Post on 21-Jan-2016
TDI TDI 入門入門
Atsushi TARUYAAtsushi TARUYA
2005/2/16~194th TAMA symposium & GW winter school
@ Osaka-city Univ.
(RESCEU, Univ. Tokyo, JAPAN)
TDI ?TDI ?
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
・・・ Time Delay Interferometry
TDI affects sensitivity curvesTDI affects sensitivity curves
X
or
Armstrong et al. (1999)
Goal of this talkGoal of this talk
Influence on signal response and sensitivity
Practical application to data analysis
How to construct noise-canceling combination
Introduction to signal processing Introduction to signal processing in space interferometer,in space interferometer, LISA LISA
From a theoretical view-point,
ContentsContents
Principle of gravitational-wave detection
Time-delay interferometry
Observational characteristics of TDI signals
Development of TDI technique
ReferencesReferences
• “Time-Delay Interferometry”, gr-qc/0409034
M.Tinto & S.V.Dhurandhar
• “Time-Delay Interferometry and LISA’s Sensitivity to Sinusoidal Gravitational Waves”,
M.Tinto, F.B.Estabrook & J.W.Armstrong
http://www.srl.caltech.edu/lisa/tdi_wp/LISA_Whitepaper.pdf
Review
ReferencesReferences
Armstrong et al., ApJ 527, 814 (1999)Armstrong et al., CQG 18, 4059 (2001)
Cornish & Rubbo, PRD 67, 022001 (2003)
Dhurandhar et al., gr-qc/0410093
Sheard et al., PLA 320, 9 (2003)
Sylvestre, PRD 70, 102002 (2004)
Estabrook et al., PRD 62, 042002 (2000)
Dhurandhar et al., PRD 65, 102002 (2002)
Tinto et al., PRD 63, 021101(R) (2001)
Sylvestre & Tinto, PRD 68, 102002 (2003)
Prince et al., PRD 66, 122002 (2002)
Dhurandhar et al., PRD 68, 122001 (2003)
Tinto & Larson, PRD 70, 062002 (2004)Cornish & Hellings, CQG 20, 4851 (2003)
Shaddock, PRD 69, 022001 (2004)Shaddock et al., PRD 68, 061303(R) (2003)
Tinto et al., PRD 69, 082001 (2004)
Vallisneri, PRD 71, 022001 (2005)
Tinto et al., PRD 67, 122003 (2003)
Shaddock et al., PRD 70, 081101(R) (2004)
Tinto et al., gr-qc/0410122
Principle of gravitational-wave detection
LISA missionLISA mission
Laser Interferometer Space Antenna
• Project on NASA, ESA
2008 LPF mission (test flight)
2013~ Launched
• Science goal:
• Schedule:
Low-frequency gravitational-wave sources
@ 1 mHz ~ 10 mHz
BH-BH coalescence, etc.
Galactic binaries : resolved, un-resolved
2/120 Hz10~strain
LISA & gravitational-wave sources LISA & gravitational-wave sources
Viewgraph by M. Ando (GW school 2004)
10–5 10–4 10–3 10–2 10–1 100 101 102 103 104
10–26
10–24
10–22
10–20
10–18
10–16
Frequency [Hz]
Str
ain
[1
/Hz1/
2]
LCGT
DECIGO ( 量子限界 ) 基線長 108 m, マス 100kg, レーザー光 10MW, テレスコープ径 3m
銀河系内連星バックグラウンド雑音
重力崩壊型超新星爆発
中性子星連星合体
大質量ブラックホール連星合体
銀河系内連星
ScoX-1 (1yr)
パルサー (1yr)
初期宇宙からの重力波 (gw=10-14)
LISA
重力場変動雑音 ( 地上検出器 )
Flight configurationFlight configuration
Arm-length: 5,000,000 km
3 spacecrafts with 6 laser-path
Circular orbit:
60 deg.SunP = 1 year
e = 0.01
a = 1 AU
Cartwheel motion
(drag-free)
(16.7 light sec)
Optical designOptical design
Optical bench (35cm×20cm×4cm)Optical bench (35cm×20cm×4cm)
Laser
Proof mass
Photodetector
1W, Nd:YAG, 1.064 m
40 mm, Au:Pt = 9:13
(drag-free sensor)
18 independent data streams :interspace(6) + intraspace(6) + USO(6)
Basic concept of LISA detectorBasic concept of LISA detector
LISA can be viewed as a large “Michelson interferometer”:
Michelson(a)
a
c
b
Lbc
Lab
Lca
“Phase-locked laser beam ”
is transferred back and forth via
“Heterodyne detection”
But, But,
actual implementation in space is very different from ground detector,
especially by using TDI technique.
