two 20-cm
test masses- Room Temperature- Vibration isolation
System-Michaelsoninterferometer○Principle Test
○The ﬁrst observation○Multi-Output system○Development of vibration
isolation system

Large-scaled- Two 10-m
test masse- Cryogenic- Environmental noise
reduction- Fabry-Perot
interferometer- High-Q suspension
and test masses○GW detection

Figure 4.1: The road map of the TOBA project.

4.2

Multi-Output System

4.2.1

MethodIn the Chapter 3, we explained about the principle to detect GWs using TOBA.
There, we assume that the bars rotate only in the horizontal plane (x − y plane).
However, the test masses rotate in horizontal plane mainly by the GWs incoming
along z axis. Considering the situation GWs coming from y axis, the test masses
do not rotate in the horizontal plane but the test mass set along x axis rotates inx − z plane (ϕ1 in Fig.4.2). Therefore, we can obtain two independent signals from asingle test mass bar by observing two rotational angular fluctuation θi and ϕi , where
i is the index of the test masses.
30

4 Phase-II TOBA

4.2 Multi-Output System

z

θ2

y
Test mass 1x

θ1

Test mass 2

Figure 4.2: Schematic picture of multi-output TOBA in the proper antenna frame.
The test masses rotate in the horizontal plane (θ1 and θ2 ) and in the vertical plane(ϕ1 and ϕ2 ).
The angular fluctuation ϕi obeys the equation of motion
I ϕ̈i (t) + γϕ ϕ̇i (t) + κϕ ϕi (t) = Fgw ,(4.1)

which is basically the same as θi . In this case, κϕ is derived as
κϕ = mgd,(4.2)

where m, g, and d are the mass of the bar, the gravitational acceleration, and the
distance between the center of mass of√the bar and the suspension point. Therefore,
the resonant frequency of ϕi , fϕi 0 = mgd/I/2π, can be also as low as that of θiby adjusting the position of the center of mass.
Moreover, since we have two test masses, we can monitor 4 modes, θ1 , ϕ1 , θ2 andϕ2 . Considering θ1 = −θ2 for the GW signals, TOBA can obtain three independent
signals:sI = (θ1 − θ2 )/2,
sII = ϕ1 ,sIII = ϕ2 ,
31

(4.3)
(4.4)
(4.5)

4 Phase-II TOBA

4.2 Multi-Output System

without changing the setup drastically but only with additional sensors for ϕ1 and
ϕ2 .

4.2.2

Antenna Pattern Function

As well as 3.21, the detector tensors for each signal is written as
DIij = nx i ny j ,
1 i j
ij
DII
=
nx nz ,
2
1 i j
ij
DIII
=
ny nz .
2

(4.6)
(4.7)
(4.8)

Therefore, the antenna pattern functions for each signal are
FI+ (Θ, Φ, Ψ) =FI× (Θ, Φ, Ψ) =FII+ (Θ, Φ, Ψ) =FII× (Θ, Φ, Ψ) =+FIII(Θ, Φ, Ψ) =×
FIII(Θ, Φ, Ψ) =1 + cos2 Θ
sin 2Φ cos 2Ψ − cos 2Φ cos Θ sin 2Ψ,2
1 + cos2 Θ
sin 2Φ sin 2Ψ + cos 2Φ cos Θ,2
1
sin 2Θ sin Φ cos 2Ψ − sin Θ cos Φ sin 2Ψ,2
1
sin 2Θ sin Φ sin 2Ψ + sin Θ cos Φ cos 2Ψ,2
1
sin 2Θ cos Φ cos 2Ψ − sin Θ sin Φ sin 2Ψ,2
1
sin 2Θ cos Φ sin 2Ψ + sin Θ sin Φ cos 2Ψ.
2(4.9)(4.10)(4.11)(4.12)(4.13)(4.14)(4.15)

Figure 4.3 is the square root of the each antenna pattern functions:()1/2
Fi = Fi+2 + Fi×2,
and their total,(
Ftotal =(4.16))1/2∑

ri Fi2.

(4.17)

i=I,II,III

Here, ri is the distance at which the source would be observed with the same SNR
in a single output. The antenna patterns shown in Fig.4.3 is pictured under the32

4 Phase-II TOBA

4.2 Multi-Output System

assumption of r1 = 1, r2 = r3 = 0.5 since the sensitivity of sI is expected to be
twice as good as sII and sIII due to the common mode noise rejection. It shows theblind direction which the conventional single-output TOBA has is vanished for themulti-output TOBA.

z
θ

y2
θ1

x

FI

FIII

FII

z

z

z
x

y

x

x
y

y

1
0.8
0.6
0.4
0.2
0

FtotalzxyFigure 4.3: Square root of the antenna pattern functions of multi-output TOBA.
The single-output TOBA has only the antenna response of FI . By adding two signals
sII and sIII with the antenna pattern function of FII and FIII , the blind direction isvanished and the region where TOBA can observe is enlarged (Ftotal ). Note that
we assume that the sensitivity of sI is twice as good as sII and sIII because of thecommon mode noise rejection, and that the sensitivity of sII and sIII are the same.

33

4 Phase-II TOBA

4.2.3

4.2 Multi-Output System

Detection Volume

As shown in Fig.4.3, the volume enclosed by the sphere of the antenna patternfunction is enlarged. It means that the area which the TOBA can observe is enlarged
and the event rate is increased.
We will estimate the increased event rate with the detection volume, [45] definedas∫∫
RN (α,δ)VN =r2 dr.

dΩ(4.18)0

RN (α, δ) is its reach of the TOBA’s observable area at each sky position α, δ:RN (α, γ) =D[FN (α, δ)]1/2 ,ρmin(4.19)

where D and rhomin are the distance to which the detector can detect GWs with aunit SNR, and the minimum value of SNR with which we can regard that GWs aredetected. FN is the square root of the pattern functions as is written in 4.16 and4.17. In this case, we have N = I, II, III, and total.
For the convenience, we take D = ρmin = 1 in order to compare the detection
volume between the single-output and the multi-output TOBAs. Then, the detection volume is derived to be VI = 1.2 for the single output, and Vtotal = 2.0 for thethree outputs. Therefore, the detection volume, i.e., the event rate will be about 1.7
times larger than the conventional configuration when we have three independent
signals.

4.2.4

Angular Resolution

The conventional GW detectors cannot identify the direction of the GW source
because of their low directivity. It is known that at least three separate detectors
are necessary for the identification. [45] Moreover, the time-of-flight method, which
is used for ground-based interferometric detectors, is not eﬀective to GWs whose
wavelength is longer than the distance between detectors any more 1 .
Since the multi-output method derives the three independent signals, it improves
the accuracy of the identification. However, because of the geometrical symmetryof the detector, the single multi-output TOBA cannot identify the source as one
point in all sky, but narrow down to one point in one quadrant, i.e., 8 points. In
The wavelength of the GWs at 1 Hz is about 3 × 108 m, which is about the distance between
the Earth and the moon. The second TOBA should be placed on the moon in order to get useful
constrains from the time-of-flight method.
134

4 Phase-II TOBA

4.2 Multi-Output Systemorder to identify the source as one point, simultaneous observation with at least two
multi-output TOBAs, or follow-up observation using the electro-magnetic waves isnecessary.
Figure 4.4 is the angular resolution of the multi-output TOBA and the singleoutput TOBA. The angular resolution is estimated using Fisher analysis [29]. The
GW source is assumed to be a monochromatic sources. It shows that the multioutput TOBA can narrow down the sky position of the GW sources of the 8 points
with the error of 4 × 10−1 steradian, while single-output TOBA cannot determined
the sky position at all if the observation time is shorter than 1 day. The singleand multi-output TOBA can determine the sky position of the GW sources if theobservation time is longer than 1 day since the signal is modulated by the Earth’s
rotation. Also, the other parameters of GW source are proven to be determinedwith multi-output TOBA even though the observation time is shorter than 1 day
[27].
101
0

10

-1

∆Ω [sr]

10

-2

10

-3

10

10-4

multiple output
single output

-5

10

2

10

10

3

4

5

10
10
T obs [sec]

10

6

7

10

Figure 4.4: The angular resolution ∆Ω vs the observation time Tobs . The frequencyof the GW source is assumed to be 1 Hz, and the amplitude is normalized by SNR= 10.
Other than the monochromatic sources, we have investigated the angular resolution for the IMBH binaries. For example, the angular resolution for the 104 M· 104 M· binaries are improved to be 6.74 × 10−2 steradian.
See [27] for more details of the Fisher analysis and the calculation. Please notethat these Fisher analysis are calculated by Kazunari Eda.
35

4 Phase-II TOBA

4.3

4.3 Design Overview

Design Overview

As described in Sec.4.1, we developed the phase-II TOBA focusing on the suspension
system.

Hexapod

Intermediate Mass
& Damping Mass
Thermal Shield

Test Masses
Optical Bench

Vacuum Tank

Figure 4.5: A picture of the hole setup.

The schematic view of the Phase-II TOBA is shown in Fig.4.5. The two test
masses which sense the GWs are suspended from a Hexapod-type active vibration
isolation table (AVIT) via an intermediate mass which is magnetically damped bya damping mass. The suspension system except the actuators and sensors of the
AVIT is in a vacuum tank. The vacuum tank has the cryogenic system so that thesuspension wires and test masses can be cooled down to 4 K in order to reduce thethermal noise, while the cryogenic system is not used for the Phase-II TOBA.
36

4 Phase-II TOBA

4.4 Suspension

4.4

Suspension

4.4.1

Test Masses

Suspension wires

Holes formirrors and magnet holders

Figure 4.6: A picture of the two test masses.
The picture of the test masses is shown in Fig.4.6. The test masses are designed
to realize the multi-output system and the common mode noise rejection. The lengthof the test mass is maximized to be 24 cm under the constraint of the size of thevacuum tank which is also used for the first prototype.
From 4.1 and√ 4.2, the resonant frequency in the vertical rotational mode iswritten as f0 = mgd/I/2π. Therefore, the suspension points of the test masses are
set to be as close as possible to their center of mass so that the resonant frequencyof the vertical rotation is minimized. In addition, the test masses have tapped holes
for counter masses at the both ends of the bar in order to adjust the position of thecenter of mass to the suspension points. Note that the suspension points should bea little bit higher (∼ 0.1mm) than the center of the mass for its stability.
The each test mass is suspended by two parallel wires so that the centers ofmass of the two test masses are at the same point in order to maximize the common√
mode noise rejection rate. The expected common mode noises rejection rate is 2
in strain at the minimum where the noise sources of the two signals are independent.
The maximum common mode noise rejection rate is expected to be more than 10
when the noises are correlated.
37

4 Phase-II TOBA

4.4 Suspension

However, the resonant frequency in the horizontal rotational mode increases dueto the suspension using the parallel two wires. When the test mass with the mass ofm suspended by two wires rotates by the angle θ in the horizontal plane, the force
applied at the suspension point is written as
F =

mgϕ
,
2

(4.20)

where ϕ = aθ/l is the inclination angle of the suspension wire (See Fig.4.7). Here, aand l is the distance between the two suspension points divided by 2 and the lengthof the suspension wire, respectively. The equation of motion in the horizontal planeisa2I θ̈ + γ θ̇ = 2F a = −mg θ,(4.21)lwhere I is the moment of inertia. Therefore, the resonant frequency in the horizontal
plane in our test mass is obtained as√
1 mg a2fθ0 =.
(4.22)2π
I l
It means that the resonant frequency is relative to the distance between the twosuspension points. In our setup, a is set to be as small as possible in order to reduce
the resonant frequency, which is limited by the size of screws under the constraintof the position of the two test masses.
The each test mass is composed of three parts: a set of wire clamps, and two
bar-shaped parts. We set clamps and suspension wires at first, then the arms areattached to the clamp. Our configuration enables us to assemble the test mases
at proper place without touching each other. Though the loss of the test mass isincreased, i.e., the thermal noise of the test mass bar is increased, the thermal noise
does not aﬀect to the sensitivity in our setup. The material of the bar-shaped parts
is aluminum since it does not magnetized and is easy to manufacture. The set ofclamps are made of stainless steel, which is harder than the suspension wires, inorder to clamp the suspension wires rigidly.
The material of the suspension wires is tungsten. Considering the weight limit,
the constraint on the diameter of the wire d is√amg,
(4.23)
d>2
πT
where a, m, g and T are the safety factor, the mass, the gravitational acceleration,and the ultimate tensile strength of the wire, respectively. Setting a = 2, the38

4 Phase-II TOBA

4.4 Suspension

φ

θ

l

F
2a

Figure 4.7: A schematic view of the rotating test mass in the horizontal plane.
diameter of the wire should be d > 7.1 × 10−5 m. Chosen from ready-made items,
the diameter of the wires for the test mass is 0.1 mm.
See Table 4.1 for the details of the test mass properties.

