TDI

2005/2/16~19
4th TAMA symposium &
GW winter school
@ Osaka-city Univ.
TDI 入門
Atsushi TARUYA
(RESCEU, Univ. Tokyo, JAPAN)
TDI ? ・・・ Time Delay Interferometry
Fundamental technique to synthesize data streams
free from the laser-frequency noise
Key ingredient to detect gravitational-wave signals
from space interferometer, LISA
Influence on response functions, sensitivity curves and S/N
Some implications to data analysis
TDI affects sensitivity curves
Armstrong et al. (1999)
a
X
z
a-b (or a - 2b + g)
Goal of this talk
From a theoretical view-point,
Introduction to signal processing
in space interferometer, LISA
How to construct noise-canceling combination
Influence on signal response and sensitivity
Practical application to data analysis
Contents
Principle of gravitational-wave detection
Time-delay interferometry
Observational characteristics of TDI signals
Development of TDI technique
References
Review
• “Time-Delay Interferometry and LISA’s Sensitivity to
Sinusoidal Gravitational Waves”,
http://www.srl.caltech.edu/lisa/tdi_wp/LISA_Whitepaper.pdf
M.Tinto, F.B.Estabrook & J.W.Armstrong
• “Time-Delay Interferometry”, gr-qc/0409034
M.Tinto & S.V.Dhurandhar
References
Armstrong et al., ApJ 527, 814 (1999)
Armstrong et al., CQG 18, 4059 (2001)
Cornish & Hellings, CQG 20, 4851 (2003)
Cornish & Rubbo, PRD 67, 022001 (2003)
Dhurandhar et al., PRD 65, 102002 (2002)
Dhurandhar et al., PRD 68, 122001 (2003)
Dhurandhar et al., gr-qc/0410093
Estabrook et al., PRD 62, 042002 (2000)
Prince et al., PRD 66, 122002 (2002)
Shaddock et al., PRD 68, 061303(R) (2003)
Shaddock, PRD 69, 022001 (2004)
Shaddock et al., PRD 70, 081101(R) (2004)
Sheard et al., PLA 320, 9 (2003)
Sylvestre & Tinto, PRD 68, 102002 (2003)
Sylvestre, PRD 70, 102002 (2004)
Tinto et al., PRD 63, 021101(R) (2001)
Tinto et al., PRD 67, 122003 (2003)
Tinto et al., PRD 69, 082001 (2004)
Tinto & Larson, PRD 70, 062002 (2004)
Tinto et al., gr-qc/0410122
Vallisneri, PRD 71, 022001 (2005)
Principle of gravitational-wave
detection
LISA mission
Laser Interferometer Space Antenna
• Project on NASA, ESA
• Schedule:
2008 LPF mission (test flight)
2013~ Launched
• Science goal:
Low-frequency gravitational-wave sources
strain ~ 10-20 Hz -1/ 2 @ 1 mHz ~ 10 mHz
Galactic binaries : resolved, un-resolved
BH-BH coalescence, etc.
LISA & gravitational-wave sources
–16
10
Strain [1/Hz
1/2
]
–18
大質量
ブラックホール連星合体
LISA
10
中性子星
連星合体
重力崩壊型
超新星爆発
銀河系内連星
–20
10
銀河系内連星
バックグラウンド雑音
–22
10
初期宇宙
からの重力波
(Wgw=10-14)
ScoX-1
(1yr)
パルサー
(1yr)
LCGT
–24
10
DECIGO (量子限界)
–26
重力場変動雑音
(地上検出器)
基線長 108 m, マス 100kg,
レーザー光 10MW, テレスコープ径 3m
10
–5
10
–4
10
–3
10
–2
–1
0
1
10 10
10 10
Frequency [Hz]
2
10
3
10
4
10
Viewgraph by M. Ando (GW school 2004)
Flight configuration
Arm-length: 5,000,000 km
(16.7 light sec)
3 spacecrafts with 6 laser-path
(drag-free)
Circular orbit:
P = 1 year
e = 0.01
a = 1 AU
Cartwheel motion
Sun
60 deg.
Optical design
Optical bench (35cm×20cm×4cm)
Laser
1W, Nd:YAG, 1.064 mm
Proof mass
40 mm,3 Au:Pt = 9:1
(drag-free sensor)
Photodetector
18 independent data streams :
interspace(6) + intraspace(6) + USO(6)
Basic concept of LISA detector
Combining 4 data streams out of the 6 interspace signals,
LISA can be viewed as a large “Michelson interferometer”:
“Phase-locked laser beam ”
Michelson(a)
is transferred back and forth via
c
Lbc
Lca
“Heterodyne detection”
a
Lab
b
But,
actual implementation in space is very different from ground detector,
especially by using TDI technique.
Basic principle of signal detection (1)
Arm-length variation caused by gravitational waves
phase difference of laser-light :  (t ' )  2 ( 0 / c)  l (t )
gravitational
wave
s/c1
laser
emit: t
s/c2
receive: t’
l  l +  l (t )
 (t ' )
 0  3 1014 Hz
(laser frequency)
Basic principle of signal detection (2)
Alternatively,
Arrival-time is delayed or advanced in presence of gravitational wave
Doppler
effect
  (t ' ) d  t ' 
   d 
Frequency shift of laser-light :
0
dt '  t 
Relation between frequency-shift and phase difference:
 (t ' )
  (t ' )  

