TDI 入門

1. 2005/2/16~19 4th TAMA symposium & GW winter school @ Osaka-city Univ. TDI 入門 Atsushi TARUYA (RESCEU, Univ. Tokyo, JAPAN)

2. Influence on response functions, sensitivity curves and S/N Some implications to data analysis 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

3. TDI affects sensitivity curves Armstrong et al. (1999) a z X a-b (or a - 2b + g)

4. 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 Goal of this talk

5. Principle of gravitational-wave detection Time-delay interferometry Observational characteristics of TDI signals Development of TDI technique Contents

6. “Time-Delay Interferometry”, gr-qc/0409034 M.Tinto & S.V.Dhurandhar • “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 References Review

7. References Tinto et al., PRD 67, 122003 (2003) Armstrong et al., ApJ 527, 814 (1999) Tinto et al., PRD 69, 082001 (2004) Armstrong et al., CQG 18, 4059 (2001) Tinto & Larson, PRD 70, 062002 (2004) Cornish & Hellings, CQG 20, 4851 (2003) Tinto et al., gr-qc/0410122 Cornish & Rubbo, PRD 67, 022001 (2003) Vallisneri, PRD 71, 022001 (2005) 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)

8. Principle of gravitational-wave detection

9. LISA mission LaserInterferometer Space Antenna • Project on NASA, ESA • Schedule: 2008 LPF mission (test flight) 2013~ Launched • Science goal: Low-frequency gravitational-wave sources @ 1 mHz ~ 10 mHz Galactic binaries : resolved, un-resolved BH-BH coalescence, etc.

10. LISA & gravitational-wave sources 大質量 ブラックホール連星合体 LISA 中性子星 連星合体 重力崩壊型 超新星爆発 銀河系内連星 銀河系内連星 バックグラウンド雑音 ScoX-1 (1yr) パルサー (1yr) 初期宇宙 からの重力波 (Wgw=10-14) 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 重力場変動雑音 (地上検出器) Viewgraph by M. Ando (GW school 2004)

11. P = 1 year e = 0.01 a = 1 AU Cartwheel motion Flight configuration Arm-length: 5,000,000 km (16.7 light sec) 3 spacecrafts with 6 laser-path (drag-free) Circular orbit: Sun 60 deg.

12. Laser 1W, Nd:YAG, 1.064 mm 3 Proof mass 40 mm, Au:Pt = 9:1 (drag-free sensor) Photodetector Optical design Optical bench (35cm×20cm×4cm) 18 independent data streams : interspace(6) + intraspace(6) + USO(6)

13. LISA can be viewed as a large “Michelson interferometer”: Michelson(a) c “Phase-locked laser beam” Lbc Lca is transferred back and forth via b a Lab “Heterodyne detection” But, actual implementation in space is very different from ground detector, especially by using TDI technique. Basic concept of LISA detector Combining 4 data streams out of the 6 interspace signals,

14. Arm-length variation caused by gravitational waves phase difference of laser-light : s/c2 gravitational wave receive: t’ s/c1 laser emit: t (laser frequency) Basic principle of signal detection (1)

15. Alternatively, Arrival-time is delayed or advanced in presence of gravitational wave Doppler effect Frequency shift of laser-light : Relation between frequency-shift and phase difference: These are both connected with path-length variation caused by gravitational waves. Basic principle of signal detection (2)

16. Gravitational wave receive: tj Unit vector and arm-length pointing from s/c i to s/c j at a time t s/c j emit: ti Laser s/c i Response function Path-length variation (Analytic formula) Cornish & Rubbo, PRD 022001 (2003)

17. Output signal of one-way Doppler tracking : Gravitational-wave signal Acceleration noise : Random forces exerted on each spacecraft Shot noise : Photon number fluctuation in laser-beam Laser-phase noise : Stability of laser-beam Noise contributions Contributions of instrumental noises:

18. Strain amplitude Acceleration noise (proof-mass noise) Shot noise (optical-path noise) Laser frequency noise ~ dn = | | | C | / f Significantly large !! Total noise budget From 「Pre-Phase A report」,

19. Michelson(a) c Lbc Lca Contribution of laser-frequency noise: b a Lab if Impact of laser-frequency noise (unit: c=1) Michelson signal (static configuration):

20. Residual noise: Fourier domain Strain amplitude : arm-length difference must be suppressed as Required accuracy To achieve the required sensitivity ,

21. 5.1 5.08 5.06 5.04 5.02 5.0 4.98 0.5 1 0 year Dhurandhar et al. gr-qc/0410093 6 [10 km] is impossible !! For actual flight configuration of LISA, Unequal armlength of LISA

22. 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. Brief summary

23. Time-delay interferometry ~ 1st generation TDI ~

24. 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 TDI • Time-domain cancellation Confronting laser-frequency noise Possible approach

25. Simple Michelson signal uses only 4 data : LISA provides 6 insterspace data, c each of which is (continuous) time-series data Lbc Lca Construct a noise-free signal using all possible combinations of time-delayed data : b a Lab (i, j =a, b, c) integer TDI ~ basic idea ~

26. Michelson(a) Consider again the Michelson signal: c Lbc Lca b a Lab Noise contribution: cancel survive X signal (1) ~ heuristic derivation ~ Non-vanishing noise contribution appears at end-point.