4.4.2

Optical Bench

An optical bench is a table on which sensors and actuators are set. The opticalbench is also suspended from the same intermediate mass. Since we measure thedistance between the sensors and test masses, the seismic motion of the sensors
couples to the signals eﬀectively even though the test masses are well isolated fromthe seismic noise. Therefore, the seismic motion of the sensors and actuators should
also be isolated from the vibration.
Here, we will explain how the seismic noise is reduced by the suspended opticalbench. The equation of motion for the test mass with mass of mTM suspended bywire with length of lTM ismẍTM (t) + γTM ẋTM (t) +mTM g(xTM (t) − X(t)) = 0,
lTM

(4.24)

where γTM and g are the damping constant and the gravity acceleration. xTM (t)and X(t) are the horizontal displacement of the test mass and the suspension point,39

4 Phase-II TOBA

4.4 Suspension

respectively. Performing Fourier transformations, we have
)
(
mTM g
mTM g
2
−ω mTM + iγTM ω +x̃TM (t) =X̃(t)(4.25)
lTM
lTM
2
ω0TM
∴ x̃TM (t) =X̃(t),(4.26)
2
ω + ω0TM−ω 2 + i ωQ0TM
TM√
where ω0TM = g/lTM is the resonant frequency and QTM = ω0TM /γTM is the Q factor. Here, tilde denotes the Fourier transformations of the corresponding variables.
The same equation is formed for the optical bench:x̃OB (t) =2
ω0OBX̃(t).
2ω+ω−ω 2 + i ωQ0OB
0OB
OB(4.27)

Therefore, the relative motion between the test mass and the optical bench isx̃TM (t) − x̃OB (t) = H(f )X̃(t),where
H(f ) =

2
2
ω0TM
ω0OB
−
.
2
2 + i ω0OB ω + ω 2−ω 2 + i ωQ0TM
ω
+
ω
−ω
0TM
0OB
Q
TM
OB

(4.28)(4.29)

H(f ) is the transfer function of the seismic motion from the suspension point tosensors. 1/H(f ) is the passive seismic vibration isolation rate.
Figure 4.8 is the graph of the H(f ). At lower frequencies than the resonant
frequencies, the test masses and the optical bench move commonly so that theseismic noise appeared in the sensor is rejected. On the other hand, the transfer
function is small above the resonant frequencies since the motion of the test masses
and the optical bench are both reduced. H(f ) gets smaller when the diﬀerence
between ω0TM and ω0OB is smaller since the transfer functions of the test masses andthe optical bench get closer. In order to reduce the seismic motion, it is eﬀective toadjust the two resonant frequencies to be close.
The material of the optical bench should be metal since the bench should be rigid
in order to avoid the resonance of the optical bench aﬀect the sensitivity. Here, we
use aluminum for the table and stainless steel for the clamps by the same reason as
the test mass clamps.
Note that the extension caused by the tensile stress does not aﬀect the operation.
In our case, the extension of the test mass wires is 4.0 × 10−4 m while the extensionof the optical bench wires is 6.7 × 10−4 mm, assuming that the wire lengths of thetest mass and the optical bench are 0.36 m and 0.38 m, respectively 2 .
2The Young’s modulus of the tungsten is E = 345 GPa.

40

4 Phase-II TOBA

4.4 Suspension

Figure 4.8: The transfer function of the horizontal motion from the suspension point
to the sensors. We assume lTM = 30 cm, lOB = 32 cm, and QTM = QOB = 100. In
this condition, the resonant frequency of the test mass and the optical bench are
f0TM = 0.91 Hz and f0OB = 0.88 Hz, respectively.

4.4.3

Intermediate Mass and Damping Mass

We have the damping system so that the peaks of the pendulum modes do not eﬀect
on the sensitivity, the control system, and so on.
The intermediate mass, which is made of aluminum, is covered by the damping
mass which has damping magnets inside. The magnets are attached by its magnetic
force since the damping mass is made of ferromagnetic stainless. When the magnets
are close enough to the intermediate mass, the movements of the intermediate massis dumped because of the loss caused by eddy current in the aluminum. In this case,the magnets are arranged in a circle so that the loss in the horizontal rotation is
reduced while the other modes are well damped.
The test masses and the optical bench are suspended from the intermediate
mass. It lets the magnetic coupling noise caused by the damping magnets smaller
than setting the damping system directly on the test masses and the optical bench.
41

4 Phase-II TOBA

4.5 Control System

Test Mass
The length of the bar
The mass
The moment of inertia (in both yaw and roll)
The quadrupole moment
The distance between the suspension points (a)The length of the suspension wires
Optical Bench
The mass
The moment of inertia (in yaw)
The moment of inertia (in roll/pitch)
The quadrupole moment (in roll/pitch)
The distance between the suspension points
The length of the suspension wires

240
0.61
3.34 ×10−3
3.20 ×10−3
17
400

mm
kg
kg m2
kg m2
mm
mm

4.4
6.17 × 10−2
3.50 × 10−2
3.08 × 10−2
45
400

kg
kg m2
kg m2
kg m2
mm
mm

Table 4.1: The main property of the suspension system.

4.5

Control System

4.5.1

Control Loop

The test masses are controlled by sensors and actuators set on the optical bench.
Servo system provides the linearity of the sensors and actuators. It enables us tocalibrate the small signals properly.
Block diagram of the control system for the test masses is shown in Fig.4.9. We
have two sensors: fiber laser interferometers for the main observation, and photosensors for the alignment. The displacement signals measured by the sensors go tocoil magnet actuators after filtered by the digital system. The digital system is a
powerful tool for the control system with multi-degrees of freedom. The details of
the digital system is described in Sec.4.8.
In this subsection, we will explain the principle of the control system. Assumingthat the GW signals h is injected in the test mass, for example, the total force x
applied on the test mass is written as
x = h − M Si Fi Ax,(4.30)

where i =LS or PS is the indicator of the servo loop. M, Si , Fi and A are the transfer
function of the test mass from force to the displacement, that of sensors, that of42

4 Phase-II TOBA

4.5 Control System
Test Masses

GW signals

h

M

A

Coil-Magnet
Actuators

Photo
Sensors

SPS

Laser

SLS Interferometers

Digital System
Switch
ADC

DAC

FPS
FLS

sFB

sERR

Figure 4.9: The control system of the test masses. We have two sensors: photosensors (PS) and laser interferometers (LS). The GW signals are injected into thetest masses and sensed by sensors via transfer function H, which is expressed in 3.17
and 3.18. The signals from the sensors go to the digital system via analog-to-digitalconverter (ADC). On and Oﬀ of the control loop is selected in the digital system.
The selected feedback signals sFB come out from the digital-to-analog converter(DAC) and go to the coil-magnet actuators.
filters, and that of actuators, respectively. Defining the open loop transfer function(OLTF) Gi as
Gi = M Si Fi A,(4.31)
x is derived asx=

1s.
1 + Gi(4.32)

Therefore, the displacement of the test mass is suppressed by 1 + Gi times where Gi
is bigger than one. We call the frequency where the absolute value of Gi is one as
the unity gain frequency (UGF). UGF represents the minimum frequency that theobject is controlled.
Meanwhile Gi should be large enough to maintain the linearity of the sensors
and actuators, Gi is required to satisfy Nyquist stability criterion [60] in order tostabilize the control system. Plotting a Nyquist diagram of Gi , where the real part43

4 Phase-II TOBA

4.5 Control Systemof Gi is plotted on the x axis and the imaginary part of Gi is plotted on the y axis
when the frequency is swept from 0 to ∞, the control system is stable if the point(−1, i0) is on the left hand side of Gi when the frequency is swept from 0 to ∞(Fig.4.10). Here, phase margin is one of the indicators for the stability of the servo
system. The phase margin is the angle of Gi from −180◦ at the UGF. It represents
how close Gi is from the point (−1, i0). If there is little phase margin, the servo
system is not stable and easily oscillate. The phase margin should be more thanabout 20◦ for the stable control.

f = -∞f=∞

f=0

Phase margin

Figure 4.10: An example of a Nyquist plot (left) and a bode plot (right). In theboth graph, a function G = 10/(s2 +s+1) is plotted as an example, where s = i2πf .
The green line shows the UGF. This system is stable since the point (-1, i0), whichis plotted as a red circle, is left hand side of G and the phase margin is about 20◦ .

4.5.2

Photo Sensors

When the test mass is swinging freely and moves a lot in total root means square(RMS), it is diﬃcult to keep the visibility stable enough for locking the interferometers since the interferometer does not have wide dynamic range and the laser beamsof the interferometer do not aligned while it has good sensitivity. Therefore, we
use location sensors called photo sensors (PSs) in order to fix the relative position
between the test masses and the optical bench (the output ports of the laser beams).
The PSs are composed of one LED and two photo detectors as shown in Fig.4.11,
and measure the distance to the object by sensing the amount of LED right reflected
at the object. They have a larger dynamic range, but worse sensitivity than interferometers.
44

4 Phase-II TOBA

4.5 Control System

Object (Mirror)

PD

LED

PD

Figure 4.11: Schematic View of the photo sensors.
In our setup, we do not use mirrors for PSs since the LED light is reflected by
the aluminum surface enough and the PSs are required to be stable only
√ for the−8
lock acquisition. The current noise level of the PSs are about 10 m/ Hz, whichis stable enough that the fluctuation of the fringe of the Michelson interferometer iswithin 5 %.

4.5.3

Laser Interferometers

The main sensors we used for the observation in Phase-II TOBA is the Michelsoninterferometric sensors. The sensor is switched from the photo sensors to interferometers after locking the test masses with photo sensors.
In order to decouple each mode and to reduce the coupling from the translational
seismic noise as seen in the first prototype, we monitor two pendulum modes as wellas two rotational modes, LONG, SIDE, YAW, and ROLL, using four interferometers,
LS1 to LS4. (See Fig.4.12 for the definition of an each mode.) We do not monitor
VERT (vertical) and PITCH mode since their motion is smaller than others due totheir high resonant frequencies.
In the Phase-II TOBA, we used the fiber optics in order to compose the Michelsoninterferometers in order to save the space. The laser fibers are suitable for our setup
in which many sensors are necessary though phase noise will be injected from fibers
vibration. The laser beam is introduced by a single bare-fiber, and the beam is split
in to 8 fibers by fiber beam splitters on the optical bench. The eight beams goes tothe fiber laser interferometers shown in Fig.4.12. The one end of LS1, LS2, and LS3
are retroreflector patch fiber cables, which work as fixed reference mirrors, and theother is the mirror attached on the test mass. The roll motion (LS4) is monitored
by measuring the diﬀerence of the distance from the output ports of laser beam tothe mirrors at the top of the test mass between the two beam paths. The interfere45

4 Phase-II TOBA

4.5 Control Systemof the returning beams are measured by the photo detectors. The photo detectors
are on the optical bench, and the signals are derived via lead cables from the opticalbench.

Type 2

LS4

VERT
YAW
ROLL

mirrors

PITCH
SIDE

LONG

test mass
LS1

ﬁber collimator

spacial beam
mirrorﬁber beam
splitter

Type2(ROLL)
Laser input

photo detector

ﬁber collimators

Type1

Type1(LONG, SIDE, YAW)
Laser input

photo detector

LS3

LS2

ﬁber optics
retroreﬂector(ﬁxed mirror)
spacial beamﬁber beam
splittermirrorﬁber collimators

Figure 4.12: The configuration of the fiber laser interferometers.
In our setup, we use the interferometers at their mid-fringe. The electromagnetic
field of the input beam is written as
Ein = E0 e−iΩt ,(4.33)

where E0 and Ω is the amplitude and the angular frequency of the laser. The input
beam goes to the fiber beam splitter (BS) and divided into two fibers. The beams
are reflected by end mirrors attached on the test mass or fiber retroreflectors, and46

4 Phase-II TOBA

4.5 Control System

Figure 4.13: The picture of the sensor module. The left one is the fiber collimator
mounted on the half inch mirror mount, and the right one is the photo sensor.
recombined on the BS. Letting the optical pass lengths from the BS to the two ends
as l1 and l2 , the electromagnetic field of the recombined beam is
EPD = E1 e−i(Ωt−2kl1 ) − E2 e−i(Ωt−2kl2 ) ,(4.34)

where E1 and E2 are the amplitudes of the returning beams. k is the wave numberof the beam. Therefore, the power of the light which the photo detector received is
PPD = |EPD |2
Pmax + Pmin Pmax − Pmin=+cos(2kδl),22(4.35)

Pmax = (E1 + E2 )2 ,Pmin = (E1 − E2 )2 ,
δl = l2 − l1 .