0
2  0
These are both connected with path-length variation caused
by gravitational waves.
Path-length variation (Analytic formula)
Cornish & Rubbo, PRD 022001 (2003)
Gravitational
wave
ˆ
W

h
emit: ti
Laser
s/c i

ˆ
 (t )  t - W  xi (t )
f ij*  [2 lij (ti )]-1
Response function
receive: tj
rˆij (t ), lij (t ) : Unit vector and arm-length
s/c j
pointing from s/c i to s/c j at a time t
j

ˆ
ˆ
r
(
t
)

r
(
t
)
1 ij i
ij i
 lij (ti ) 
:  d h ( )
ˆ
2 1 - W  rˆ (t )
ij
i
i

~
ˆ
 lij (ti )  df  dW D( f , ti , W) : h ( f ) ei 2 f  (ti )

 f
 i 2 f ij [1-Wˆ rˆij (ti )]
1
ˆ )  [rˆ (t )  rˆ (t )] sinc 
ˆ  rˆ (t )]  e
D( f , t i , W
[1 - W
ij i
ij i
ij i 

2f
2
 ij

f
Noise contributions
Output signal of one-way Doppler tracking :
Gravitational-wave signal
  ij (t j )  20   lij (ti )
 accel
 accel
shot
+
n
+rˆij (ti )  n j (t j ) - ni (ti ) ij (t j ) + Ci (ti ) - C j (t j )
Contributions of instrumental noises:
Acceleration noise : Random forces exerted on each spacecraft
Shot noise
:
Laser-phase noise :
Photon number fluctuation in laser-beam
Stability of laser-beam
Total noise budget
Strain amplitude
From 「Pre-Phase A report」,
Acceleration noise
(proof-mass noise)
Shot noise
(optical-path noise)
Laser frequency noise
~
|  | | C | / f
( l / l )
2 1/ 2
3 10
-15
2
m / s Hz
-1/ 2
1.5 10
30 Hz  Hz
Significantly large !!
 f 


 mHz 
4 10 -21
2 10-11 m Hz -1/ 2
-1/ 2
- 20
[Hz -1/ 2 ]
10
-12
 f 


 mHz 
-1
-2
Impact of laser-frequency noise
(unit: c=1)
Michelson signal (static configuration):
Michelson(a)
c
  Michelson
( t )    ab ( t - Lab ) +   ba ( t )
a
-   ac ( t - Lca ) -   ca ( t )
Lbc
Lca
a
Lab
Contribution of laser-frequency noise:
Michelson
 freq
( t )  Ca ( t - 2Lab ) - Cb ( t - Lab ) + Cb ( t - Lab ) - Ca (t )
- Ca ( t - 2Lca ) - Cc ( t - Lca )  - Cc ( t - Lca ) - Ca (t ) 
 Ca ( t - 2Lab ) - Ca ( t - 2Lca )  0
if Lab  Lca
b
Required accuracy
Michelson
 freq
 Ca ( t - 2Lab ) - Ca ( t - 2Lca )
Residual noise:
 2 C a ( t - Lab ) L
Fourier domain
~
~ Michelson
|   freq
( f ) |  4 f | Ca ( f ) | L
f -1 |  |
Strain amplitude :
~Michelson
 Lfreq
L
L  Lab - Lca
2
 L
L

