Time-delay Interferometry (TDI) for LISA

Transcription

1 Time-delay Interferometry (TDI) for LISA Sanjeev Dhurandhar IUCAA, Pune TAMA Symposium 19 Feb 5

2 Plan of the Talk Introduction: LISA GW sources for LISA Laser phase noise cancellation problem Theoretical Framework: Solution space - Module of Syzygies Stability issues, changing armlengths and TDI TAMA Symposium 19 Feb 5 1

3 Introduction Ground-based Laser Inteferometric Detector Network: LIGO, VIRGO, TAMA, GEO & AIGO - 1 Hz to KHz. Ground-based gravitational wave detectors have started acquiring scientific data for the past few years. TAMA3 first to take scientific data. LIGO, TAMA, GEO detectors have completed several Science Runs. The Space Antenna LISA: Why go to Space? Low frequency searches :1 5 Hz to 1 Hz Complementary to groundbased detectors - just as the different astronomies, optical, radio, etc. complement each other. Gauranteed Gravitational Wave Sources The LISA Project: ESA & NASA TAMA Symposium 19 Feb 5

4 Introduction Ground-based Laser Inteferometric Detector Network: LIGO, VIRGO, TAMA, GEO & AIGO - 1 Hz to KHz. Ground-based gravitational wave detectors have started acquiring scientific data for the past few years. TAMA3 first to take scientific data. LIGO, TAMA, GEO detectors have completed several Science Runs. The Space Antenna LISA: Why go to Space? Low frequency searches :1 5 Hz to 1 Hz Complementary to groundbased detectors - just as the different astronomies, optical, radio, etc. complement each other. Gauranteed Gravitational Wave Sources The LISA Project: ESA & NASA TAMA Symposium 19 Feb 5

5 Introduction Ground-based Laser Inteferometric Detector Network: LIGO, VIRGO, TAMA, GEO & AIGO - 1 Hz to KHz. Ground-based gravitational wave detectors have started acquiring scientific data for the past few years. TAMA3 first to take scientific data. LIGO, TAMA, GEO detectors have completed several Science Runs. The Space Antenna LISA: Why go to Space? Low frequency searches :1 5 Hz to 1 Hz Complementary to groundbased detectors - just as the different astronomies, optical, radio, etc. complement each other. Gauranteed Gravitational Wave Sources The LISA Project: ESA & NASA TAMA Symposium 19 Feb 5

6 The LISA Project TAMA Symposium 19 Feb 5 3

7 LISA sources and sensitivity Short period binaries: WD/WD, LMXBs - 4U18-3, RX etc. 1 1M inspiral into a massive blackhole LISA sensitivity curve Orbit eccentric, maps Kerr geometry, test no-hair theorem cycles for a 1M BH falling into a 1 6 M BH - SNR Few per year at 1 Gpc (Sigurdson and Rees, Phinney) vibration noise shot noise armlength TAMA Symposium 19 Feb 5 4

8 Merger of Super Massive Blackholes No templates needed! solar mass BH, SNR ! High SNR allows time and position determination of the coalescence at early times TAMA Symposium 19 Feb 5 5

9 Laser Frequency Noise Frequency of Laser: Frequency noise: ν Hz ν 1 Hz / Hz h noise ν ν But h sens orders of magnitude Optical bench motion noise: 1nm/ Hz TAMA Symposium 19 Feb 5 6

10 Unequal-arm Interferometer and TDI Equal arms: This noise is automatically cancelled φ(t): Laser phase noise φ 1 (t) = φ(t L 1 ) φ(t) = (E 1 1)φ(t) φ (t) = φ(t L ) φ(t) = (E 1)φ(t) φ(t) = φ 1 (t) φ (t) X(t) = [φ 1 (t L ) φ 1 (t)] [φ (t L 1 ) φ (t)] = [(E 1)(E 1 1) (E 1 1)(E 1)]φ = Just LCM! Commutative Algebra TAMA Symposium 19 Feb 5 7