(4.37)(4.38)(4.39)(4.36)

where

Since cos(2kδl) ∼ 0 around the mid-fringe, we obtain
PPD ∼Pmax + Pmin Pmax − Pmin 4π
+
δl
2
2
λ

(4.40)

by taking the first order of δl in 4.36. Here, λ is the wavelength of the laser. Though
E1 = E2 for the ideal case, E1 and E2 are not the same in the actual setup. Then,47

4 Phase-II TOBA

4.6 Active Vibration Isolation Table

we define the visibility VLS as
Pmax − Pmin.
(4.41)
Pmax + Pmin
When VLS = 1, E1 = E2 and there are no loss in the two beam paths. VLS get
smaller as the optical loss is larger.
In our setup, the wavelength of the laser is 1550 nm. The power of the laser is
500 mW in total. Since the laser is equally divided into 8 interferometers, the input
power of the each interferometers is 62.5 mW.
Figure 4.13 is the picture of the fiber collimator (the output port of the laserbeam) and the PS. The bases is original for the Phase-II TOBA instead of theready-made pedestal bases in order to save the space.
VLS =

4.5.4

Coil-Magnet Actuators

We use the coil-magnet actuators for the test mass control, which is one of the mosttypical actuators used in GW detectors. The coil-magnet actuator consists of a pin
magnet attached on a test mass and a coil set on the optical bench. The magnetic
field produced by the coil applies the force on the test mass via the magnet.
Compared to other actuators such as an electrostatic actuator [43], the coilmagnet actuators can produce relatively strong force as well as is easy to handle
with. However, the environmental magnetic field also applies the force on the test
mass and would reduce the sensitivity. In our setup, the coil-magnet actuators are
adopted since the priority is given to the simplicity than the magnetic coupling noise
reduction.
In our setup, the pin magnets are neodymium magnets with the length of 5mm and the diameter of 1 mm. The coil has the self-inductance of 0.14 mH andthe resistance of 11 Ω. The bobbin of the coil is made of Polyether ether ketone(PEEK) in order to avoid the eddy current flowing and suppress deterioration of adegree of vacuum.

4.6

Active Vibration Isolation TableIn the previous sections, we explain about the passive vibration isolation systemusing the pendulum. However, as shown in Fig.4.8, the seismic noise around theresonant frequencies of the pendulum mode is not reduced by the passive vibration isolation system. Therefore, we introduced the Hexapod-type active vibration
isolation table (AVIT) in order to reduce the seismic noise around 1 Hz.
48

4 Phase-II TOBA

4.6 Active Vibration Isolation Table

The AVIT is a table with six legs made of piezoelectric elements (PZT). The
position of the table can be controlled by applying the voltage on the PZTs. The
seismic motion of the table is suppressed by a feedback control using the seismometerset on the table as the sensors, and the PZTs as the actuators. The table is compact
such as the diameter of 45 cm, but has a good vibration isolation rate around afew Hz, while a passive vibration isolation system needs a large setup, such as
long suspension wires and GAS filters [17] and so on, in order to obtain a highvibration isolation rate at low frequencies. Also, the AVIT can suppress the vibration
introduced from other than the ground, such as the vibration of a cryogenic anchor.
Using the passive vibration isolation system, another intermediate mass with large
mass to which the anchor is attached is necessary in order to reduce the introduced
vibration of the cryogenic cooler as planned in bKAGRA [18].
A schematic view of the AVIT is shown in Fig.4.14. Hexapod-type AVIT has six
legs with piezoelectric elements in them so that the length of the legs are changeable.
Seismometers and the photo sensors are used for monitoring the vibration and the
DC position of the table. These signals are fed back to the piezoelectric elements sothat the seismic motion is canceled.

Seismometers

Vibration
isolatedtable

Six actuators

GroundFigure 4.14: The schematic view of the Hexapod-type AVIT. It has six piezoelectric
actuators as legs, which move the top table in any directions.

49

4 Phase-II TOBA

4.6 Active Vibration Isolation Table
AVIT

x0

+

H

x

Seismometers

Mo

Fs

Ms

Filters

Input
matices

Fp

Mp

Ss

xtop

Sp

xtop - x0
Photo sensors

Figure 4.15: The control loop of the hexapod. Note that it describes the control
loop only in one degree of freedom for the simplification.

4.6.1

Control System

Control Loop
The block diagram of the control loop is shown in Fig.4.15. In this control loop, two
sensors, PSs and seismometers, are used at the same time. The seismometers areused for sensing the seismic motion of the table, while the PSs are used for sensing
the absolute distance between the controlled table and the ground. The outputsof the sensors are gathered into the digital system and converted to the signals ofeach degree of freedom. Details of the method how the signals are converted are
discussed in Appendix A. The output signals of the seismometer are the absolute
vibration of the top table (xtop ) while the PSs sense the relative movements between
the top table and ground (xtop − x0 ). Each signal is filtered, then combined. The
feedback signal are sent to the PZTs after being converted to actuation signals byan output matrix Mo .
In this control loop, the open loop transfer functions of each sensors are
G s = Ss M s F s
Gp = Sp Mp Fp ,(4.42)(4.43)and overall open loop transfer function is
G = Gs + Gp .
50

(4.44)

4 Phase-II TOBA

4.6 Active Vibration Isolation Table

The motion of the top table xtop can be written as{

xtop = x0 − x
x = Gs xtop + Gp (xtop − x0 )(4.45)(4.46)

1 + Gpx0 .
1 + (Gs + Gp )(4.47)∴ xtop =Therefore, the vibration reduction rate R isR=

1 + (Gs + Gp ).
1 + Gp(4.48)The gain of the control loop is adjusted so as to be R > 10 at 1 Hz in order toreduce the seismic motion by 10 times.
Seismometers
L4 seismometers produced by Sercel are used for the seismic sensors 3 . It is a passive
seismic sensor made composed of a coil attached to the housing and a suspended
magnet (Fig.4.16).
L4 housingX

magneticﬁeldsuspendedmagnetx

Φ( x − X )CoilVsig = Rc I

Induced currentI ∝ (x& − X& )Figure 4.16: The system of the L4 seismometer.
3

See the web page for the details of the product.

51

4 Phase-II TOBA

4.6 Active Vibration Isolation Table

By analogy of 4.26, relative motion of the coil and the magnet is written as
)
(
ω02
x−X =
− 1 X,(4.49)−ω 2 + i ωQ0 ω + ω02
where X and x are the position of the ground and the magnet, respectively. ω0 is
the resonant frequency of the magnet, and Q is the Q value of the suspension. Since
the magnetic field inside the coil is relative to x − X, the induced electromotive
force relative to ẋ − Ẋ is applied on the coil. The calibration function from theL4 output to the seismic displacement is shown in Fig.4.17. Above the resonant
frequency, which is 1 Hz in the case of L4, the output of the seismometer is relative
to the velocity of the ground.

Figure 4.17: The calibration function of L4. Note that the calibration gain includes
pre-amplifier
The noise level of L4 is shown in Fig.4.18. This noise is non-correlated signals
between the three L4 seismometers set next to each other. Their sensitivity is limitedby the amplifier circuit.
Photo Sensors
Since L4 sensors do not have sensitivities at low frequencies, the position of the table
would drift if controlled only by L4 since the outputs of the seismometers are the52

4 Phase-II TOBA

4.6 Active Vibration Isolation Table

5

Figure 4.18: The seismic signal and the noise level of the seismometer. The redsolid line is the seismic motion of the ground and the purple dashed line is the noiseof the seismometer. The noise is calculated as an uncorrelated signal between thethree seismometers set next to each other.
circuit noise below 0.1 Hz. Therefore, the table is controlled by the photo sensors
at lower frequencies where the seismometers do not have good sensitivity.
The photo sensors used here are the same as described in Sec.4.5.2.
Piezoelectric Elements
We used six piezoelectric elements produced by PI (product number: P.844-30) 4 .
The piezoelectric element is a device which convert the voltage to the force. It has adynamic range of 45 µm with the input voltage of 0 to 100 V. The dynamic range isestimated from the order of the√ seismic motion. The seismic motion of the ground−7
around 1 Hz is about 10 m/ Hz, PZTs are chosen so that the seismic motion iscanceled out within the dynamic range of the PZTs.
The piezoelectric materials are set inclined by 45 degrees to the gravity force,
though they are generally week if the burden perpendicular to the actuating direction
is applied. In order to reduce the burden of the material, the piezoelectric elements
4

See the web page for the details of the product.

53

4 Phase-II TOBA

4.7 Vacuum Systemand the moving parts are connected with a point, not with a surface. Additionally,
we have hinges at the both ends of the devices in order to reduce the load on the
PZTs caused by the distortion of the legs (See Fig.4.19). Their load limit is 15 kg
per one piezoelectric element. Therefore, the total load limit of the AVIT is 90 kg.

72mm

Slit
Figure 4.19: A picture of PZT. The length of the PZT is 72 mm, and it has slits at
both end for the burden reduction.

4.7

Vacuum System

The setup described above, except the actuators and sensors of the AVIT is installed
in a vacuum tank. It prevents the setup from the acoustic noise and loss causedby the residual gas. The chamber is evacuated using a scroll pump and a turbomolecular pump. The achieved degree of vacuum is about 10−3 Pa. In order toavoid the vibration of the pump induced, the pumps are connected to the vacuumtank through the damper and long flexible tube.
The vacuum tank is originally the same as the one used in the first prototype.
The biggest diﬀerence is that the cryogenic system is installed in the chamber for
the future application.

4.8

Digital System

All the processes of servo filters, matrices described, and data taking mentioned
above are done in a digital system. The digital system is a powerful tool, especially54

4 Phase-II TOBA

4.9 Principle Sensitivity

for a system which needs many signals, multi-dimensional matrices, and fine tuningof filtering parameters.
The digital system we used is the same system as one used in LIGO, Virgo,
and KAGRA. The systems [65], such as filters, matrices, and their connections,are constructed with simLink, the tool in Matlab. This system has tools for basic measurement: oscilloscope, spectrum measurement, open loop transfer function
measurement, and so on.
The data is processed with the sampling rate of 4 kHz. The data for the longobservation is stored with the sampling rate of 256 Hz.
Also, we have third order low pass filters with the cutoﬀ frequency 10 kHz of Hz
for the anti-aliasing/imaging filters before ADC and after DAC.

4.9

Principle Sensitivity

Figure 4.20 is the principle sensitivity in the Phase-II TOBA. We assume that thenoise reduction rate by the AVIT as 10 and noise reduction rate with decoupling as
10. Thanks to the seismic isolation systems, the sensitivity above 1 Hz is expected
to be improved more than 100 times than the first prototype.
Note that the magnetic coupling noise is not considered here since the test massand the suspension system is diﬀerent from the previous detector.

55

4 Phase-II TOBA

4.9 Principle Sensitivity

Figure 4.20: The principle sensitivity expected to be obtained with Phase-II TOBA.
The bold black line is the total sensitivity. The blue, green, and magenta lines are
the seismic noise, shot noise, and the suspension thermal noise. The red line is thesensitivity obtained with the first prototype.

56

5

Chapter
Experiments
In this chapter, we explain about the characteristics of the Phase-II TOBA. We
constructed the Phase-II TOBA according to the design explained in the previous
chapter. As a result, the three independent signal are successfully obtained, andthe eﬀect of the vibration isolation system is evaluated. We will explain about thecalibration, its sensitivity, and its noise source.

5.1

Setup

The Phase-II TOBA is constructed in the room B207, Science Building 1, Department of Physics, University of Tokyo, which is at Hongo campus in Tokyo.
The picture of the vacuum tank with the suspension system installed is Fig.5.1.
The two test masses which sense the GWs are suspended from a Hexapod-type active
vibration isolation table via an intermediate mass which is magnetically damped bya damping mass. The suspension system except the Hexapod is in a vacuum tank.
The vacuum tank has the cryogenic system so that the suspension wires and test
masses can be cooled down to 4 K in order to reduce the thermal noise, though thecryogenic system is not use in this thesis.
Also, the test masses and the optical bench without the sensors and actuators
are shown in Fig.5.2. Figure 5.3 is a picture of the AVIT integrated in the vacuum
tank. The AVIT is set at the top of the vacuum tank in order to suspend theintermediate mass and the damping mass directly. The center of the vibration
isolated table, including the suspension points of the masses, is in the vacuum while
the sensors and actuators are in the air. The vibration isolation table is connectedto the vacuum tank below and fixed table above with bellows. The table has airholes so that both the upper side and underside of the table are in the vacuum.
Keeping the air pressure the same above and below the table prevents piezoelectricelements from being broken by the air pressure. Note that we used welded bellows
for the connectors which have small spring constants in order to actuate the table57

5 Experiments

5.1 Setup

without non-linearity.

Active Vibration
Isolation Part

Vacuum Chamber

Damping Mass/
Intermediate Mass

Thermal Shield

Optical Bench/
Test Masses

Fiber Optics

Figure 5.1: A picure of the detector.

58

5 Experiments

5.1 Setup

Suspension Wires

Mirrors forinterferometers

Test Masses
Optical Bench

Adjusting mass
Magnets for
actuators
Figure 5.2: A picure of the detector.