 2.2 10 -13
[Hz -1/ 2 ]
0 L
L
To achieve the required sensitivity (~ 10-20 Hz -1/ 2 ) ,
arm-length difference must be suppressed as
L / L 10-7
Unequal armlength of LISA
5.1
L23 L31
5.08
6
[10 km]
5.06
L12
5.04
5.02
5.0
4.98
0
0.5
1
year
Dhurandhar et al. gr-qc/0410093
For actual flight configuration of LISA, L / L 10-7 is impossible !!
Brief summary
LISA measures the graviational-wave signal through the phase
measurement in optical bench of each spacecraft.
6 independent signals

  ij (t j ) lij (ti )
~
i 2f  ( t )
ˆ

df
d
W
D
(
f
,
t
,
W
)
:
h
(
f
)
e
i
20
c  
i
 f
 i 2 f ij [1-Wˆ rˆij (ti )]
1
ˆ  rˆ (t )]  e
[rˆij (ti )  rˆij (ti )] sinc   [1 - W
ij i 
2f
2
 ij

f
Noise contributions to the phase measurement
Laser-frequency (phase) noise is 3~5 order of
magnitude larger than the GW signals.
Time-delay interferometry
~ 1st generation TDI ~
Confronting laser-frequency noise
Possible approach
Reduction of laser-freq. noise :
Improving laser-frequency stability by introducing new
technique
Sheard et al. (2003); Sylvestre (2004)
Cancellation of laser-freq. noise
• Frequency-domain cancellation
• Time-domain cancellation
TDI
TDI ~ basic idea ~
Simple Michelson signal uses only 4 data :
  ab ( t - Lab ),  ba ( t ),   ac ( t - Lca ),  ca ( t )
LISA provides 6 insterspace data,
each of which is (continuous) time-series data
c
Construct a noise-free signal using all
possible combinations of time-delayed data :
 aij  ij ( t - lij Lab - mij Lbc - nij Lca )
Lbc
Lca
i, j
(i, j =a, b, c)
integer
a
Lab
b
X signal (1) ~ heuristic derivation ~
Michelson(a)
Consider again the Michelson signal:
c
  ac
  Michelson
( t )    a b ( t ) -   a  c ( t ) ;
a
Lbc
Lca
 ab (t )    ab (t - Lab ) +  ba ( t )
 ac (t )    ac ( t - Lca ) +  ca ( t )
a
Lab
b
  ab
Noise contribution:
  freq
a b (t )  Ca (t - 2 Lab ) - Ca ( t )
Non-vanishing noise
contribution appears
at end-point.
  freq
a c (t )  Ca ( t - 2 Lca ) - Ca ( t )
survive
cancel
X signal (2) ~ heuristic derivation ~
cancel
Consider the following path:
 ac (t - 2Lab )
－  a b (t - 2Lca )
Laser-freq.
noise
Ca ( t - 2Lab - 2Lca ) - Ca ( t - 2Lab )
Ca (t - 2Lab - 2Lca ) - Ca ( t - 2Lca )
non-vanishing, but same as
the residual of Michelson
  Xa ( t )    a b ( t ) -   a c ( t )
c
X
+   a c ( t - 2Lab ) -   a b ( t - 2Lca )
Lca
Lbc
laser-frequency noise cancelled !!
a
“X signal”, or “unequal-arm Michelson”
Lab
b
Sagnac signal
Recall that residual laser-freq. noise appears at end-points of path:
 aSagnac ( t )    acba ( t ) -   abca ( t ) ;
 acba (t )    ac (t - Lab - Lbc ) +  cb ( t - Lab ) +  ba ( t )
 abca (t )    ab ( t - Lbc - Lca ) +  bc ( t - Lca ) +  ca ( t )
Noise contribution:
cancel !!
a
c
 freq
a c ba (t )  Ca (t - Lab - Lbc - Lca ) - Ca (t )