11 Schematic LISA LISA: Interferometric triangle with unequal arms S. Dhurandhar, R. Nayak, J-Y Vinet () V 1 * 1 L 3 U n^ 3 L ^ 1 n 1 U 3 V All L i unequal L i km 17 sec. 1* Data Streams: U V 3 1 n^ U 1, U, U 3, V 1, V, V 3 L 3 3* TAMA Symposium 19 Feb 5 8

12 The six elementary data streams In LISA the data are recorded as fractional Doppler shifts The laser frequency noise component: C i (t) = ν i(t) ν One such data stream: U 1 (t) = C 3 (t L ) C 1 (t) + U 1 GW + U 1 opt + U 1 pf We use the algebra of finite difference operators E k, k = 1,, 3: E k (f(t)) = f((t L k )) Six elementary data streams (only laser noise terms displayed): U 1 = E C 3 C U = E 3 C 1 C +... U 3 = E 1 C C V 1 = C 1 E 3 C +... V = C E 1 C V 3 = C 3 E C TAMA Symposium 19 Feb 5 9

13 Examples of data combinations cancelling laser phase noise Combinations cancelling the laser phase noise were found by Estabrook, Tinto, Armstrong, JPL, Pasadena, U.S. - adhoc methods used (1999-1). Symmetric Sagnac: ζ = E 1 V 1 + E V + E 3 V 3 + E 1 U 1 + E U + E 3 U 3 E 1 C 1 E 1 E 3 C +E C E E 1 C 3 +E 3 C 3 E 3 E C 1 +E 1 E C 3 E 1 C 1 +E E 3 C 1 E C +E 3 E 1 C E 3 C 3 = Polynomial vectors in the delay operators: ζ = (E 1, E, E 3, E 1, E, E 3 ) X = (1 E,, E (E 3 1), 1 E 3, E 3(E 1), ) X: Michelson - Sensitivity curve of LISA We give a general method to generate ALL data combinations for cancelling the laser noise. TAMA Symposium 19 Feb 5 1

14 A general data combination: The General Method X = 3 p i V i + q i U i i=1 where p i and q i are the polynomials in E 1, E, E 3. Laser frequency noise cancellation condition: 3 p i V i + q i U i = i=1 Only laser frequency noise terms involving C i (t) are considered TAMA Symposium 19 Feb 5 11

15 Explicit conditions on the polynomial vector For arbitrary C i (t) the polynomials p i, q i satisfy: p 1 q 1 + E 3 q E p 3 = p q + E 1 q 3 E 3 p 1 = p 3 q 3 + E q 1 E 1 p = TAMA Symposium 19 Feb 5 1

16 General Method (continued) Gaussian elimination of p 1, p leads to one equation, (E 1 E E 3 1)p 3 + (E 1 E 3 E )q 1 + E 1 (1 E 3)q + (1 E 1)q 3 = Solutions are 4-tuples: polynomial vectors {p 3, q 1, q, q 3 } Kernel of a module homomorphism Solution space: Module - first module of syzygies. Resubstitution of p 1 and p - (isomorphic) module of 6-tuple polynomial vectors {p 1, p, p 3, q 1, q, q 3 } The theorems gaurantee that the full module is generated TAMA Symposium 19 Feb 5 13

17 Generators of the Module of Syzygies Using the software Macaulay, we obtain one generating set: X (1) = (E 1 E 3 E,, E 3 1,, E E 3 E 1, E 3 1) X () = (E 1, E, E 3, E 1, E, E 3 ) X (3) = (1, E 3, E 1 E 3, 1, E 1 E, E ) X (4) = (E 1 E, 1, E 1, E 3, 1, E E 3 ) Another generating set: α = (1, E 3, E 1 E 3, 1, E 1 E, E ) β = (E 1 E, 1, E 1, E 3, 1, E E 3 ) γ = (E, E E 3, 1, E 1 E 3, E 1, 1) ζ = (E 1, E, E 3, E 1, E, E 3 ) TAMA Symposium 19 Feb 5 14