59

5 Experiments

5.1 Setup

Figure 5.3: Pictures of the AVIT installed in the vacuum tank. The AVIT is set on
the stainless table fixed above the vacuum tank. Also the other fixed stainless table
is set above the AVIT. The tank and the AVIT table, and the AVIT table and theupper table are connected by bellows, respectively.

60

5 Experiments

5.2 Calibration and Sensitivity

5.2

Calibration and Sensitivity

5.2.1

Calibration from the voltage to the angular fluctuation

We calibrate the output signal of the interferometer to the angular displacement
using the fringe of the interferometer and the distance between the two mirrors.
The calibration factor derived here corresponds to S, S1 , and S2 in Fig.5.4 andFig.5.7.
The calibration from the output voltage of the interferometer to the displacement
is calculated according to 4.40. Pmax and Pmin are measured just before the test
masses are locked by the control system using the interferometer. The residual
displacement of the test masses is bigger than the wavelength of the laser when thetest masses are controlled by PSs whose noise level is bigger than the wavelength ofthe laser.
The displacement signals are calibrated to the angular fluctuation δθ as
δθ =

δldiﬀ
,
d

(5.1)

where δdiﬀ is the diﬀerential between the displacement of the two mirrors, and d isthe distance between the centers of the two mirrors. Note that the calibration error
is caused by the uncertainty of d since the beam spot of the laser is not exactly at
the center of the mirror. Considering that the diameter of the mirror is 12.7 mm (ahalf inch) and d = 190 mm, the calibration error caused here is estimated as about
7 %.

5.2.2

Calibration from the feedback signal to the out-of-loop
signalThe angular fluctuation of the test mass is derived from the error signal or the
feedback signal in the servo system. In this subsection, we will explain how tocalibrate the feedback signal to the angular fluctuation θ and ϕ.
ROLL Mode Calibration
The angular fluctuation in ROLL mode ϕ is controlled by the servo system shownin Fig.5.4.
61

5 Experiments

5.2 Calibration and Sensitivity

S

LS

b
F

θ
test massactuatorA
Figure 5.4: The block diagram of the servo system in the vertical plane. Here, we
use the single laser interferometer in order to control the ROLL motion.
The displacement disturbance injected to the control loop was suppressed by
1 + G times as described in Sec.4.5.1. Therefore, the out-of-loop signal is calibratedass = (1 + G)serr .
(5.2)
The OLTF G is measured by injecting the excitation signal into the control loop.
The measured OLTF of ROLL mode is shown in Fig.5.5. The UGF is around 30 Hz
and the phase margin is about 30 degree.
The OLTF is not measured with good coherence since the injected excitation
signal is largely suppressed at lower frequencies than 1.5 Hz. In order to guarantee
the OLTF value G = SF A, the transfer functions S, F, and A are measured individually. The transfer function of the laser sensor S is obtained from the fringe of theinterferometers as mentioned in the previous subsection. The transfer function of
the filter F is obtained from the setup in the digital system. The transfer function of
the actuator A is measured PSs. Figure 5.6 is the transfer function of the actuator,i.e., the transfer function from the input of the actuator to the output of the PSs.
It is measured without controlling the test masses. The peak around 1 Hz is thecoupling from the longitudinal degree of freedom.
The green line in Fig.5.5 is the OLTF calculated using the value of S, F , and A
measured individually. It fits well to the data of the overall OLTF measurement.
The phase delay of the OLTF at higher frequencies above 40 Hz is considered to becaused by the phase delay of the digital system.
62

5 Experiments

5.2 Calibration and Sensitivity

Figure 5.5: The OLTF of the ROLL mode. The blue dots are measured values
and the green line is the OLTF calculated using the value of S, F , and A measured
individually. The gain of the theoretical OLTF was fitted to the measured values.

Figure 5.6: The transfer function of the coil-magnet actuator. The blue dots are themeasured data and the green line is the fitted line.
63

5 Experiments

5.2 Calibration and Sensitivity

YAW Mode Calibration
The angular fluctuation in YAW mode θ and the translational fluctuation in x
axis is controlled by the servo system shown in Fig.5.7. Note that the control
system is diﬀerent from that of the ROLL mode since the translational mode andthe rotational mode are controlled together by two servo systems. It is necessary tocontrol both the translational mode and the yaw mode at the same time with thissensor configuration since the both interferometers should be locked at their mid
fringes.

Fh+n1
-

S2
actuator
x LS2 b2

sER2

θ

F2

a1

sFB1
F1
sER1

nexc2

+

b1 LS1

test massactuator

a2

S1

+

nexc1

sFB2
-

A1

A2

Fh+n2

Figure 5.7: The block diagram of the servo system in the horizontal plane. The
test mass in the figure is pictured from the top view. The two laser interferometers
monitors the displacement b1 and b2 and each signal is fed back to the actuators set
at the opposite of the each interferometers.
Defining the force applied at the two coil-magnet actuators as a1 and a2 , respectively, the equations of motion in the translational mode and the rotational mode
represented in the frequency domain are
)
(
ωT0
2
2
(5.3)
m ω +iω + ωT0 x = a1 − a2 ,
QT
(
)
ωR0
2
2
(5.4)
I ω +iω + ωR0 θ = (a1 + a2 ) l,
QR
64

5 Experiments

5.2 Calibration and Sensitivity

where m, I and l are the mass, the moment of inertia of the test mass, and the
distance from the center of the test mass to the coil-magnet actuator. x is thetranslational displacement of the center of the mass, and θ is the angle of the bar
around the center of the mass. ωT0 and ωR0 are the resonant angular frequencies of
the test mass in the translational mode and the rotational mode, respectively. QT
and QR are the Q-value in the translational mode and the rotational mode.
Here, b1 and b2 are the displacement of the mirrors set on the test mass writtenasb1 = −x + lθ,
b2 = x + lθ.

(5.5)
(5.6)

Substituting 5.3 and 5.4 into 5.5 and 5.6, we get
b1 =
=
b2 =
=

−T (a1 − a2 ) + R(a1 + a2 )(−T + R)a1 + (T + R)a2 ,T (a1 − a2 ) + R(a1 + a2 )(T + R)a1 + (−T + R)a2 ,

(5.7)
(5.8)
(5.9)
(5.10)

where
T =

(

1

(

l2

),

(5.11)).

(5.12)2m ω 2 + i ωQT0
ω + ωT0TR =I2
ω + ωR0ω 2 + i ωQR0
R

Considering the control loop where the displacement disturbance from GWs h
are injected, the relation between a1 , a2 , b1 , and b2 are)( ) ( )
( )
(
a1
h
b1
−T + R T + R(5.13)= −+b2
T + R −T + R
a2
h
(
)(
)( ) ( )h−T + R T + R
B1 0
b1
+
,
(5.14)= −b2h
T + R −T + R
0 B2
where Bi = Si Fi Ai for i = 1, 2 . Therefore, the force applied on the test mass
including noise is written as
( ) [
(
)(
)] ( )h−T + R T + R
B1 0
b1
,
(5.15)= I+b2
h
T + R −T + R
0 B2
65

5 Experiments

5.2 Calibration and Sensitivity

where I is the identity matrix. The signal s′1 is derived by adding the two signals:
s′1 = 2h = (1 + 2RB1 )b1 + (1 + 2RB2 )b2 .

(5.16)The transfer function B1 and B2 are measured using the excitation signals nexc1
and nexc2 . The measured function isG̃i =Si b i,
Si bi + nexci(5.17)

where i = 1, 2. Here, we define the vectors and matrix as follows:
( )
b1
⃗b =
b2
(
)
nexc1 /S1
⃗nexc =
nexc2 /S2
)(
)
(
B1 0
−T + R T + R
.
G =
0 B2
T + R −T + R(5.18)(5.19)(5.20)

Assuming h ≪ nexc1 , nexc2 , ⃗b + ⃗nexc is derived as()⃗b = G ⃗b + ⃗nexc∴ ⃗b = (I − G)−1 G⃗nexc∴ ⃗b +⃗nexc = (I − G)−1 ⃗nexc .

(5.21)(5.22)(5.23)

Substituting 5.20 to 5.22 and 5.23,()
B2 (T + R)
⃗b = X B1 (−T + R) + 4B1 B2 T R⃗n (5.24),
B1 (T + R)
B2 (−T + R) + 4B2 B2 T R exc)(
B2 (T + R)
⃗b + ⃗nexc = X 1 − B2 (−T + R)⃗n ,(5.25)
B1 (−T + R)
1 − B1 (−T + R) exc
where
X=

1
.
1 − B1 (−T + R) − B2 (−T + R) − 4B1 B2 T R(5.26)

Since excitation signal is injected one by one, nexc2 = 0 when nexc1 ̸= 0 and nexc1 = 066

5 Experiments

5.2 Calibration and Sensitivity

when nexc2 ̸= 0 Therefore, the measured OLTF G̃i is
B1 (−T + R) + 4B1 B2 T R
1 − B2 (−T + R)
(
)2
T −R
G2
= G1 +
G1 ,T +R
1 − G2B2 (−T + R) + 4B1 B2 T R=
1 − B1 (−T + R)
(
)2
T −R
G1
= G2 +
G2 ,T +R
1 − G1G̃1 =G̃2(5.27)(5.28)(5.29)(5.30)

where G1 = B1 (−T + R) and G2 = B2 (−T + R).
The measured transfer function G̃1 and G̃2 are in Fig.5.8 and 5.9.

Figure 5.8: The transfer function G̃1 . The blue dots are the measured data and thegreen line is the fitted line.

67

5 Experiments

5.2 Calibration and Sensitivity

Figure 5.9: The transfer function G̃2 . The blue dots are the measured data and thegreen line is the fitted line.

5.2.3

Calibration to GW amplitude

According to 3.12 and 3.13, the three GW signals are derived from the out-of-loop
angular fluctuations as
1 1(θ1 − θ2 )
H×1 2
1
=
ϕ1
H×2
1
ϕ2
=
H×3

s1 =

(5.31)

s2(5.32)

s3(5.33)(5.34)

where H×i (i = 1, 2, 3) are defined as
OB
TM
.
− H×i
H×i = H×i

(5.35)OBTM
are the transfer functions from the GW signals to the angularand H×i
Here, H×i
fluctuations of the test masses and the optical bench, respectively. H+i are considered to be zeros since q+ = 0. Note the optical bench is also sensitive to GWs since
it is suspended in our setup.

68

5 Experiments

5.2 Calibration and Sensitivity

ROLL Mode Calibration
Assuming the test mass as a rectangular solid, the transfer function of the test massTM
H×2,3
is derived as
TM
q×1
TM
H×1 ∼ TM ∼ 0.48,(5.36)
2I1
above the resonant frequencies of the test masses in our setup.
OB
Note that the transfer functions of the optical bench H×2,3
are not zero in ourcase since the optical bench has also the rectangular shape from the side view.
OB
Calculating q×iand IiTM around the horizontal axis for i = 1, 2, we obtainOB
q×i∼ 0.44.
2IiOB(5.37)

This is calculated from the design value. The fiber optics and photo detector set
on the optical bench are not included in this calculation. Therefore, Hi used tocalibrate s2 and s3 is calculated as shown in Fig.5.10. The peak at 0.15 Hz is theresonant frequency of the test mass, and the peak at 1.1 Hz is the resonant frequencyof the optical bench.
YAW Mode Calibration
TM
The transfer function of the test mass is the same as that of ROLL mode H×1,2. In
the case of YAW, the transfer function of the optical bench is considered to be zero
since the optical bench has the circular shape from the top or bottom view.
The transfer function H×1 is shown in Fig.5.11. The peak at 0.1 Hz is theresonant frequency of the test mass. The absolute value of H×1 is about 0.5 above
the resonant frequency.

5.2.4

Sensitivity

The obtained typical sensitivity of this detector is shown in Fig.5.12. The sensitivityof s1 is improved by about 100 times at the maximum compared to the first prototype
at the higher frequencies where the seismic coupling noise limits the sensitivity in thefirst prototype. Al lower frequencies than 1 Hz, the sensitivity is not improved since
the resonant frequencies of the test mass are higher than before. The sensitivityof s2 and s3 is worse by about 1,000 times than s1 , because of the seismic noise as
described in Sec.5.3 and the calibration factor about 10 times diﬀerent from that ofs1 .
69

5 Experiments

5.2 Calibration and Sensitivity

Figure 5.10: The calibration factor from the rotational angle to the amplitude of
GWs in s2 . The resonant frequency of the test mass is set to be 0.15 Hz, and thatof the optical bench is 1.1 Hz. The quality factor is assumed to be 10.

Figure 5.11: The calibration factor from the rotational angle to the amplitude of
GWs in s1 .
70

5 Experiments

5.3 Seismic Noise

Figure 5.12: The GW strain equivalent noise spectra. The blue, green, and red lines
are the sensitivity of s1 , s2 , and s3 , respectively. The yellow line is the sensitivity
obtained from the first prototype of the TOBA.
In the following sections, their noise source and other characteristics will be
evaluated.