Combining 4 data streams out of the 6 interspace signals,
Basic principle of signal detection (1)Basic principle of signal detection (1)
Arm-length variation caused by gravitational waves
s/c1
s/c2
laser
receive: t’gravitational
wave
emit: t
)(tlll
phase difference of laser-light :phase difference of laser-light : )()/(2)'( 0 tlct
Hz103 140
(laser frequency)
)'(t
Basic principle of signal detection (2)Basic principle of signal detection (2)
Alternatively,
Arrival-time is delayed or advanced in presence of gravitational wave
Frequency shift of laser-light :Frequency shift of laser-light :
'
0 '
)'( t
t
ddt
dt Doppler
effect
00 2
)'()'(
tt
Relation between frequency-shift and phase difference:
These are both connected with path-length variation caused by gravitational waves.
Path-length variationPath-length variationCornish & Rubbo, PRD 022001 (2003)
)](ˆˆ1[2)](ˆˆ1[
2sinc)](ˆ)(ˆ[
2
1)ˆ,,(D
iijij
trf
fi
iijij
iijiiji etrf
ftrtrtf
�Response function
)(h:)(ˆˆ1
)(ˆ)(ˆ
2
1)(
j
i
dtr
trtrtl
iij
iijiijiij
�
:)(),(ˆ tltr ijij Unit vector and arm-length pointing from s/c i to s/c j at a time t
)(2)(h~
:)ˆ,,(D)( itfiiiij eftfddftl
�)(ˆ)( txtt i
1* )](2[ iijij tlf
h�
s/c i
s/c j
Laser
receive: tjGravitational
wave
emit: ti
(Analytic formula)
Noise contributionsNoise contributions
Output signal of one-way Doppler tracking :
)(2)( 0 iijjij tlt Gravitational-wave signal
Contributions of instrumental noises:
)()()(ˆ accelacceliijjiij tntntr
Acceleration noise : Random forces exerted on each spacecraft
Shot noise : Photon number fluctuation in laser-beam
)(shotjij tn
Laser-phase noise : Stability of laser-beam
)()( jjii tCtC
Total noise budgetTotal noise budget
From 「 Pre-Phase A report 」 ,
2/1215 Hzs/m103
2/111 Hzm102
2/1HzHz30
2/12/ ll ]Hz[ 2/1
Strain amplitude
220
mHz105.1
f
21104
112
mHz10
f
Significantly large !!
Shot noise
(optical-path noise)
Acceleration noise
(proof-mass noise)
Laser frequency noise
fC /|~
|||
Impact of laser-frequency noiseImpact of laser-frequency noise
Michelson signal (static configuration): Michelson(a)
a
c
b
Lbc
Lab
Lca)()( tLt cacaac
)()()(Michelson tLtt baababa
Contribution of laser-frequency noise:
)()()()2()(Michelsonfreq tCLtCLtCLtCt aabbabbaba
)()()()2( tCLtCLtCLtC acaccaccaa
0)2()2( caaaba LtCLtC if caab LL
(unit: c=1)
Required accuracyRequired accuracy
Residual noise: )2()2(Michelsonfreq caaaba LtCLtC
LLtC aba )(2
caab LLL
]Hz[102.22~
2/113
0
Michelsonfreq
L
L
L
L
L
L
Strain amplitude :
LfCff a |)(~
|4|)(~
| Michelsonfreq
Fourier domain
||1 f
arm-length difference must be suppressed as 710/ LL
To achieve the required sensitivity , )Hz10~( 2/120
Unequal armlength of LISAUnequal armlength of LISA
Dhurandhar et al. gr-qc/0410093
5.1
5.08
5.06
5.04
5.02
5.0
4.98
23L 31L
12L
year0 10.5
[10
km
]6
For actual flight configuration of LISA, 710/ LL is impossible !!