freq
a b c a
Lca
(t )  Ca ( t - Lbc - Lca - Lab ) - Ca (t )
“Sagnac signal” (a-type)
a
Lbc
Lab
b
Fully symmetric Sagnac
+
 a ,bc (t )
 b,ca (t )
 c ,ab (t )
c
c
c
Lbc
Lca
a
Lab
‐
b
Ca ( t - Lac ) - Cb ( t - Lbc )
‐
Lca
a
Lbc
Lab +
b
Cb ( t - Lab ) - Cc ( t - Lca )
Lca
a
Lbc
Lab ‐
+
b
Cc ( t - Lbc ) - Ca ( t - Lab )
Noise-canceling combination:
 z (t )    a,bc ( t - Lab ) +  b,ca ( t - Lbc ) +  c,ab ( t - Lca )
“Fully symmetric Sagnac” ( z )
Family of TDI signals ~ summary ~
6-pulse combination
Sagnac ( a, b, g )
Symmetric Sagnac ( z )
8-pulse combination
Unequal-arm Michelson ( X, Y, Z )
Beacon ( P, Q, R )
Monitor ( E, F, G )
Relay ( U, V, W )
Armstrong et al.(1999), Estabrook et al.(2000)
Algebraic relationship
All the TDI variables presented above are related with each other
and can be expressed in terms of the Sagnac signals :
(Armstrong et al. 1999)
z - z ,123  a ,1 - a , 23 + b, 2 - b,31 + g ,3 - g ,12
X ,1  a ,32 - b , 2 - g ,3 + z
E  a - z ,1
shortcut notation
U  g ,1 - b
X ,ij    X (t - ti - t j )
(i, j = 1,2,3)
P  z - a ,1
where
( t1 , t2 , t3 )  ( Lbc , Lab , Lca )
Mathematical background (1)
Dhurandhar et al. (2002)
There are fundamental set of TDI signals, which generate all the
other combinations canceling the laser-frequency noise.
Delay operator Ek
:
Ek f (t )  f (t - tk )
( t1 , t2 , t3 )  ( Lbc , Lab , Lca )
General form of signal combination :
(t )   pij ( E1 , E2 , E3 )   ij (t )
i, j

freq
(i, j =a, b, c)


ij
(t )    mn ({Ei }) pij ( E1 , E2 , E3 )  Cn (t )
n  a ,b , c  i , j

given function
Noise-canceling condition :
ij
m
 n ({Ei }) pij ( E1 , E2 , E3 )  0
i, j
Mathematical background (2)
Recalling that delay operator Ek forms a ring of polynomial,
Noise-canceling condition forms 1st module of syzygies.
Generator of module of syzygies
Fundamental set of TDI signals
Computational
commutative algebra
Sagnac signals ( a, b, g, z )
can be regarded as a fundamental set of TDI.
For details, → next talk by Prof. Dhurandhar.
Extension (1) ~ practical setting ~
Estabrook et al. (2000)
Practical setting envisaged for LISA :
Additional noises: Optical-bench motion noise, Optical-fiber noise
Additional signals: Intra-spacecraft data communicating with
adjacent optical bench
s/c 2
s/c
3
Further,
lasers are not
necessarily locked.
Extension (2) ~ noise contribution ~
4 phase measurements in each spacecraft:
Inter-s/c
data
Intra-s/c
data
GW
s31 (t )  s31
(t ) + [ p13 - 13 ], 2 - [ p31 + 31 ] + 2 31 + n31
GW
s21 (t )  s21
(t ) + [ p12 + 12 ],3 - [ p21 -  21 ] + 2 21 + n21
 21 (t )  p31 - p21 + 231 + 2 31 + m1
 31 (t )  p21 - p31 + 2 21 + 2 21 + m1
s/c 2
pij :
laser-phase noise
 ij :
optical-bench motion noise
 ij :
proof-mass noise
mi :
optical-fiber noise
No GW signals
s/c 3
Extension (3) ~ canceling s/c motion ~
Defining new signals combination with intra-s/c and inter-s/c data :
Noise function including
Optical-bench motion noise
Laser-frequency noise
The same TDI combinations as presented previously can be applicable,
eliminating both optical-bench motion and laser-frequency noises.
note Acceleration and shot noises still remain non-vanishing.
Brief summary
unequal-armlength
1st generation TDI variables :
static configuration
Sagnac ( a, b, g ) + Symmetric Sagnac ( z )
 Sagnac
a
Unequal-arm Michelson ( X, Y, Z )
Mathematical background :
Systematic method to derive TDI with a help of
computational commutative algebra
Extention for practical setting :
c
Lca
a
Lbc
Lab
 Xa ( t )
c
Lca
a
Canceling s/c motion effects
without any recourse of previous TDI combination.
b
Lbc
Lab
b
Observational characteristics of
TDI signals
Sensitivity curves
Depending on the signal combinations,
Response to the GW signals
changes significantly.
noise contribution
sensitivity curves
Roughly,
(noise spectrum)
1/2
[Hz
RMS of response function
‐1/2
]
Strain amplitude (1)
phase :
 (t )    GW (t ) +   noise(t )