5.3

Seismic Noise

5.3.1

Passive Vibration Isolation

Since the test masses and the optical bench are suspended, the seismic noise isreduced by the passive vibration isolation. The passive vibration isolation rate, i.e.,
the transfer function from the top table to the signals are measured by shaking the
top table of the AVIT.
LONG to YAW coupling
Fig.5.13 is the transfer function from the displacement of the suspension point tothe signals of the interferometers LS1 and LS2 when the suspension point is shaken
to the LONG direction by the AVIT. The black line is the theory of the transfer
function described in Sec 4.4.2. In this plot, we assume that the coupling rate of71

5 Experiments

5.3 Seismic Noise0.1. It shows that the transfer functions to the LS1 and the LS2 signals are close to
the theory, though the peak was largely damped. The cause is under investigation
while it is considered to be because of the control loop.

Figure 5.13: Transfer function from the displacement at the top table to longitudinalmotion of the test mass. The blue dots represent the transfer function to the horizontal rotational signal. The black lines are the theoretical line with the assumptionof the decoupling rate of 10.

YAW to YAW transfer function
Fig.5.14 is the transfer function from the horizontal rotation of the top table to thehorizontal rotation of the test mass. The green line is the theory of the transfer
function from rotational motion of the suspension point to the rotational motion of
the test mass. The red line is the transfer function from the translational motion to
the rotational motion. The cyan line is the sum of the two transfer functions.
The transfer function from the translational motion to the rotational motion
should be considered since the motion of the AVIT is not perfectly decoupled. Even
though the output matrix of the AVIT is adjusted, the translational motion would
be excited when the rotational motion is excited because of the small asymmetry of
the AVIT.
72

5 Experiments

5.3 Seismic Noise

Other than the peaks, the gain of the transfer function at low frequencies also
does not fit to the theory. Though it is under investigation, it shows that thesensitivity is not aﬀected by the rotational motion of the suspension point.

Figure 5.14: Transfer function from the rotation at the top table to the rotation of
the test mass.

SIDE to ROLL transfer function
Assume that the coordinates of the suspension point and the test mass are X(t) andx(t), respectively (Fig.5.15), the equation of motion of the rotational angle θ(t) isI θ̈(t) + γ θ̇(t) = mgd(ϕ + θ),(5.38)

where ϕ = (X(t) − x(t))/l. I, m, g, d, and l are the moment of the inertia of thetest mass, the mass of the test mass, the gravity acceleration, the distance between
the suspension point and the center of the mass of the test mass, and length of thesuspension wire, respectively. Performing Fourier transformation and substituting
4.26, we obtain
)
(
ωθ0
ωTM0
X̃
θ̃(ω) =
1−,
(5.39)
ωθ0
ωTM0
2
2
2
−ω + i Q ω + ωθ02
−ω + i Q ω + ωTM0 l
73

5 Experiments

5.3 Seismic Noise

where ωθ0 = mgd/I. The same equation is applied to the optical bench. The
observed transfer function is the diﬀerence between the rotational angle of the test
mass and that of the optical bench.

X

φ
F=mg
d
θ

center of mass

Figure 5.15: A schematic view of the test mass excited by translational displacement.
The black line in Fig.5.16 shows the transfer function calculated as above. We
assume that the resonant frequency of the ϕ(s2 ) of the test mass as 0.15 Hz, oneof the ϕ of the optical bench as 1.1 Hz, which is calculated in case that the opticalbench do not have any optics and the center of the mass is at the suspension points.
The peak appeared at about 0.7 Hz is generated by the pendulum mode from thependulum mode including the intermediate mass. The resonant frequency of thependulum mode only below the intermediate mass is about 1 Hz. Note that Q
values are assumed to be 30 for the optical bench.

5.3.2

Seismic noise level

From the measurement above, how largely the seismic noise contributes to the sensitivity is calculated. Figure 5.17, 5.17, and 5.17 show the sensitivity and the seismic
74

5 Experiments

5.3 Seismic Noise

Figure 5.16: The transfer function from the translational displacement at the top
table to the signals of the rotational mode. The blue points represent the transfer
function to the two outputs of the interferometers, LS4, from which the ROLL signal
is derived. (See Fig.4.12.)75

5 Experiments

5.4 Decoupling

noise contribution. They show that the seismic noise limits the sensitivities around1 - 10 Hz.

Figure 5.17: The aﬀect of the seismic noise on the sensitivity. The blue line is theangular displacement of the YAW mode and the cyan line is the contribution of theseismic noise.

5.4

Decoupling

When the signals are calibrated from the signal to GW-equivalent noise spectrum,
the signals of four interferometers are diagonalized to four degrees of freedom.
Though the rotational angle θ = sLS1 − sLS2 in principle, the eﬃciency of the two
sensors are slightly diﬀerent. Here, sLS1 and sLS2 represent the signals from LS1 andLS2, which are converted to the rotational angle. The diﬀerence between the two
eﬃciency is, for example, caused by the diﬀerence of the distance between the centerof the mass and the point on which the laser beam hit. Therefore, we adjusted thecoeﬃcient of the decoupling according to the height of the peaks appeared becauseof the translational motion as shown in the previous section. Figure 5.20 shows theeﬃciency of the decoupling. The blue line is the GW strain equivalent noise to which
the raw signal from LS1 is calibrated. The green line is the signal of sLS1 − sLS2 ,i.e., the signal decoupled with non-adjusted coeﬃcients. The GW signals are not76

5 Experiments

5.4 Decoupling

Figure 5.18: The aﬀect of the seismic noise on the sensitivity. The green line is theangular displacement of the ROLL mode, which corresponds to s2 , and the yellow
line is the contribution of the seismic noise.

Figure 5.19: The aﬀect of the seismic noise on the sensitivity. The red line is theangular displacement of the ROLL mode, which corresponds to s3 , and the magenta
line is the contribution of the seismic noise.
77

5 Experiments

5.5 Common Mode Noise Reduction

cancelled by the decoupling since the signals used for the decoupling is confirmed tobe the translational motion from the seismic noise measurement mentioned above.
The red line is the signal of rotational motion decoupled with adjusted coeﬃcients.
They show the coeﬃcient adjustment improves the sensitivity about 5 times at the
maximum.

Figure 5.20: The GW equivalent noise spectra of sLS1 , sLS1 + sLS2 , and csLS1 + c′ LS2.
c and c′ are the adjusted coeﬃcients. It shows he careful decoupling improves thesensitivity.

5.5

Common Mode Noise Reduction

In our setup, the motion of the two test masses are measured. The signal s1 isderived from the diﬀerential rotation of the two test masses. It reject the rotational
noise which aﬀect on the test masses commonly. Figure 5.21 shows the eﬀect of thecommon mode noise rejection. The blue line and the green line are the sensitivity
derived from the rotational fluctuation of the single test mass. The red line is thediﬀerential rotation of the two test masses. It shows the noise at low frequencies,
mainly below 1 Hz are reduced by about 5 times at maximum.
From these observations, it is shown that the sensitivities of s2 and s3 are limitedby the seismic noise at almost all the frequencies. Also, the sensitivity of s1 below
78

5 Experiments

5.6 ADC/DAC Noise

Figure 5.21: The GW equivalent noise spectra of θ1 , θ2 , and (θ1 −θ2 )/2. The common
mode rejection between the two angles are eﬀective at lower frequencies.
about 2 Hz is limited by the seismic noise even though its eﬀect is reduced by thedecoupling and common mode rejection.

5.6

ADC/DAC NoiseADC and DAC has the quantization noise since the analog signals are converted todiscrete signals and vise versa. The noise level of ADC and DAC is shown in Fig
5.22.

5.7

Sensor Noise

5.7.1

Shot NoiseThe shot noise is one of the fundamental noise for laser interferometers which is
caused by the Heisenberg’s uncertainty principle. A photo detector counts the number of photons. Here, the quantum mechanics yields the fluctuation of the numberof the photons. Since the number of photons has Poisson distribution, the current
has uncertainty called the shot noise.
79

5 Experiments

5.7 Sensor Noise

Figure 5.22: The noise spectrum of the ADC and the DAC. The blue line is thesensitivity of s1 . The green line is the contribution from the ADC, and the red line
is from the DAC.
When an averaged current iDC is emitted from the photo detector, the spectrumof the shot noise current is written as√
ishot = 2eiDC ,(5.40)
where e is the elementary charge. The DC photo current produced by the laser withpower of P iseηi=
(5.41)P,ℏΩ
where η and ℏ are the quantum eﬃciency of the photo diode and the reduced Planck
constant, respectively. The noise in the optical pass length caused by the shot noiseisλ2δlshot =Pshot .
(5.42)
4π Pmax − Pmin
Assuming Pmin = 0 for the convenience, equivalent optical pass noise is obtained as√ℏcλ,(5.43)
δlshot =4πP
where P = Pmax /2.
80

5 Experiments

5.7.2

5.7 Sensor Noise

Radiation Pressure Noise

Radiation pressure noise arises from the Heisenberg’s uncertainty principle as wellas the shot noise. When photons are reflected by a mirror, back reaction force isapplied to the mirror. The back reaction force also fluctuates since the number ofphotons hitting on the mirror also fluctuates.
The applied force is written as√
2πℏP.
(5.44)
δFrad =
cλTherefore, the angle fluctuation according to the quantum back action is√
1
2πℏPδθ = 2,ω I
cλ(5.45)

where I is the moment of inertia of the test mass bar.
The shot noise and radiation pressure noise in the GW strain equivalent noise isshown in 5.23.

Figure 5.23: The shot noise and the radiation pressure noise. The blue, green, andred lines are the sensitivity of the signals s1 , s2 , and s3 , respectively. The magenta
and the cyan lines are the shot noise and the radiation pressure noise. The shotnoise and the radiation pressure noise in s1 are diﬀerent from those in s2 and s3
since the calibration factors are diﬀerent. Note that the calibration factors of s2
and s3 are assumed to be the same, though they are slightly diﬀerent because of thediﬀerent resonant frequencies.

81

5 Experiments

5.7.3

5.7 Sensor Noise

Intensity NoiseThe amplitude of the laser is not perfectly stable. The intensity noise of our laser
source is shown in Fig.5.24. It is measured with the laser beam just before injected to a beam splitter, just after the initial beam is split into 8 beams for the 8interferometers on the optical bench.

Figure 5.24: The noise caused by the intensity noise and the phase noise from thelaser interferometers. The blue line is the sensitivity in s1 . The green, and the redlines are the noise from the intensity fluctuation of the laser, and the phase noise
injected from the interferometers, respectively.

5.7.4

Phase Noise

The total noise caused by the fiber interferometer is also shown in Fig.5.24. You can
see that the sensitivity is limited by the phase noise of the interferometers or close
to the phase noise at all frequencies, while some peaks appeared in the spectrum of
s1 is caused by the seismic noise.
It is the spectrum of the output of the interferometers when the test masses are
fixed. It includes the shot noise, the dark current noise, the intensity noise, and thephase noise induced from the fiber interferometers. Since the noise level is larger82

5 Experiments

5.8 Performance of the AVIT

than the other noise described above, the sensitivity of the interferometers is limitedby the phase noise caused by the fiber interferometer part.
Our noise level of the phase noise above 1 Hz is almost the same as the sensitivityof the interferometers with fixed mirrors not on the optical bench but on the optical
table. [61, 62]. It means that the cause of the noise of our interferometers is not thestructure of the optical bench, but the configuration of the fiber interferometers.

5.8

Performance of the AVIT

5.8.1

Open Loop Transfer Function

The open loop transfer function of the AVIT control in the translation direction isshown in Fig.5.25.
It shows out vibration isolation region is about 0.5 Hz to 7 Hz. The vibration
isolation rate is limited by the dynamic range of the piezoelectric elements at low
frequencies. The seismic displacements has a bump around 0.5 Hz called the micro
seismic noise, which is mainly caused by ocean waves beating the shore. The micro
seismic noise is too large to compensate the displacement with the piezoelectric
devices we used. At higher frequencies, the vibration isolation rate is limited by
the peaks at 7 and 9 Hz. These peaks are resonance of the structure of the poles
supporting the AVIT which is set around the vacuum tank. Since these poles are
originally made for the support of the superconductor in the first prototype, they
are not thick enough.
The blue dots do not fit to the theoretical values at lower frequencies in x and y
directions. It is because of the tilt coupling of the seismometers. Since the relative
position of the coil and the suspended magnets in the seismometers also changes
according to the tilt of the seismometers, the careful adjustment of the output
matrix is necessary in order not to rotate the top table when the table is actuatedin translational direction. Even though the careful adjustment, the tile couplings
still remain.

5.8.2

Vibration Isolation Performance

The vibration isolated seismic displacement by the AVIT controlled by the open looptransfer function as shown above is Fig.5.26. The blue, red, and green lines are theseismic displacement in x, y, and z directions. The dashed lines represent the seismic
displacement when the AVIT is not controlled, while the solid lines represents the83

5 Experiments

5.8 Performance of the AVIT

x

y

z

Figure 5.25: The open loop transfer functions of the AVIT in x, y, and z directions.
The red line and dots are the total open loop transfer function of the control system(G in 4.42 and 4.43). The blue and the green lines and dots are the open looptransfer function of the seismometers (Gs ) and the photo sensors (Gp ), respectively.
The dots are measured data, while the lines are theoretical values.