Brief summaryBrief summary
)(2
0
)(h~
:)ˆ,,(D)(
2
)(itfi
iiijjij eftfddf
c
tlt
�
)](ˆˆ1[2)](ˆˆ1[
2sinc)](ˆ)(ˆ[
2
1 iijij
trf
fi
iijij
iijiij etrf
ftrtr
LISA measures the graviational-wave signal through the phase measurement in optical bench of each spacecraft.
6 independent signals
Noise contributions to the phase measurement
Laser-frequency (phase) noise is 3~5 order of magnitude larger than the GW signals.
Time-delay interferometryTime-delay interferometry
~ 1st generation TDI ~
Confronting laser-frequency noiseConfronting 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
• Time-domain cancellation
• Frequency-domain cancellation
TDITDI
TDI ~ basic idea ~TDI ~ basic idea ~
LISA provides 6 insterspace data,
each of which is (continuous) time-series data
a
c
b
Lbc
Lab
Lca
Simple Michelson signal uses only 4 data :
)(),(),(),( tLttLt cacaacbaabab
ji
caijbcijabijijij LnLmLlta,
)(
Construct a noise-free signal using all possible combinations of time-delayed data :
(i, j =a, b, c)integer
XX signal (1) ~ heuristic derivation ~ signal (1) ~ heuristic derivation ~
;)()()(Michelson ttt cabaa
)()()( tLtt cacaacca
)()()( tLtt baababba
Noise contribution:
)()2()(freq tCLtCt acaaca
)()2()(freq tCLtCt aababa
Michelson(a)
a
c
b
Lbc
Lab
Lca
ca
ba
Consider again the Michelson signal:
cancel
Non-vanishing noise contribution appears
at end-point.
survive
XX signal (2) ~ heuristic derivation ~ signal (2) ~ heuristic derivation ~
Consider the following path:
)2( abca Lt
)2( caba Lt
)2()22( abacaaba LtCLLtC
)2()22( caacaaba LtCLLtC Laser-
freq. noise-
cancel
non-vanishing, but same as the residual of Michelson
)()()(X ttt cabaa
)2()2( cabaabca LtLt
laser-frequency noise cancelled !!a b
c
Lbc
Lab
Lca
X
“X signal”, or “unequal-arm Michelson”
Sagnac signalSagnac signal
Recall that residual laser-freq. noise appears at end-points of path:
;)()()(Sagnac ttt acbaabca
)()()()( tLtLLtt cacabccabcabacba
)()()()( tLtLLtt baabcbbcabacabca
Noise contribution:
)()()(freq tCLLLtCt aabcabcaacba
)()()(freq tCLLLtCt acabcabaabca
cancel !!
a b
c
Lbc
Lab
Lca
“Sagnac signal” (-type
Fully symmetric SagnacFully symmetric Sagnac
)()( bcbaca LtCLtC )()( cacabb LtCLtC )()( ababcc LtCLtC
)(, tcba
a
c
b
Lbc
Lab
Lca+ ‐
)(, tacb
a
c
b
Lbc
Lab
Lca
+
‐
)(, tbac
a
c
b
Lbc
Lab
Lca+
‐
)()()()( ,,, cabacbcacbabcba LtLtLtt
Noise-canceling combination:
“Fully symmetric Sagnac” (
Family of TDI signals ~ summary ~Family of TDI signals ~ summary ~
6-pulse combination
8-pulse combination
Sagnac
Symmetric Sagnac
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 relationshipAlgebraic relationship
All the TDI variables presented above are related with each other
and can be expressed in terms of the Sagnac signals :
12,3,31,2,23,1,123,
3,2,32,1,X
1, E
1,U
1, P
shortcut notation
)(, jiXij tttX
),,(),,( 321 caabbc LLLttt
(i, j = 1,2,3)
where
(Armstrong et al. 1999)
Mathematical background (1)Mathematical background (1)
There are fundamental set of TDI signals, which generate all the other combinations canceling the laser-frequency noise.