strain :

h(t , xk ) 
s(t)
=
h(t)
+
1
2  0 L
n(t)

~
ˆ  x )
i
2

f
(
t
W
A
k
  df  dWˆ D( f , Wˆ ) :e (Wˆ ) hA ( f , Wˆ ) e
A  + ,
Combination of one-way Doppler signal multiplied by the phase factor:
f ˆ
f
ˆ rˆ ]
-i  W
rˆik
i
[1- W

ij




1
f
f ik
2 f ij
ˆ
ˆ
ˆ
D i  j ( f , W)  e
; Di  j ( f , W)  [rˆij  rˆij ] sinc   [1 - W  rˆij ]  e
2f

2
ij


Strain amplitude (2)
phase :
 (t )    GW (t ) +   noise(t )

strain :
s(t)
=
n(t )   df n~( f ) ei 2 f t
h(t)
+
1
2  0 L
n(t)
Sum of noise terms associated with
combination of one-way Doppler tracking
- i 2 f Lij
proof
opt
~
~
~
n ( f )  [nij ( f ) + nij ( f )] e
+
Non-vanishing contribution (secondary noise) is
proof-mass and optical-path noises.
Statistical averaging
s 2 (t )

h 2 (t )
+
n 2 (t )
2 ˆ ˆ
~
~

ˆ )h * ( f ' , W
ˆ ' )   ( f - f ' ) (W, W' )  S ( f )
hA ( f , W
A'
AA' h
4
n~ijproof ( f )n~kl*proof ( f ' )   ( f - f ' )  ik jl S proof ( f )
n~ijopt ( f )n~kl*opt ( f ' )   ( f - f ' )  ik jl Sopt ( f )
h (t )   df S h ( f ) R
2
*
ˆ 
A
d
W
ˆ
( f ) ; R ( f )
[D( f , W) :e ][D ( f , Wˆ ) :e A ]
A 4
n 2 (t )   df [ A ( f ) S proof ( f ) + B ( f ) Sopt ( f )]   df S n ( f )
Strain sensitivity
Time-domain
S/N=1
2
2
h
(t )
S
 
   2
n (t )
N
Fourier-domain
heff ( f )   S h ( f ) 
1/ 2

2
Sh ( f ) R ( f )
S
  (f)
Sn ( f )
N
Sn ( f )
[Hz -1/ 2 ]
R (f)
Note ―.
• Both R ( f ) and Sn ( f ) depend on signal combination.
• In S n ( f ) , main contributions are optical-path and proof-mass noises.
-2
-1
L
 f  

1/2
1/2
-1 / 2
Sopt
( f )  4 10-21 [Hz -1/ 2 ] , S proof
( f )  1.5 10 - 20 
[
Hz
]
 

6
 mHz   5 10 km 
Sensitivity curve for X-signal (1)
Equal armlength case ( Lab  Lbc  Lca )
Noise spectrum

(
f* 
c
2 L
 10 mHz
)
SnX ( f )  4 sin 2 ( f / f* ) 4 Sopt ( f ) + 8 1 + cos 2 ( f / f* ) Sproof ( f )
f 2 Sproof  f -2
( f  f* )
Sopt  f 0
( f  f* )
Detector response
R ( f )  R ( f / f* )
f2
( f  f* )
f -2
( f  f* )