84

5 Experiments

5.8 Performance of the AVIT

seismic displacement when the AVIT is active. It shows that the seismic vibration
is isolated about 10 times at 1 Hz.
5

Figure 5.26: The seismic noise with the AVIT oﬀ and on. The dashed lines represent
the seismic displacement with AVIT oﬀ, and the solid lines represent one with the
AVIT on.
Also, the spectrogram of the seismic displacement during the AVIT on is shownin Fig.5.27. It shows that the AVIT is able to stably isolate the vibration in all thetranslational directions at the same time for more than two days. Note that it ismeasured without the suspension systems.

5.8.3

Sensitivity Improvement

In order to see the impact of the AVIT, we compared the sensitivity of Phase-II
TOBA with the AVIT on and oﬀ. In Fig.5.28, the solid lines are the sensitivity
with the AVIT on, while the dashed lines are the sensitivity with the AVIT oﬀ. The
blue, green and red lines represent the sensitivity of s1 , s2 , and s3 with AVIT on,
respectively. The sky-blue, right green, and pink dashed lines are the sensitivity ofs1 , s2 , and s3 with AVIT oﬀ. The sensitivity around 1 Hz is improved in the every
signals according to the vibration isolation rate obtained by the AVIT. However, the85

5.8 Performance of the AVIT

Frequency [Hz]

-6.5
-7
-7.5
-8
-8.5

y

-7
-7.5
-8
-8.5

-9

-9

Time [H]-6

Frequency [Hz]

-6.5
-7
-7.5
-8
-8.5

Displacement (log) [m/rtHz]Time [H]

z

-6
-6.5

Frequency [Hz]-6

Displacement (log) [m/rtHz]x

Displacement (log) [m/rtHz]

5 Experiments-9Time [H]Figure 5.27: The spectrogram of the seismic displacement during the AVIT is on,which is plotted in the logarithm of 10. The top left, the top right, and the bottom
left are the spectrogram of x, y, and z, respectively. The left color-bar on the left ofthe each spectrogram shows the seismic displacement with the AVIT oﬀ.

865 Experiments

5.8 Performance of the AVIT

noise around 7 Hz is raised when the AVIT is on. It is because of the deteriorationof the seismic noise at high frequencies. Since the phase margin around the peaksin 7 Hz and 9 Hz due to the resonance of the supporting table for the AVIT is notenough, the seismic noise gets worse around 7 Hz when the AVIT is controlled.

Figure 5.28: The sensitivity of signals with the AVIT on and oﬀ.The blue, green
and red lines represent the sensitivity of s1 , s2 , and s3 with AVIT on, respectively.
The sky-blue, yellow, and pink dashed lines are the sensitivity of s1 , s2 , and s3 with
AVIT oﬀ.

87

6

Chapter
Observation and Analysis

Using the Phase-II TOBA we developed, the observational run had been performed
as a part of the end-to-end test. As a demonstration, we have searched for GWs
from the inspiral phase of the compact binary coalescences around 1 to 10 Hz using
matched filtering analysis. The target source is the intermediate mass black holebinaries with the mass of 200 M⊙ , which had never been searched for previously. In
this chapter, we will explain about the source and how to analyze the data obtained
from the Phase-II TOBA.

6.1

Intermediate-Mass Black Hole Binaries

The compact binary coalescences (CBCs) are the most promising source for the GW
detectors. The frequencies of GWs emitted from the binary system depend on theirmass. At lower frequencies where TOBA has good sensitivity, such as around 1 Hz,
GWs from the intermediate-mass black hole (IMBH) inspiral with the masses of
102 M⊙ ≲ m ≲ 103 M⊙ are the main observation target.

6.1.1

Observational Evidences of Intermediate-Mass Black
Holes

Recently, hyper-luminous X-ray source 1, HLX-1, in the galaxy M82 was provento be the IMBH from the observation with Chandra and Swift [42]. The IMBHs
are considered to be mainly in globular clusters and formed by the core collapse oflarge star clusters [58]. These IMBHs formed in the cluster would fall into their hostgalaxy. When they interact with other IMBHs on the way to the center of their
host galaxy, they would form a binary system, which emit energy in the form of the
GW. Since they lose their energy because of the gravitational radiation, they would
merge and form a single super-massive black hole at the center of the galaxy. Thisis one of the hypotheses of the formation of the super massive black holes.
89

6 Observation and Analysis

6.1 Intermediate-Mass Black Hole Binaries

Besides the observations with electro-magnetic waves, the GW detectors are
expected to observe such IMBH coalescence directly and estimate how often the
IMBHs merge and form a super massive black hole. The direct observation of
IMBHs with GWs will reveal how the super massive black holes are evolving, andhow the galaxies are evolved.

6.1.2

Waveform

CBCs are composed of three dynamical phases: inspiral, merger, and ringdown. In
the inspiral phase, the chirp signal is obtained as described in Sec.2.2.1. The waveforms of the inspiral are described accurately by the post-Newtonian approximation
to the Einstein equation [16]. The signal in the merger phase is modeled only withnumerical simulations since there is a nonlinear eﬀect [31]. The ringdown is thephase that the merged objects are resting. In our analysis, we searched for only the
chirp signals. In the following, we will explain about the chirp waveform in details.
The GW signal as observed by the detector is written as
h(t) = F+ h+ (t) + F× h× (t),(6.1)

as explained in 3.19. Here, the pattern functions are assumed to be constant since
the eﬀect of the Earth spin is negligible for the GW signals with the frequencies of
1 - 10 Hz. The chirp waveform can be written as
GM⊙(1 + cos2 ι)hc (t),
c2 r
2GM⊙
h× (t) = − 2
cos ιhs (t),cr
h+ (t) = −

with

(
)
2
µ
hc (τ ) = 2(πGM f (t))2/3 cos Φ(t),
c M⊙
(
)
2
µ
(πGM f (t))2/3 sin Φ(t).
hs (τ ) = 2
c M⊙

(6.2)
(6.3)

(6.4)
(6.5)

Here, G, M⊙ , c, r and ι are the gravitational constant, the solar mass, the speed oflight, luminosity distance from the Earth to the source, and the inclination angle.
µ is called the reduced mass of the binary defined as
µ=

m1 m2
,
m1 + m2
90(6.6)

6 Observation and Analysis

6.1 Intermediate-Mass Black Hole Binaries

where m1 and m2 are the mass of the binary objects. M is the total mass of thebinary defined as M = m1 + m2 . The GW frequency f (t) is function of time since
the period of the binary system varies. f (t) and Φ(t) is calculated through thepost-Newtonian formalism [16]. At leading order,
(
)3/8
c3
5GM
f (t) =,
(6.7)
8πGM c3 η(tc − t)where η is the symmetric mass ratio η = m1 m2 /M 2 and tc is the coalescence time.
The orbital phase Φ(t) is derived as∫
Φ(t) = f (t)dt − ϕ0 ,(6.8)
where ϕ0 is an initial orbital phase.
Also, it is useful to derive the waveform in the frequency domain. Using thestationary phase approximation, the chirp signals in the frequency domain is derived[56, 25] as()1/2 (
)1/3
(
)−1/6
5µ
M
G
2GM⊙
−7/6
h̃c (f ) =feiΨ(f ;M,η) ,
(6.9)
c2 r
96M⊙
π 2 M⊙
M⊙
with

[
(
)
π
3715 55
3−5
Ψ(f ; M, η) = 2πf tc − 2ϕ0 − +x ++ η x−3 − 16πx−2
4 128η
756
9()]
15293365 27145
3085 2++η+η x−1 ,
508032504
72(6.10)

where x = (πM f G/c3 )1/3 . Since hs (t) and hc (t) are orthogonal to each other, h̃s (f )is obtained ash̃s (f ) = ih̃c (f ).
(6.11)
Note that the Post-Newtonian expansion is applicable only in a flat background.
Therefore, the waveform is not correct when the two objects are close, i.e., in thephase of merger. The minimum radial distance of the binary system beyond which
the approximation is no more correct is called Innermost Stable Circular Orbit(ISCO). The frequency of the GW signal when the ISCO is approached is derived
as
c3
1
.
(6.12)
fISCO = √6 62π GM91

6 Observation and Analysis

6.1 Intermediate-Mass Black Hole Binaries

Substituting the numerical values, fISCO is(
fISCO ∼ 5.5 Hz

400M⊙
M

)
.

(6.13)

The waveform used for the matched filtering was stopped at fISCO since the waveform
at higher frequencies than fISCO is no longer in the phase of chirp but merger. Figure
6.1 shows the spectrum of the waveform.

Figure 6.1: The spectrum of the waveform and the detector sensitivity. The blueline is the sensitivity of s1 . The red and green dashed lines are the spectrum of the
GW chirp signal. The red line is the signal from the binary system with the mass of
200M⊙ − 200M⊙ at 0.01 pc away from the Earth, while the green line is the signal
from the binary system with the mass of 200M⊙ − 200M⊙ at 10 pc away from the
Earth. The dots show the fISCO , where the chirp waveform stops.

92

6 Observation and Analysis

6.1.3

6.2 Analysis Pipeline

Previous Results

LIGO and Virgo have searched for the IMBH previously. However, the GWs from
IMBH inspiral appear only in lower frequencies than the frequency where the interferometric GW detectors have the best sensitivity. Therefore, they searched for GWs
from IMBH inspiral with the masses of 100 M⊙ at the maximum [5]. Note that they
searched for GWs including IMBH merger and ringdown. The most latest result isthat they have set the upper limit on the event rate as 3.3 × 10−7 times/Mpc3 /yr
for the IMBHs mergers with the masses of 19 - 28 M⊙ [21].
Considering GWs from IMBH merger and ringdown, they have searched for the
IMBH mergers with the total mass of up to 500M⊙ [2]. They recently set an upper
limit on the event rate 6.9 × 10−8 mergers/Mpc3 /yr for the total mass of 100 -150
M⊙ as the result of the ringdown search [1].

6.2

Analysis Pipeline

The flowchart of the signal processing is shown in Fig.6.2. In our analysis, the mostsensitive signal s1 is mainly used for the analysis, though we have the other signals s2
and s3 . s2 and s3 are used only for the coincidence check when s1 has the significant
signal. In other words, we perform the analysis using only s1 and the other signals
s2 and s3 are used only for the consistency check.

6.2.1

Segmentation and Calibration

At first, we divide the raw signal into the several segments with the length of Tseg .
Each segment is analyzed independently after transformed into the frequency domain
and calibrated according to the description in Sec.4.5.1. In the Fourier transformation, we use the Hanning window as a window function. Therefore, we take theoverlap of 50 % between the two segments next to each other so that information is
not lost because of the window function.
After the Fourier transformation is performed, the data are calibrated into the
GW strain equivalent amplitude as described in Sec.4.5.1. Using calibrated data,
the data selection is performed as a next step.

6.2.2

Data Selection

The noise level over the observation time is not stable. Sudden noise, such as
spike-like noise is temporarily observed. Such noise will aﬀect the analysis badly.
936 Observation and Analysis

6.2 Analysis Pipeline

Devide the row data into the segmentswith the length of
・・・・

・・・・

Fourier transform
・・・・

・・・・

RMS vetoθ1 + θ2 veto
・・・・
・・・・

・・・・

Matched Filtering
・・・・
・・・・

・・・・

Coincidence test
with
and

Non-Detection
Signal Injection
Test

Detection

Upper Limit

Parameter
Estimation
Figure 6.2: The flowchart of the signal processing.

