Laser Interferometer Space Antenna (LISA) Time-Delay Interferometry - PowerPoint PPT Presentation

Laser Interferometer Space Antenna (LISA) Time-Delay Interferometry with Moving Spacecraft Arrays Massimo Tinto Jet Propulsion Laboratory, California Institute of Technology 8 th GWDAW, Dec. 17-20, 2003, Milwaukee, Wisconsin GSFC � JPL

W.M. Folkner et al., C.Q.G ., 14 , 1543, (1997)

Unequal-arm Interferometers φ = + − − + ( t ) h ( t ) p ( t 2 L ) p ( t ) n ( t ) L 3 2 2 2 2 φ = + − − + ( t ) h ( t ) p ( t 2 L ) p ( t ) n ( t ) φ 3 (t) 3 3 3 3 P.D φ i = two-way phase measurements Laser L 2 φ = + − + ( t ) h ( t ) [ D D I ] p ( t ) n ( t ) ν 0 , p(t) 2 2 2 2 2 P.D φ 2 (t) φ = + − + ( t ) h ( t ) [ D D I ] p ( t ) n ( t ) 3 3 3 3 3 Ψ ≡ Ψ − where : D ( t ) ( t L ) i i S.V. Dhurandhar, K.R. Nayak, and J-Y. Vinet , Phys. Rev. D , 65 , 102002 (2002). ≡ − φ − − φ X ( t ) [ D D I ] ( t ) [ D D I ] ( t ) 3 3 2 2 2 3 = − + − − + [ D D I ][ h ( t ) n ( t )] [ D D I ][ h ( t ) n ( t )] 3 3 2 2 2 2 3 3 + − − − − − [( D D I )( D D I ) ( D D I )( D D I )] p ( t ) 3 3 2 2 2 2 3 3 M. Tinto, & J.W. Armstrong , Phys. Rev. D , 59 , 102003 (1999).

Time-delay Interferometry 2 • It is best to think of LISA as a 2* L 1 closed array of six one-way delay l r r n n lines between the test masses. 1 3 3* o l • This approach allows us to L 3 3 l reconstruct the unequal-arm L 2 r n Michelson interferometer, as well 2 1 as new interferometric 1* combinations, which offer advantages in hardware design, in robustness to failures of single links, and in redundancy of data. M. Tinto : Phys. Rev. D, 53 , 5354 (1996); Phys. Rev. D, 58 , 102001 (1998) J.W. Armstrong, F.B. Estabrook, and M. Tinto : Ap. J., 527 , 814 (1999) F.B. Estabrook, M. Tinto, & J.W. Armstrong, Phys. Rev. D, 62 , 042002 (2000) M. Tinto, D.A. Shaddock, J. Sylvestre, & J.W. Armstrong : Phys. Rev. D 67 , 122003 (2003)

Unequal-arm Interferometers (Cont.) = φ + φ − φ + φ X ( t ) [ ( t ) D D ( t )] [ ( t ) D D ( t )] 3 3 3 2 2 2 2 3 • One can actually regard X as given 3 by the interference of two beams that propagate within the two arms 2 of LISA, each experiencing a delay L 2 L 3 equal to (2L 2 + 2L 3 ) . • X is actually a zero-area Sagnac Interferometer, synthesized by properly combining measurements from each arm. 1 D.A. Shaddock, M. Tinto, F.B. Estabrook & J.W. Armstrong , Phys. Rev. D , 68 , 061303 (R) (2003).

Six-Pulse Data Combinations . 2 α, β, γ, ζ L 3 L 1 . . 3 1 L 2 -- α = η + η + η − η + η + η ( ) ( ) 31 23 , 2 12 , 12 21 32 , 3 13 , 13 ? ? ζ = η − η + η − η − η + η ( ) ( ) 31 , 1 32 , 2 12 , 2 13 , 3 23 , 3 21 , 1 = − − − + + ( p p ) ( p p ) ( GW Secondary noises ) 1 , 23 1 , 1 1 , 23 1 , 1 D.A. Shaddock, M. Tinto, F.B. Estabrook & J.W. Armstrong , Phys. Rev. D , 68 , 061303 (R) (2003).

Eight-Pulse Data Combinations . . 2 2 (X, Y, Z) L 3 L 3 L 1 Unequal-arm Michelson (P, Q, R) . . . . Beacon 3 1 3 1 L 2 L 2 . . 2 2 L 3 L 3 (E, F, G) L 1 L 1 (U, V, W) . . . . Relay Monitor 3 3 1 1 L 2 L 2

Moving spacecraft Arrays and Clocks Synchronization • The analysis above assumed the clocks onboard the LISA S/Cs to be synchronized to each other in the frame attached to the LISA array. • In a rotating reference frame, the Sagnac effect prevents the implementation of the Einstein’s Synchronization Procedure, i.e. synchronization by transmission of electromagnetic signals (GPS is a good example of this problem!) • To account for the Sagnac effect, one introduces an hypothetical inertial reference frame, and time in this frame is the one adopted by the spacecraft clocks! • In other words, the onboard receivers have to convert time information received from Earth to time in this inertial reference frame (SSB). N. Asbby , “The Sagnac effect in the GPS System”, http://digilander.libero.it/solciclos/ M. Tinto, F.B. Estabrook, & J.W. Armstrong , gr-qc/0310017, October 6, 2003

Moving spacecraft Arrays and Clocks Synchronization (Cont.) • In the SSB frame, the differences between back-forth delay times are very much larger than has been previously recognized. • The reason is in the aberration due to motion and changes of orientation in the SSB frame. • With a velocity V=30 km/s, the light-transit times of light signals in opposing directions (L i , and L’ i ) will differ by as much as 2VL (a few thousands km) • They will also change in time due to rotation (0.1 m/s); this however is significantly smaller than the spacecraft relative velocity (10 m/s).