Delay operator )()( kk ttftfE kE :),,(),,( 321 caabbc LLLttt
ji
ijij tEEEpt,
321 )(),,()( (i, j =a, b, c)
General form of signal combination :
)(),,(})({)(,, ,
321freq tCEEEpEt n
cban jiiji
ijn
given function
Noise-canceling condition : 0),,(})({,
321 ji
ijiijn EEEpE
Dhurandhar et al. (2002)
Mathematical background (2)Mathematical background (2)
For details, → next talk by Prof. Dhurandhar.
Noise-canceling condition forms 1st module of syzygies.
Recalling that delay operator forms a ring of polynomial, kE
Generator of module of syzygiesFundamental set of TDI signals
Computational commutative algebra
Sagnac signals ( )can be regarded as a fundamental set of TDI.
Extension (1) ~ practical setting ~Extension (1) ~ practical setting ~Estabrook et al. (2000)
Practical setting envisaged for LISA :
s/c 2 s/c 3
Optical-bench motion noise, Optical-fiber noiseAdditional noises:
Additional signals: Intra-spacecraft data communicating with adjacent optical bench
lasers are not necessarily locked.
Further,
Extension (2) ~ noise contribution ~Extension (2) ~ noise contribution ~
4 phase measurements in each spacecraft:
s/c 2 s/c 3
No GW signals
313131312,13133131 2][][)()( npptsts GW
212121213,12122121 2][][)()( npptsts GW Inter-s/c
data
13131213121 22)( ppt
12121312131 22)( ppt
Intra-s/cdata
:ijp laser-phase noise
:ij optical-bench motion noise
:ij proof-mass noise
:i optical-fiber noise
Extension (3) ~ canceling s/c motion ~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.
Acceleration and shot noises still remain non-vanishing.note
Brief summaryBrief summary
Extention for practical setting :
Canceling s/c motion effects without any recourse of previous TDI combination.
Mathematical background :
Systematic method to derive TDI with a help of computational commutative algebra
a b
c
Lbc
Lab
Lca
Sagnaca
a b
c
Lbc
Lab
Lca
)(X ta
1st generation TDI variables :
Sagnac Symmetric Sagnac ( )
Unequal-arm Michelson ( X, Y, Z )
unequal-armlength
static configuration
Observational characteristics of Observational characteristics of TDI signalsTDI signals
Sensitivity curvesSensitivity curves
Depending on the signal combinations,
changes significantly.Response to the GW signals
noise contribution
sensitivity curves
RMS of response function
(noise spectrum)1/2Roughly,
[Hz ]‐1/2
Strain amplitude (1)Strain amplitude (1)
)()()( noiseGW ttt phase :
L02
1
s(t) = h(t) + n(t)strain :
)ˆ(2
,
)ˆ,(~
)ˆ(:)ˆ,(ˆ),( kxtfi
AA
Ak efhfddfxth
�
eD
]ˆˆ1[2]ˆˆ1[
2sinc]ˆˆ[
2
1)ˆ,(D
ijij
rf
fi
ijij
ijijji erf
frrf
�
Combination of one-way Doppler signal multiplied by the phase factor:
;)ˆ,(Dˆˆik
ik
rf
fi
ji ef
�
Strain amplitude (2)Strain amplitude (2)
)()()( noiseGW ttt phase :
L02
1
s(t) = h(t) + n(t)strain :
tfiefndftn 2)(~)( Sum of noise terms associated with combination of one-way Doppler tracking
ijLfiijij efnfnfn 2optproof )](~)(~[)(~
Non-vanishing contribution (secondary noise) is proof-mass and optical-path noises.
Statistical averagingStatistical averaging
)(2 ts )(2 th )(2 tn
;)()()(2 ffSdfth h R ]:[]: )ˆ,(ˆ4
ˆ *),([)( AA
Affdf ee DD
��
R
)()]()()()([)( optproof2 fSdffSffSfdftn nBA
)()'()'(~)(~proof
proof*proof fSfffnfn jlikklij
)()'()'(~)(~opt
opt*opt fSfffnfn jlikklij
)(4
)'ˆ,ˆ()'()'ˆ,'(
~)ˆ,(
~'
2*' fSfffhfh hAAAA
Strain sensitivityStrain sensitivity
S/N=1 ]Hz[)(
)()()( 2/12/1
eff
f
fSfSfh n
h R
)(
)()()(
2
fS
ffSf
N
S
n
h R
)(
)(2
22
tn
th
N
S
Time-domain Fourier-domain
Note ―.