94

6 Observation and Analysis

6.2 Analysis Pipeline

Sometimes they are detected as signals in error, or make the upper limit worse.
Therefore, we select the data to be analyzed and condition the data quality. It is
important not to remove the GW signals intentionally. In our analysis, we applied
two vetoes in order to remove the noisy segment.
RMS Veto
When the noise level gets worse, the root mean square (RMS) gets larger. Therefore,
we remove the segments where the RMS value is large. At first, the data is whitenedass̃ij (f )s̃ijwt =,(6.14)|s̃i (f )|where i is the indexes of the output signals and j is the indexes of the segments.
¯ )||˜ i (fs is the averaged spectrum over the whole segments. This procedure enables
us to evaluate the level of the noise floor by reducing the eﬀect at the peaks appear
at the specific frequencies.
The RMS is calculated only the outside of the frequency range where the matchedfiltering is performed. It prevents GW signals from being removed intentionally. In
our analysis, two frequency band is used to calculate the RMS. One is below 0 - 1
Hz, the other is 10 - 100 Hz. In other words,
v
u 0.1
100
u∑
1 ∑
2
t
X=
|sijwt (f )| +|sij (f )|,(6.15)
90 f =10
f =0where the coeﬃcient 90 is the normalizing factor, so that the weight of the RMS at
two frequency band is equal.
θ1 + θ2 Veto
The data is also selected using the actual data which do not have sensitivity to GWs,
i.e., θ1 + θ2 . The signal s1 = θ1 − θ2 , which has the information about the noise andGWs, is used in order to derive the SNR ρ in the main analysis. Here, the SNRρveto calculated by the matched filter analysis using the veto signal sveto = θ1 − θ2
is used for a veto indicator. The segments with large ρveto are removed by this vetoprocedure.
While the GW signals are rejected in sveto , sveto has the information about thenoise level. Therefore, this is the eﬀective veto system which exploits the information about the noise at the analyzed frequencies while the GW signals are notintentionally removed.
95

6 Observation and Analysis

6.2.3

6.2 Analysis Pipeline

Matched Filtering

Using the survived data, we perform the matched filtering analysis. In order todetect the small GW signal h̃(f ) in the detector output s̃(f ) = h̃(f ) + ñ(f ), whereñ(f ) is the detector noise, we take the correlation between h̃(f ) and s̃(f ) as thesignal:z(t) = (s|h),(6.16)
where (a|b) is the inner product defined by∫(a|b) = 4Reã(f )∗ b̃(f ) 2πif tedf.
Sn (f )

(6.17)

Here, the data are weighted by the noise power spectral density of the detector
output Sn (f ). ∗ denote a complex conjugate. Since h consists of two orthogonal
signals hc (t) and hs (t), the signal z is computed by√(6.18)z = (s|hc )2 + (s|hs )2 .
The variance of (s|h) in the presence of a signal is⟨⟩ ⟨⟩σ 2 = [(s|h) − (h|h)]2 = [(n|h)]2= (h|h).

(6.19)

Therefore, the squared SNR ρ2 is written as|z|2
ρ = 2,
σ2(6.20)

which is analytically maximized over the amplitude and the phase of the GW waveform. Note that ρ2 has χ2 distribution with 2 degrees of freedom when the detector
noise is Gaussian.

6.2.4

Signal Detection

The presence of the GW signal is determined according to the values of the square
SNR ρ21j . According to the Neyman-Pearson criterion, the decision rule is given as• if ρ1j ≤ ρth , the signal is present in the j-th segment,• if ρ1j < ρth , the signal is absent in the j-th segment.
96

6 Observation and Analysis

6.2 Analysis Pipelineρth is called ‘detection threshold,’ which is determined from the probability distribution of ρ1j and the false alarm rate. The false alarm rate is the probability that
we decide the signal is present, though the signal is absent in reality. Defining theprobability distribution of ρ1j when the signal is absent as p(ρ|0), the false alarm
rate α is written as∫ +∞α=p(ρ|0)dρ.
(6.21)
ρth

In the case of the matched filtering, p(ρ|0) is χ2 distribution with two degrees of
freedom.
If the signal is detected, the consistency with s2 , s3 , or the environmental signal
is checked in order to ascertain the detected signal is not noise but GWs. Here, theGW parameters such as their amplitude, i.e., the distance to the source, is estimatedfrom the value of ρ. When the signal is undetected, the injection test is processedin order to set the upper limit.

6.2.5

Upper Limit

We performed the signal injection test in order to set an upper limit. This analysis
derives the distance to the GW sources from which the GW signals are detectablewith our detector. The procedure is as follows:• Make the simulated signal in the frequency domain from the GWs source apartfrom the Earth by the distance r. The sky position of the source of the mocksignal, polarization angle, and the inclination angle are randomly decided.
• Inject the signal into a randomly chosen segment and calculate the SNR of theinjected segment.
• Repeat the first and the second procedure.
• Calculate a false dismissal rate (FDR), which is the probability that we missthe GW signals, from the obtained distribution of the SNR.
• Change the distance r and repeat the procedure above. Then the FDR functionof r is derived.
The distance r where the FDR is β is the upper limit with the confidence level of β.
The Fig.6.3 is the position of the mock signal in the sky. Here, we assume thatthe sources are not localized, i.e., isotropic. It shows the position of the mock signal
is equally random.
97

6 Observation and Analysis

6.3 Observation

1.5

1.0

0.5

0.0

−0.5

−1.0

Figure 6.3: Blue dots are the direction of the mock signal. It is set on the spherewith the radius of 1.
−1.5

1.5

1.0

0.5
0.0

6.3

−1.5

Observation
−1.0

−0.5

−0.5
0.0

0.5

−1.0
1.0

1.5

−1.5

We performed the observation from 8:50 UTC, December 10, 2014 to 8:50 UTC,
December 11, 2014. The control system stably continued for more than 24 hours.
The latitude and the longitude of the cite are (35.42, 139.45). The two test mass
bars lie almost north and south, and east and west. One is oblique in the northwest
to the southeast direction by about 10 degrees from the north-south line, and theother.
The spectrogram of the signals during the observational run is shown in Fig.6.4.
The control systems continued stably, and are disturbed mainly by the earthquakes.

6.4

Parameters

In our analysis, we searched for the GWs from the binary of equal-mass IMBHs
with the mass of 200M⊙ . Though the waveform template bank is necessary in order
to search for binaries with various parameters, we performed the matched filtering
only with the fixed parameters for the simplification.
98

6 Observation and Analysis

6.4 Parameters

s2

s1

1

6

11

16

11

16

1

6

11

16

s3

1

6

Figure 6.4: The spectrogram of the three signals during the observation.

99

6 Observation and Analysis

6.5 Results

The obtained data were divided into the segments with the length of 200 seconds,
and we take the overlap between the segments as 50 %. The length of the segment islonger than the time of the chirp signal, which is calculated to be about 150 seconds
using 2nd Post-Newtonian expansion in this case. In the data selection pipeline, weremoved the 10 % of the data in the RMS veto and the θ1 + θ2 veto respectively.
We have set the False Alarm Rate (FAR) as 10−5 events/year, which means
α = 3.17 × 10−11 . Considering that the distribution of ρ2 obeys χ2 distribution, thedetection threshold is determined as ρ2th = 48.3.

6.5

Results

6.5.1

Data Selection

The values of RMS and ρveto is shown in Fig.6.5 and Fig.6.6. The lines are the valuesof the RMS and ρveto , and the circles represents the removed segments according to
the each veto. The dots indicate the segments removed by the other veto. It shows
the value of RMS increased as the observation continues, while ρbeto is relatively
stable over the observation.

Figure 6.5: The value of RMS and the removed segments. The cyan circles are theremoved segments by the RMS veto. The magenta dots are the segments removedby the θ1 + θ2 veto.
100

6 Observation and Analysis

6.6 Observational Boundary

Figure 6.6: The value of ρveto and the removed segments. The magenta circles are
the removed segments by the θ1 + θ2 veto. The cyan dots are the segments removedby the RMS veto.

6.5.2

Signal to Noise Ratio

The time series of the ρ is shown in Fig.??, and the histogram of the ρ2 obtained
from the matched filter analysis is shown in Fig.6.8. The red line is the theoreticalvalue of the χ2 distribution. Though the values of SNR have tails, which exceed theχ2 distribution, no signal which exceed ρth was found. Therefore, we concluded that
no GW signal are detected.

6.6

Observational Boundary

Since we detected no GW signal, we have set an upper limit on the distance where
IMBH does not exist. Figure 6.9 is the FDR derived from the signal injection. Here,
we injected data for 74000 times, which is the number of survived segments times
100. The distance where the FDR is 10 % is derived by collinear approximation
using the data close to 10 %, which is independent from the assumption of the IMBH
distribution. The fitted linear line is shown in the red line. Consequently, we have an101

6 Observation and Analysis

6.6 Observational Boundary

Figure 6.7: The time vs the SNR. The blue dots are the value of ρ. The dots at 0
represents the segments removed by the data selection.

Figure 6.8: The blue dots are histogram of ρ. The green line is the theoretical lineof χ2 distribution.
102

6 Observation and Analysis

6.6 Observational Boundary

90 % confidence upper limit that no IMBH binaries exist where r < (2.1−0.7)×10−4
pc from the fitted parameters with the fitting error.
The systematic error arises mainly from the uncertainty in calibration: the amplitude of the fringe of the interferometers, the beam spot position and fluctuation of the eﬃciency of coil-magnet actuators. We estimate these systematic error
as 10 %. Therefore, our upper limit with 90 % confidence including the error isr < (2.1 − 0.7 − 0.2) × 10−4 pc.

Figure 6.9: The false dismissal rate derived from the signal injection. The blue dots
are derived FDR from the simulation. The red line is fitted linear line using 5 points
around FDR = 10 %.

103

7

Chapter
Summary
7.1

Summary and Conclusion

We developed the detector called Phase-II TOBA. The TOBA is GW detector which
has two bar-shaped test masses. The two bars rotate due to the tidal force of GWs.
Its main feature is that TOBA has a sensitivity at lower frequencies such as around 1
Hz even on the ground thanks to the low resonant frequencies of the rotational mode.
Our scientific targets are GWs from IMBH binaries or a stochastic GW background.
These observations will provide us the information about the Universe, which is notable to be obtained with the electro-magnetic waves. Assembling several techniques,
such as high-Q suspension for 10-m scaled large test mass bars, and cryogenic system,
TOBA is expected to detect GWs from mergers of IMBH binaries about 10 Gpc
apart from the Earth.
Previously, the first prototype TOBA had been developed. The principle test
and the first GW observation around 0.2 Hz were performed as the first detector.
In this thesis, we have developed the Phase-II TOBA, which is the new detector forthe technical verification and the sensitivity improvement. We have focused on thevibration isolation technologies and the common mode noise reduction technologies,
which are realized by a suspension system and an active vibration isolation table(AVIT). The Phase-II TOBA is an important step towards the detector with enoughsensitivity for the GW astronomy.
Besides, we introduced the multi-output system in the Phase-II TOBA for the
first time. The multi-output mode provides three independent signals from a single
detector without the drastic change of the setup configuration. The only modules we
add are the sensors for the vertical rotation monitoring. In our research, we pointed
out that the test masses rotate in the vertical plane as well as the horizontal plane.
Using the three signals, not only the event rate, but also the angular resolution
would be improved since the three outputs have diﬀerent response function to the
GWs. Assuming the three signals have the same sensitivities, the event rate would
be improved by about 1.7 times. Also, the sky positions of the GW sources could be105

7 Summary

7.1 Summary and Conclusion

determined with only a pair of detectors set apart, and also narrowed down to the 8points even with a single detector, while three conventional detectors are necessary
in order to determine the sky positions of the GW sources. It means that TOBA
could evolve the GW astronomy with fewer detectors. The multi-output systemenhances the value of TOBA as a simple and inexpensive ground-based detector for
low-frequency GWs.

In our√ setup, the sensitivity in the horizontal rotation is achieved to be about
10 [1/ Hz] in the GW strain at 7 Hz. The sensitivity was improved especiallyaround 1 - 10 Hz by about 100 times than the previous detectors because of thepassive vibration isolation using the suspension system, AVIT which reduces theseismic noise about 10 times at 1 Hz, and common mode noise rejection. As a result,the current sensitivity is limited by the residual seismic noise and sensor noise whichis induced from fiber laser interferometers. Our result shows the eﬃciency of thevibration isolation system. The seismic noise has been reduced to be close to thesensor noise already. Especially, the AVIT is the powerful tool for the low-frequency
vibration isolation system not only for TOBA but also other experiments whichneeds precise measurement.
−10

Also, we successfully obtained the three signals at the same time and establishedthe multi-output system. However, the sensitivity in the vertical rotation is worse
by about 103 times than that in horizontal. The optical bench where the sensors
and the actuators are set is also sensitive to GWs incoming from the horizontal
direction. It reduces the sensitivity of the vertical monitor by about 10 times. The
sensitivity is limited by the seismic noise over almost all the frequencies since thetranslational seismic displacement excites the rotational motion.