TDI with Moving spacecraft Arrays • The “first-generation” TDI expressions do not account for: – The Sagnac Effect – Time-dependence (velocity) of the arm lengths in the TDI expressions (the “Flex-effect”) • Both effects prevent the perfect cancellation of the laser frequency fluctuations in the “first-generation” TDI combinations. • With a laser frequency stability of 30 Hz/Hz 1/2 the remaining laser frequency fluctuations could be as much as 30 times larger than the secondary noise sources. D.A. Shaddock , Phys. Rev. D: to appear; gr-qc/0306125 Cornish & Hellings , Class. Quantum Grav. 20 No 22 (21 November 2003) 4851-4860 D.A. Shaddock, M. Tinto, F.B. Estabrook & J.W. Armstrong , Phys. Rev. D , 68 , 061303 (R) (2003).

The Sagnac Effect and the Sagnac Combinations . 2 (α, β, γ) • In presence of rotation, the amount of time spent by a beam to propagate clockwise is different by L 3 L 1 the time it spends to propagate counterclockwise . . along the same arm => (L 1 , L 2 , L 3 , L’ 1 , L’ 2 , L’ 3 ). • The Sagnac effect prevents the perfect 3 1 L 2 cancellation of the laser frequency fluctuations in the existing expressions of the Sagnac . combinations ( α, β, γ, ζ ). 2 @ 10 -3 Hz the laser frequency fluctuations • remaining in ( α, β, γ, ζ ) would be about 30 times L 3 L 1 L’ 1 . larger than the secondary noise sources. . L’ 3 L’ 2 3 1 L 2 r r + + − + + = Ω ⋅ Α ≅ ' ' ' | ( L L L ) ( L L L ) | 4 | | 14 km 1 2 3 1 2 3 α 1 , α 2 , α 3 α, β, γ

“Flexy” 3 2 1 D.A. Shaddock, M. Tinto, F.B. Estabrook & J.W. Armstrong , Phys. Rev. D , 68 , 061303 (R) (2003).

Systematic Approach • Is there a general procedure for deriving the “2 nd generation” TDI combinations? • YES! Ψ = Ψ − − − ≠ Ψ D D ( t ) ( t L ( t ) L ( t L )) D D ( t ) i j ' i j ' i j ' i Ψ ≅ Ψ − − − [ D , D ] ( t ) ( t L L )( V L V L ) i j ' i j ' j ' i i j ' M. Tinto, F.B. Estabrook, & J.W. Armstrong , gr-qc/0310017, October 6, 2003

Systematic Approach (Cont.) η η η η ( t ), ( t ), ( t ), ( t ) η + η = − [ D D I ] p 12 21 13 31 21 12 ; 3 ' 3 ' 3 1 . η + η = − [ D D I ] p 2 31 13 ; 2 2 2 ' 1 L 3 L’ 3 . . L’ 2 = − η + η − − η + η X [ D D I ]( ) [ D D I ]( ) 3 1 L 2 2 2 ' 21 12 ; 3 ' 3 ' 3 31 13 ; 2 = − − − − − [ D D I ][ D D I ] p [ D D I ][ D D I ] p = 0 2 2 ' 3 ' 3 1 3 ' 3 2 2 ' 1 η + η + η + η = − ( ) ( ) [ D D D D I ] p 31 13 ; 2 21 12 ; 3 ' ; 2 ' 2 2 2 ' 3 ' 3 1 η + η + η + η = − ( ) ( ) [ D D D D I ] p 21 12 ; 3 ' 31 13 ; 2 ; 33 ' 3 ' 3 2 2 ' 1 = − η + η + η + η X [ D D D D I ][( ) ( ) ] 1 3 ' 3 2 2 ' 31 13 ; 2 21 12 ; 3 ' ; 2 ' 2 − − η + η + η + η ≅ [ D D D D I ][( ) ( ) ] 0 2 2 ' 3 ' 3 21 12 ; 3 ' 31 13 ; 2 ; 33 '

How Does the LISA Sensitivity Change? • Once the laser frequency fluctuations are removed, the corrections to the signal and the secondary noises (optical path, proof-mass, etc.), introduced by the extra delays due to the Sagnac (14 km) and flexy (~ 300 m) effects, are many orders of magnitude below the signals and secondary noises determined by the “1 st generation” TDI expressions: = − − − X ( t ) X ( t ) X ( t 2 L 2 L ) 1 2 3 ~ ~ π + = − 2 if ( 2 L 2 L ) X ( f ) X ( f ) [ 1 e ] 2 3 1