18 Sensitivity curves for the generators SNR(f; θ, φ) = h X(f; θ, φ) SX (f) Sensitivity: T obs = 1 year, f = 1 mhz: Σ(f; θ, φ) = 5 h X (f; θ, φ) S X (f) T obs θ 1 θ φ 4 φ 4 X X 6 TAMA Symposium 19 Feb 5 15

19 Σ(f; θ = π/4, φ = π/4): Sensitivity of generators (contd) 1e-13 1e-14 1e-15 X 4 X 3 X X 1 1e-16 1e-17 Σ(f) 1e-18 1e-19 1e- 1e-1 1e- 1e-3 1e-4 1e-5 1e Frequency f TAMA Symposium 19 Feb 5 16

20 Results General method for generating data combinations for cancelling laser phase noise in LISA Method can be applied to more general configurations LISA follow-on missions Mathematical theorems guarantee that ALL data combinations are generated Formulation allows simple expressions/notations - Important for coding Method can be extended in a straight forward way for bench motion noise, USO noise. SNR maximisation - improvement by to 3 R. Nayak, A. Pai, SVD, J-Y Vinet (3) Optimally track a source: improvement in sensitivity from 34 % at low frequencies to 9 % at mhz R. Nayak, A. Pai, SVD, J-Y Vinet (4) TAMA Symposium 19 Feb 5 17

21 Stability of LISA, changing armlengths, TDI CW Frame and Equations CW: Clohessy-Wiltshire eqns. && x y& 3 && y + x& =, && z + z =. x =, TAMA Symposium 19 Feb 5 18

22 The general solution of CW equations Reference orbit circular: = constant. sin cos ) ( ), ( ) 3( cos )sin 4 (6 ) ( ), ( )cos (3 sin ) ( t z t z t z x y t y x t x t y x t y y x t y x t x t x + = = = & & & & & & & & Offset terms: +, x y y x & &,,,,, z y x z y x o & & & Initial conditions: TAMA Symposium 19 Feb 5 19

23 1. No offsets : Stability requirements y& x& x + =, y =. Constant distance from the origin of the CW Frame where 1 x( t) = y( t) = z( t) = ± ρ = ρ 4x cos( t φ ρ sin( t φ), 3 ρ cos( t φ + y, tanφ ), ), y = x Orbits must lie in a plane making ± 6 angle with the ecliptic SVD, R. Nayak, S. Koshti, J-Y Vinet (5) TAMA Symposium 19 Feb 5

24 Main Results Orbits must lie in a plane making ± 6 angle with the ecliptic in the CW approximation (first order in eccentricity). The whole plane rotates rigidly distances between points on the plane remain constant. The constellation can consist of arbitrary number N of spacecraft and can have arbitrary shape. For LISA with N=3, the equilateral triangular configuration is not the only one. N=4 square configuration has been proposed for detecting stochastic GW background. TAMA Symposium 19 Feb 5 1

25 Arm Length (million Kms.) Breathing modes For Exact Keplerian orbits % Time (years) TAMA Symposium 19 Feb 5

26 Previous analysis for stationary LISA Future directions in TDI TDI needs to be generalised to take into consideration, moving LISA, changing armlengths, etc.: Second Generation TDI Sagnac Effect: Light travel times in two directions around the Sagnac circuit are different: L i L i, (L 1 + L + L 3 ) 4Aω/c 8 km 6 operators are needed: E i, E i, i = 1,, 3. Time-dependent armlengths: L i 1 m/sec. The operators no longer commute E 1 E E E 1 : φ(t L 1 L (t L 1 )) φ(t L L 1 (t L )) A general relativistic model of LISA/optical links is required to account for Sagnac effect, changing armlengths, gravitational red-shift, etc - future TDI should include these effects TAMA Symposium 19 Feb 5 3