Using our setup, we have performed the observational run for 24 hours andsearched for the chirp GW signal from mergers of IMBHs with the mass of 200M⊙ .
This is the first search for the IMBH black hole binaries with the mass larger than
100M⊙ . Our system was stable enough to continue the observation for more than
one day without losing its control. As a result of matched filtering analysis, no
GW signals are detected and we exclude the presence of IMBH merger at r <(2.1 − 0.7 − 0.2) × 10−4 pc with 90 % confidence. Here the analysis pipeline for the
IMBH search was successfully established in our detector.
106

7 Summary

7.2 Future Plans and Applications

7.2

Future Plans and Applications

7.2.1

Sensitivity

In order to improve the sensitivity more, the noise to get rid of at first is theseismic noise and laser sensor noise. As for the sensor noise, it is necessary to reveal
the mechanism from which the sensor noise come. Though it is relatively easy toassemble many monitors with the fiber optics, more study is required in order toreduce the phase noise in the fiber optics enough. For the next upgrade, the sensors
are necessary to be upgraded to the Fabry-Perot interferometers in terms of theadvanced technology. Fabry-Perot interferometric sensors are expected to improve
the sensitivity due to an optical spring eﬀect [54]In order to reduce the seismic coupling noise further, the much larger suspension
system is required. However, since the resonant frequency of the pendulum mode isaround 1 Hz, the peaks of the pendulum resonance would decrease the sensitivity.
Still, it is fundamentally possible to reduce the seismic noise more by adjusting theposition of the center of the mass of the optical bench and so on.
The vibration isolation rate of the AVIT could also be improved by replacing thepiezoelectric elements to those with a larger dynamic range. In order to improve
the performance at higher frequencies, the support table for the AVIT should be
buttressed since the resonance of the supporting table limit the open loop transfer
function of the control system.
In order to improve the sensitivity in lower frequencies, it is necessary to decrease
the resonant frequency of the rotational mode. Since the resonant frequencies in our
setup are relatively high, such as 0.1 Hz, due to the two suspension wires per one
test mass, the test masses do not have a good response around 0.1 - 1 Hz. In orderto reduce the resonant frequency, it is necessary to suspend the test masses with asingle wire even though it is diﬃcult to set the two centers of the mass at the same
point.
Although the thermal noise is not observed in our experiment, research on the
low-loss suspension system and on the cryogenic system are required at a following
step. As for the cryogenic system, we already have the pulse-tube cooler. The
cooling test and vibration reduction test of the heat anchor is the possible test for
the next plan.
Also, the gravity gradient noise, called Newtonian noise [44], would limit thesensitivity in the future. It is caused by the gravity gradient caused by the seismic
displacement of the ground, or the motion of the objects around the detectors. It
is one of the fundamental noise for the TOBA since it aﬀects the detectors largely107

7 Summary

7.3 Applications

at low frequencies. Since the Newtonian noise eﬀects on the test masses throughgravity, it cannot be shielded out. Therefore, the Newtonian noise has to be canceledout by the data processing using environmental sensors or multiple detectors [26, 15].

7.2.2

Multi-Output System

In our setup, we succeeded in controlling the test mass in 4 degrees of freedom at
the same time and derive the 3 independent signals. However, it is revealed that the
new signals largely couples to the translational seismic motion. Since the transfer
function from the translational seismic displacement to the angular fluctuation isrelative to f −2 above the resonant frequency of the rotational motion, the seismic
coupling noise could be reduced when the resonant frequency of the optical bench
is reduced to the same order of that of the test masses, as an analogy of the seismic
noise in the horizontal rotation.
Also, it is necessary to re-design the shape of the optical bench so that the
optical bench do not rotate according to the GWs, i.e., the quadrupole moment of
the optical bench should be negligible.

7.3

Applications

Other than the gravitational astronomy, torsion pendulum has many applications
to explore physics. One is the precise force measurement since a torsion pendulum
behave as a free-falling mass at low frequencies even on the ground. Many precise
measurement experiments have been performed with torsion pendulums, such as thetest of inverse square low of the gravity, measurement of gravity constant, measurement of the Casimir force, test of the equivalent principle and so on. Our setup
could be also applied to such measurement with the optimization of the parametersof the pendulum.
Other than that, the TOBA is expected to be applied to the prompt earthquake
detection [63]. As mentioned in the last of Sec.7.2.1, the test mass of the TOBA
is sensitive to the gravity gradient. When the large earthquake occurs, the gravitational potential changes [28]. The change of the gravity potential propagates withthe speed of light, faster than the seismic waves: p-waves and s-waves. Therefore, if
the change in the gravitational potential is detected with the TOBA, the earthquake
is expected to be alerted earlier by about 5 seconds. In order to detect the gravitational potential√ fluctuation due to the large earthquake, the sensitivity of the orderof 10−15 [1/ Hz] is required, which is less sensitive than the 10-m scaled TOBA as108

7 Summary

7.3 Applications

shown in Sec.3.3. Since the TOBA is relatively low-cost compared to other GW
detectors, it is feasible to set several TOBAs at various points. Then the center ofthe earthquake can be also detectable.

109

A

Chapter
Decoupling matrices for Active Vibration
Isolation System

The default matrices which converts signals to and from the each degrees of freedom,
Min and Mout are written as.

0 −1/4 1/2
0
0
1/3
1/3

0
0
1/2
0


 0
1/2
0
0
0
1/3


Min = 
(A.1)

1/3
0
0
−1/4
1/2
0


 0 −1/4 −1/2
0
0
1/3
1/3
0
0
−1/4 −1/2 0


1
0 1 0
1
1
 −1
0 1 0
1
−1


−0.5 1 1 1 −0.5 1 Mout = (A.2)
 0.5 −1 1 1 −0.5 −1
−0.5 −1 1 −1 −0.5 1 
0.5
1 1 −1 −0.5 −1
where the matrices are defined as




Sx
SH1
 Sy 
SV1 
 Sz 
SH2 




(A.3)
SV2  Min = Spitch  Sroll SH3 
Syaw
SV3
(
)
(
)
Sx Sy Sz Spitch Sroll Syaw Mout = SP1 SP2 SP3 SP4 SP5 SP6(A.4).
See Fig.C.4 for the definition of the label of sensors and actuators, and directions.
Those matrices are adjusted so that the signals in the other degrees of freedom
111

A Decoupling matrices for Active Vibration Isolation System

do not couple to the main signal. Especially, the tilt movement largely couples
to the horizontal seismometers’ signals at low frequencies. Therefore, Mout should
be carefully adjusted so that the transfer functions from the actuator input to thehorizontal seismometers outputs fit well to the theory even at low frequencies around
0.1 Hz.
V1

P6

H3

P1

H1

Vacuumarea

P5

Air vent
holes

Suspension
points

V3

P2

V2

y
P4
z
x

P3
H2

Figure A.1: The setup of the AVIT. P1, P2, . . . , and P6 are the piezoelectricelements below the table. V1, V2, and V3 are the seismometers which sens thevertical seismic motion, and H1, H2, and H3 are the ones which sens the horizontal
seismic motion. Photo Sensors are set below the seismometers so that each of themmeasures the DC position of the table in the same direction as the seismometer
above it. In the Vacuum area, the table has air vent holes in order to protect thepiezoelectric elements from the pressure of the air.

112

B

Chapter
Q measurement
Q values of the test mass pendulum is measured from the damped oscillation. Figure
B.1 are the graph of the damped oscillation of longitudinal mode measured by thePSs. Note that the data was filtered by the resonant filter in order to get rid of
the eﬀect of the other noise. They are measured without the control system. The
obtained Q values of the LONG, ROLL, and YAW modes are in Table B.1. The
errors on the Q values are the fitting errors.
10
5

Amplitude [AU]

Amplitude [AU]

10
5

0
−5
−10
30

0

35

40
Time [sec]

45

50

−5
−10
0

50

100
150
Time [sec]200

Figure B.1: The dumped oscillation of the longitudinal mode of the test mass. The
blue line is the measured line, and the green line is the fitted line.

113

B Q measurement

Pendulum mode
LONG
YAW
ROLL (s2 )ROLL (s3 )Resonant frequency [Hz]
0.88
0.10
0.19
0.14

Q value1709 ± 34800 ± 10
311.1 ± 0.4
376.1 ± 4.6

Table B.1: The resonant frequencies and Q values of the test mass 1, which is along
north-south direction.

114

C

Chapter
Circuits
1k

AD587(5V reference)

1k

-

30k

BUF634(current buﬀer)
PS HEAD
Vcc

+

PDsig

Output

+

Vcc

PDsig
82

L1915-02
(LED)

-

3.9k
S2386-18K(Si PD)S2386-18K(Si PD)

Figure C.1: A circuit for the photo sensor. It drives the LED and convert the photodetector current to the voltage.
We checked the analysis pipeline using the simulated detector output. The simulated data is just the Gaussian noise shown in Fig.??.
115

C Circuits
3.3k
1k

input

-

+

82
OP27

output

BUF634(current buﬀer)

Figure C.2: A circuit for coil-drivers. It emit current enough for the coil-magnet
actuators. BUF634 emit 200 mA at maximum.
10k

100k
IN1

IN2

-

+

-

+

1k

-

+

OP470

1k

-

OP470

+

1k
OP470

output
OP470

100k

330k

Figure C.3: A circuit for the seismometer amplifier. It has diﬀerential input ports
and the amplifier gain of 1000.
499
IN(+)

-

+

1k

1k

AD8622

10k

1k
24n

499
IN(-)

-

+

10k

1k
AD8622

4.7n
-

24n

24n

+

1k

THS4131
1k

4.7n

499

100

100

100

100

24n

24n

100

100

100

24n

AD8622-+24n 499

OUT(+)
10

AD8622-+OUT(-)
10

Figure C.4: A circuit for the anti-aliasing and anti-imaging filter. It is third order
10 kHz cut-oﬀ low-pass filters with notch at 65536 Hz, which is the frequency of thetiming signal. They are placed before the ADC input and after the DAC output.
116

Acknowledgement
This work could not be accomplished without lots of supports and advices frommany people.
I am deeply grateful to Prof. Masaki Ando for the constant support. When wewere designing our detector, he patiently advised me in spite of my poor understandings. He has also watched over and supported me in the construction phase.
Also, advices from Prof. Kimio Tsubono is indispensable for my thesis. I have agreat honor to have him as my former supervisor. It was a great time to talk withhim about the physic and the future plans. The former assistant professor in ourlab, Prof. Yoichi Aso, also give me a lot of advice especially about the developmentof the AVIT. The AVIT developed in this thesis is originally the active vibration isolation system for a cryogenic optical cavity for the frequency-stabilized laser source
which Prof. Aso proposed.
I was greatly helped by co-researchers. I am obliged to Mr. Kazunari Eda andProf. Yosuke Ito for their calculation of the multi-output system. Their work reveal
the significance of my proposal. Mr. Yuya Kuwahara have been working on theexperiment with me and complemented our thesis. He can point out the essential
issues and made enormous contribution. I would also like to express my gratitude
to Mr. Kodai Tejima for designing the pendulum with us.
Also, our experiments are not accomplished without Mr. Shigemi Otsuka, Mr.
Yoshikatsu Nanjo, and Mr. Takeshi Abe, the engineers in the Science Faculty. They
finely manufactured the parts of our setup in spite of our immature blueprints. Also
their warm encouragement cheers me up when I was tired during the experiment.
I also appreciate Prof. Osamu Miyakawa. He told me everything about thedigital systems. Whenever we had troubles, he kindly tell me what to do. Our work
was not accomplished without Osamu.
Additionally, several companies have been involved in this work. I appreciate Mr.
Yoshitaka Takahashi (Cosmotec Corporation) for the construction of the vacuum
system. Mr. Yasuhiro Higuchi, Mr. Tsutomu Nakanishi, and Mr. Shuichi Goto
(Jecc Torisha Co., Ltd.) also helped us for the cryogenic system.
I would like to express my gratitude to our group members. It was really lucky
to me that I started my research career at the same time with Mr. Yuta Michimura
who is a current assistant professor in our lab. He is one of the most dependable
researcher who can take care of the lab members. I am greatly honor to have him
117

C Circuits

as a comrade from now on. Dr. Nobuyuki Matsumoto often tells me interesting researches on the optical mechanics. Though I do not know about the opt-mechanics
well, I admire his deep understandings and insights into the optical physics. Mr.
Takafumi Ushiba is a researcher who can work steadily with objective view. WhenI worked with him on the experiment for the frequency-stabilized laser source, hehelped me a lot of work without getting down even though the process was troublesome. Mr. Kentaro Komori is also a promising student in our group. I am reallylooking forward to hearing that his enthusiastic work has been accomplished. I also
would like to thank my former group mate, Prof. Koji Ishidoshiro, Mr. Kakeru
Takahashi, Mr. Kenshi Okada, and Mr. Kazunori Shibata. Especially, Prof. Koji
Ishidoshiro and Mr. Kenshi Okada supported me with the inspect of the previous
TOBA experiments. Other than the Ando’s group member, I would like to express the deepest appreciation to Dr. Noriaki Ohmae, who is a co-researcher of thefrequency-stabilized laser source experiment. He always gives me fruitful advices
especially on the development of the AVIT. Ms. Miki Ueda, Ms. Michiko Kondo,
and Ms. Ami Ito has helped me with the oﬃcial work.
In addition, I am grateful to Prof. Makoto Sasaki, Prof. Kazuo Makishima,
Prof. Hiroshi Fukuyama, Prof. Naoki Yoshida, and Prof. Takeo Moroi, members ofa thesis committee. They gave me a lot of useful advices during the defense of mythesis.
Finally, I would like to oﬀer my special thanks to my parents, and my husband.
They, especially my husband, never blamed on me even though I neglect the houseworks. He always warmly encouraged and watched over me when I was depressed.
I believe that I could overcome anything only if I am with him.

118

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