27. cancel Laser-freq. noise － non-vanishing, but same as the residual of Michelson X c Lbc Lca a b Lab “X signal”, or “unequal-arm Michelson” X signal (2) ~ heuristic derivation ~ Consider the following path: laser-frequency noise cancelled !!

28. a c Noise contribution: cancel !! Lbc Lca a b Lab “Sagnac signal” (a-type) Sagnac signal Recall that residual laser-freq. noise appears at end-points of path:

29. c c c ‐ ‐ + + Lbc Lbc Lbc Lca Lca Lca ‐ + b b b a a a Lab Lab Lab Noise-canceling combination: Fully symmetric Sagnac “Fully symmetric Sagnac” ( z )

30. 6-pulse combination Sagnac ( a, b,g ) Symmetric Sagnac ( z ) 8-pulse combination Unequal-arm Michelson ( X, Y, Z ) Family of TDI signals ~ summary ~ Beacon ( P, Q, R ) Monitor ( E, F, G ) Relay ( U, V, W ) Armstrong et al.(1999), Estabrook et al.(2000)

31. shortcut notation (i, j = 1,2,3) where 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)

32. Delay operator : General form of signal combination : (i, j =a, b, c) given function Noise-canceling condition : 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.

33. 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 ) Recalling that delay operator forms a ring of polynomial, can be regarded as a fundamental set of TDI. Mathematical background (2) For details, → next talk by Prof. Dhurandhar.

34. Optical-bench motion noise, Optical-fiber noise Additional noises: Intra-spacecraft data communicating with adjacent optical bench Additional signals: Further, lasers are not necessarily locked. Extension (1) ~ practical setting ~ Estabrook et al. (2000) Practical setting envisaged for LISA : s/c 2 s/c 3

35. Inter-s/c data No GW signals Intra-s/c data laser-phase noise optical-bench motion noise proof-mass noise optical-fiber noise Extension (2) ~ noise contribution ~ 4 phase measurements in each spacecraft: s/c 2 s/c 3

36. Defining new signals combination with intra-s/c and inter-s/c data : Noise function including Optical-bench motion noise Laser-frequency noise note Acceleration and shot noises still remain non-vanishing. Extension (3) ~ canceling s/c motion ~ The same TDI combinations as presented previously can be applicable, eliminating both optical-bench motion and laser-frequency noises.

37. unequal-armlength static configuration 1st generation TDI variables : Sagnac ( a, b, g ) + Symmetric Sagnac ( z ) Unequal-arm Michelson ( X, Y, Z ) c c Mathematical background : Lbc Lca Lbc Lca Systematic method to derive TDI with a help of computational commutative algebra b a b a Lab Lab Extention for practical setting : Canceling s/c motion effects without any recourse of previous TDI combination. Brief summary

38. Observational characteristics of TDI signals

39. Depending on the signal combinations, Response to the GW signals changes significantly. noise contribution sensitivity curves Roughly, 1/2 (noise spectrum) ‐1/2 [Hz ] RMS of response function Sensitivity curves

40. phase : strain : s(t) + n(t) = h(t) Combination of one-way Doppler signal multiplied by the phase factor: Strain amplitude (1)

41. phase : strain : s(t) + n(t) = h(t) Sum of noise terms associated with combination of one-way Doppler tracking Non-vanishing contribution (secondary noise) is proof-mass and optical-path noises. Strain amplitude (2)

42. ; Statistical averaging

43. Time-domain Fourier-domain S/N=1 Note ―. • Both depend on signal combination. • In main contributions are optical-path and proof-mass noises. Strain sensitivity

44. Equal armlength case Noise spectrum Detector response Sensitivity curve for X-signal (1)

45. X-signal = Michelson Sensitivity curve for X-signal (2)

46. 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. Instrumental noise is dominant at low-frequency regime. Sensitive curves for Sagnac signals (1)

47. T= 1 year z a, b, g Armstrong et al. (2001) 1/2 h (25/T) eff Sensitive curves for Sagnac signals (2) z-signal may be useful for real-time monitoring of instrumental noise. (Tinto et al. 2000; Sylvestre & Tinto 2003)

48. Combining fundamental TDI set ( a, b, g ), signals optimized for proper observation can be constructed: : 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) Optimization of TDI signal

49. Orthogonal modes with uncorrelated noise : A, E, T can be regarded as “independent” signals. Particularly useful for study of stochastic GW background Uncorrelated-noise combination (1) Prince et al. (2002)

50. T A, E X Uncorrelated-noise combination (2) Prince et al. (2002)