Sensitivity curve for X-signal (2)
f* 
c
2 L
 10 mHz
X-signal
= Michelson
f*
heff ( f ) 
Sn ( f )
R (f)
f -2
f
1
( f  f* )
( f  f* )
Sensitive curves for Sagnac signals (1)
Sagnac ( a, b, g )
Behaviors at low-/high-frequency are qualitatively the same
as X-signal.
Symmetric Sagnac ( z )
Detector response is insensitive to the low-frequency GW.
R ( f ) ~ f 4 ( f  f* )
Instrumental noise is dominant at low-frequency regime.
Sensitive curves for Sagnac signals (2)
1/2
(25/T) h eff
 f
 f -3
-2
T= 1 year
z
a, b, g
Armstrong et al. (2001)
z-signal may be useful for real-time monitoring of instrumental noise.
(Tinto et al. 2000; Sylvestre & Tinto 2003)
Optimization of TDI signal
Combining fundamental TDI set ( a, b, g ), signals optimized
for proper observation can be constructed:
 ~
 ~
 ~
 ( f )  a1 ( f ,  ) a ( f ) + a2 ( f ,  ) b ( f ) + a3 ( f ,  ) g ( f )

 : optimazation parameters
Optimal TDI signals free from the noise correlation
Prince et al. (2002)
Optimizing SNR for known binaries with unknown polarization
Nayak, Dhurandhar, Pi & Vinet (2003)
Zero-signal solution that has zero response to GW signal
Tinto & Larson (2004)
Uncorrelated-noise combination (1)
Prince et al. (2002)
Orthogonal modes with uncorrelated noise :
A
1
g - a 
2
E
1
a - 2 b + g
6
1
a + b + g
T
3


A, E, T can be regarded as “independent” signals.
Particularly useful for study of stochastic GW background
Uncorrelated-noise combination (2)
Prince et al. (2002)
 f -3
 f -2
A, E
X
T
Zero-signal solution (1)
Tinto & Larson (2004)
Sky pattern of detector’s response depends on
both the signal combination and geometry of detector configuration

Zero response to GW at a particular direction   (s , s ) :
 ~
 ~
 ~
 ( f )  a1 ( f ,  ) a ( f ) + a2 ( f ,  ) b ( f ) + a3 ( f ,  ) g ( f ) = 0
×
×
f = 10 mHz
Source position :
( s , s )  ( 32 , 107 )
Zero-signal solution (2)
Tinto & Larson (2004)

  (31.5 ,106.5 )

  (s , s )
Slightly mismatching
Perfect matching
ZSS technique may be useful for
accurate determination of source location.
Brief summary
Sensitivity to GW and noise contributions depend on
signal combination of TDI.
Most of TDI signals :
heff ( f ) 
Sn ( f )
R (f)
f -2
f
1
( f  f* )
( f  f* )
Symmetric combination such as “z” can change low-freq. behavior.
Optimization of signal combination :
Uncorrelated-noise combination ( A, E, T )
Zero-signal solution
Development of TDI
Evolution of TDI technique
For practical implementation for LISA,
static configuration that has been assumed so far is invalid :
Orbital motion (Sagnac effect)
Flexing motion
Modification/improvement of 1st generation TDI :
Modified TDI,
2nd generation TDI
Orbital motion (Sagnac effect)
Cornish & Hellings (2003)
Shaddock (2004)
Violation of direction symmetry due to cartwheel motion:
L  L12 + L23 + L31 - ( L13 + L32 + L21 )
 
4W  A

 14.4 km
c
Imperfect cancellation of laser-frequency noise :
~
 Lfreq
L
dominate the
secondary noises !!
 L
-13 L
2

