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)