,)( fSn
)(and)( fSf nR depend on signal combination.• Both
main contributions are optical-path and proof-mass noises.• In
]Hz[km105mHz
105.1)( 2/11
6
2201/2
proof
LffS,]Hz[104)( 2/1211/2
optfS
Sensitivity curve for Sensitivity curve for XX-signal (1)-signal (1)
mHz102*
L
cf
)()/(cos18)(4)/(sin4)( proof*2
opt*2 fSfffSfffS X
n
Noise spectrum
Equal armlength case )( cabcab LLL
Detector response
)/()( *fff RR
)( *ff
)( *ff
2proof
2 fSf
0opt fS
2f
2f
)( *ff
)( *ff
Sensitivity curve for Sensitivity curve for XX-signal (2)-signal (2)
mHz102*
L
cf
)(
)()(eff f
fSfh n
R
2f
1f
)( *ff
)( *ff
*f
= MichelsonX-signal
Sensitive curves for Sagnac signals (1)Sensitive curves for Sagnac signals (1)
Sagnac ( )
Behaviors at low-/high-frequency are qualitatively the same as X-signal.
Symmetric Sagnac ( )
Detector response is insensitive to the low-frequency GW.
)(~)( *4 ffff R
Instrumental noise is dominant at low-frequency regime.
Sensitive curves for Sagnac signals (2)Sensitive curves for Sagnac signals (2)
Armstrong et al. (2001)
(25/
T)1/
2h
eff
2 f3 f T= 1 year
-signal may be useful for real-time monitoring of instrumental noise. (Tinto et al. 2000; Sylvestre & Tinto 2003)
Optimization of TDI signalOptimization of TDI signal
Combining fundamental TDI set ( ), signals optimized for proper observation can be constructed:
)(~),()(~
),()(~),()( 321 ffaffaffaf
: optimazation parameters
Optimal TDI signals free from the noise correlation
Zero-signal solution that has zero response to GW signal
Prince et al. (2002)
Tinto & Larson (2004)
Optimizing SNR for known binaries with unknown polarization
Nayak, Dhurandhar, Pi & Vinet (2003)
Uncorrelated-noise combination (1)Uncorrelated-noise combination (1)Prince et al. (2002)
3
1T
2
1A
26
1E
Orthogonal modes with uncorrelated noise :
A, E, T can be regarded as “independent” signals.
Particularly useful for study of stochastic GW background
TA, E
X
Uncorrelated-noise combination (2)Uncorrelated-noise combination (2)Prince et al. (2002)
2 f
3 f
Zero-signal solution (1)Zero-signal solution (1)
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
)(~),()(~
),()(~),()( 321 ffaffaffaf
= 0
:),( ss
Tinto & Larson (2004)
×
×
f = 10 mHz
)107,32(),( ssSource position :
Zero-signal solution (2)Zero-signal solution (2)Tinto & Larson (2004)
),( ss
Perfect matching
)5.106,5.31(
Slightly mismatching
ZSS technique may be useful for accurate determination of source location.
Brief summaryBrief summary
Sensitivity to GW and noise contributions depend onsignal combination of TDI.
Most of TDI signals :
)(
)()(eff f
fSfh n
R
2f
1f
)( *ff
)( *ff
Symmetric combination such as “”can change low-freq. behavior.
Optimization of signal combination :
Uncorrelated-noise combination ( A, E, T )
Zero-signal solution
Development of TDIDevelopment of TDI
Evolution of TDI techniqueEvolution of TDI technique
Orbital motion (Sagnac effect)
Flexing motion
static configuration that has been assumed so far is invalid :
For practical implementation for LISA,
Modification/improvement of 1st generation TDI :
Modified TDI, 2nd generation TDI
Orbital motion (Sagnac effect)Orbital motion (Sagnac effect)
Violation of direction symmetry due to cartwheel motion:
)( 213213312312 LLLLLLL
km4.144
c
A
]Hz[102.22~
2/113
0
freq
L
L
L
L
L
L
Imperfect cancellation of laser-frequency noise :
Shaddock (2004)
]Hz[10~ 2/118
dominate the secondary noises !!