 2.2 10
[Hz -1/ 2 ] ~ 10-18 [Hz -1/ 2 ]
0 L
L
This effect particularly affects the Sagnac-type TDI signal.
Flexing effect
Cornish & Hellings (2003)
Shaddock et al. (2004)
In reality, armlengh between s/c varies in time :
Lij (t +  )  Lij (t )
Lij (t )  3 - 15 m s -1
~ L2 /(TaE )
Time-delay operation does not commute:
Pre-Phase A report
 [t - Lab (t ) - Lbc (t - Lab )]   [t - Lbc (t ) - Lab (t - Lbc )]
Imperfect cancellation of laser-freq. noise
This effect affects both Sagnac and unequal-arm Michelson signals.
2nd generation TDI
Shaddock et al. (2004)
Tinto et al. (2004)
Modification of 1st generation TDI to account for
orbital-motion and flexing effects.
Generalized X-signal
Generalized Sagnac-signal
s/c b
s/c c
s/c c
s/c b
s/c a
X1
s/c a
a1
Outcome of 2nd generation TDI
Shaddock et al. (2004)
Tinto et al. (2004)
For orbital-motion effect,
exact cancellation of laser-frequency noise becomes possible.
For flexing effect,
first order correction in non-commutative time-delay operation:
 L L
   (t - L (t ) - L (t - L (t ) ))   + 
;i j
j
i
j
,i j
,i j
i
j
can be cancelled for X-signal, however, residual frequency noise
remains for Sagnac signals.
Residual noise contribution
( f / f )2 S ( f )
*
Even if the exact cancellation is impossible, residual laser-frequency
noise in 2nd generation TDI is now well below the secondary noises.
Simulated by
synthetic LISA
(Vallisneri 2004)
Requirement of TDI technique
Tinto, Shaddock, Sylvestre & Armstrong (2003)
Further practical issues to implement TDI for LISA :
Accurate armlength determination
~30 m (~100 ns)
Synchronization of clocks onboard the 3 spacecrafts
Timing accuracy / sampling rate
Data digitization with high dynamic range
~50 ns
100 ns / 10 MHz
~36 bits
Numerical values are estimated in the case of 1st generation TDI.
Post-processed TDI
Shaddock et al. (2004)
Instead of real-time
signal processing,
TDI signal is
constructed at the
Earth as postprocessing.
With Implementing TDI as post-processing,
• Phase measurement data with arbitrary timing accuracy can be
reconstructed by interpolating a low-sampled data ( ~10Hz ).
Shaddock et al. (2004)
• Accurate determination of armlength ( L ~ 3-5m ) (as well as
clock-synchronization) can be achieved by the new variational
procedure called “TDI ranging”.
Tinto, Vallisneri & Armstrong (2004)
Summary
最後は日本語でおさらい
TDI って何だったっけ？
LISAで実装される予定の周波数雑音キャンセル法
６つのデータを組み合わせて、シグナルを構成
( X, Y, Z ), ( a, b, g, z ), ...
それって重要？
重力波応答へ影響します（→ 感度曲線）
組み合わせで、重力波応答、あるいは雑音特性を最適化
( A, E, T ), ゼロシグナル解
おさらい（続き）
懸案事項
TDI 法の実装可能性・残された技術的課題
TDI シグナルを使ってデータ解析する際の影響
DECIGO でも TDI を使うべきか？
Appendix
Noise spectra
~ analytic expressions ~
Equal armlength case ( Lab  Lbc  Lca )

(
)
f
c
fˆ  ; f* 
f*
2 L
SnX ( f )  4 sin 2 fˆ 4 Sopt ( f ) + 8 1 + cos 2 fˆ Sproof ( f )
 4 sin 2 fˆ  SnMichelson ( f )
 ( )
( f ) + 24 sin ( fˆ / 2)S

( )
Sna ( f )  6 Sopt ( f ) + 8 sin 2 3 fˆ / 2 + 2 sin 2 fˆ / 2 Sproof ( f )
Snz ( f )  6 Sopt
2
proof
(f)

SnA ( f )  SnE ( f )  8 sin 2 ( fˆ / 2) ( 2 + cos 2 fˆ ) Sopt ( f )
+ 2 ( 3 + 2 cos fˆ + cos 2 fˆ ) S

SnT ( f )  2 (1 + cos fˆ ) 2 Sopt ( f ) + 4 sin 2 ( fˆ / 2) Sproof ( f )

proof
(f)

Response function
~ analytic expressions ~
Equal armlength case ( Lab  Lbc  Lca )
X-signal
1
4
Sagnac-signal
a
1/2
(5/T) h eff
Sensitivity curves ~ other signals ~
Tinto et al. (LISA white paper)
Explicit expression for X 1
Tinto et al. (2004)
2
L3 '
L1 '
L1
L3
L2
1
L2 '
3
Explicit expression for a 1
Shaddock et al. (2004)
Roughly, a1 (t )  a (t ) - a ( t - L1 - L2 - L3 )

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