Cornish & Hellings (2003)
This effect particularly affects the Sagnac-type TDI signal.
Flexing effectFlexing effect
In reality, armlengh between s/c varies in time :
)()( tLtL ijij
Time-delay operation does not commute:
)]()([)]()([ bcabbcabbcab LtLtLtLtLtLt
Imperfect cancellation of laser-freq. noise
Pre-Phase A report
1sm153)( -ij tL
)/(~ 2ETaL
Cornish & Hellings (2003)Shaddock et al. (2004)
This effect affects both Sagnac and unequal-arm Michelson signals.
22ndnd generation TDI generation TDI
Modification of 1st generation TDI to account for orbital-motion and flexing effects.
Shaddock et al. (2004)Tinto et al. (2004)
1
s/c cs/c b
s/c a1X
s/c c s/c b
s/c a
Generalized X-signal Generalized Sagnac-signal
Outcome of 2Outcome of 2ndnd generation TDI generation TDIShaddock et al. (2004)
Tinto et al. (2004)
exact cancellation of laser-frequency noise becomes possible.
For orbital-motion effect,
For flexing effect,
first order correction in non-commutative time-delay operation:
)()(; tLtLtLt jijji jijiji LL,,
can be cancelled for X-signal, however, residual frequency noise remains for Sagnac signals.
Residual noise contributionResidual noise contribution
(Vallisneri 2004)
Simulated by synthetic LISA
( f /
f ) *
2S
( f )
Even if the exact cancellation is impossible, residual laser-frequency noise in 2nd generation TDI is now well below the secondary noises.
Requirement of TDI techniqueRequirement of TDI technique
Further practical issues to implement TDI for LISA :
Accurate armlength determination
Synchronization of clocks onboard the 3 spacecrafts
Timing accuracy / sampling rate
Data digitization with high dynamic range
Tinto, Shaddock, Sylvestre & Armstrong (2003)
~30 m (~100 ns)~30 m (~100 ns)
~50 ns~50 ns
100 ns / 10 MHz100 ns / 10 MHz
~36 bits~36 bits
Numerical values are estimated in the case of 1st generation TDI.
Post-processed TDIPost-processed TDIShaddock et al. (2004)
With Implementing TDI as post-processing,
• Phase measurement data with arbitrary timing accuracy can be reconstructed by interpolating a low-sampled data ( ~10Hz ).
• 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)
Shaddock et al. (2004)
Instead of real-time signal processing, TDI signal is constructed at the Earth as post-processing.
SummarySummary
最後は日本語でおさらい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 ), (
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)おさらい(続き)
懸案事項懸案事項
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?
TDI 法の実装可能性 ・ 残された技術的課題
AppendixAppendix
Noise spectraNoise spectra
Equal armlength case )( cabcab LLL
)(ˆcos18)(4ˆsin4)( proof2
opt2 fSffSffS X
n
)(2/ˆsin22/ˆ3sin8)(6)( proof22
opt fSfffSfSn
)(ˆsin4 Michelson2 fSf n
L
cf
f
ff
2;ˆ
**
)()ˆ2cos2()2/ˆ(sin8)()( opt2 fSfffSfS E
nAn
)()ˆ2cosˆcos23(2 proof fSff
)()2/ˆ(sin4)()ˆcos1(2)( proof2
opt2 fSffSffST
n
)(2/ˆsin24)(6)( proof2
opt fSffSfSn
~ analytic expressions ~~ analytic expressions ~
Response function Response function
41
~ analytic expressions ~~ analytic expressions ~
Equal armlength case )( cabcab LLL
X-signal
Sagnac-signal
Sensitivity curvesSensitivity curves
Tinto et al. (LISA white paper)
(5/T
)1/2h
eff
~ other signals ~~ other signals ~
Explicit expression for Explicit expression for XX 1
Tinto et al. (2004)
1L
'1L
3L
'3L
2L
'2L1
2
3
Explicit expression for Explicit expression for 1
Roughly, )()()( 3211 LLLttt
Shaddock et al. (2004)