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Synthetic LISA simulating time-delay interferometry in a model LISA

Synthetic LISA simulating time-delay interferometry in a model LISA. (presenting) Michele Vallisneri (in absentia) John W. Armstrong LISA Science Office, Jet Propulsion Laboratory 12/17/2003. lisa.jpl.nasa.gov. Why Synthetic LISA?.

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Synthetic LISA simulating time-delay interferometry in a model LISA

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  1. Synthetic LISAsimulating time-delay interferometryin a model LISA (presenting) Michele Vallisneri (in absentia) John W. Armstrong LISA Science Office, Jet Propulsion Laboratory 12/17/2003 lisa.jpl.nasa.gov

  2. Why Synthetic LISA? • Simulate LISA fundamental noisesat the level of science/technical requirements • Higher level than extended modeling (no spacecraft subsystems) • Lower level than data analysis tools (do time-domain simulation of TDI; include removal of laser frequency fluctuations) • Provide streamlined module to filter GWs through TDI responses, for use in developing data-analysis algorithms • Include full model of TDI(motion of the LISA array, time- and direction-dependent armlengths,causal Doppler observables, 2nd-generation TDI observables) • Use directly or to validate (semi)analytic approximations • Make it friendly and fun to use GWDAW 2003: Michele Vallisneri on Synthetic LISA

  3. GW sources for plane waves, work from k, h+(t), hx(t) at SSB Doppler yij LISA geometry inter-spacecraft relative frequency fluctuations spacecraft positions photon propagation armlengths Doppler zij intra-spacecraft relative frequency fluctuations LISA noises laser freq. fluctuations, (optical bench),proof mass, optical path A LISA block diagram (very high level!) TDI observables time-delayed combinations of yij and zij laser-noise and optical-bench-noise free 3 independent observables GWDAW 2003: Michele Vallisneri on Synthetic LISA

  4. GW sources for plane waves, work from k, h+(t), hx(t) at SSB Doppler yij LISA geometry inter-spacecraft relative frequency fluctuations spacecraft positions photon propagation armlengths Doppler zij intra-spacecraft relative frequency fluctuations LISA noises laser freq. fluctuations, (optical bench),proof mass, optical path A LISA block diagram (very high level!) TDI observables TDI observables time-delayed combinations of yij and zij laser-noise and optical-bench-noise free 3 independent observables time-delayed combinations of yij and zij laser-noise and optical-bench-noise free 3 independent observables GWDAW 2003: Michele Vallisneri on Synthetic LISA

  5. GW sources for plane waves, work from k, h+(t), hx(t) at SSB TDI observables Doppler yij photon propagation vector LISA geometry inter-spacecraft relative frequency fluctuations GW TT tensor time-delayed combinations of yij and zij laser-noise and optical-bench-noise free 3 independent observables spacecraft positions photon propagation armlengths Doppler zij intra-spacecraft relative frequency fluctuations geom. projection factor LISA noises laser freq. fluctuations, (optical bench),proof mass, optical path A LISA block diagram (very high level!) Doppler yij inter-spacecraft relative frequency fluctuations Doppler zij Doppler shift due to GWs (Wahlquist-Estabrook response) measured for reception at spacecraft r and emission at spacecraft s(laser travels along arm l) intra-spacecraft relative frequency fluctuations GW buffeting of spacecraft s at emission (t-Ll) GW buffeting of spacecraft r at reception (t) wavefront retard.; pi are spacecraft pos. geom. projection factor GWDAW 2003: Michele Vallisneri on Synthetic LISA

  6. GW sources for plane waves, work from k, h+(t), hx(t) at SSB TDI observables Doppler yij LISA geometry inter-spacecraft relative frequency fluctuations time-delayed combinations of yij and zij laser-noise and optical-bench-noise free 3 independent observables spacecraft positions photon propagation armlengths Doppler zij intra-spacecraft relative frequency fluctuations LISA noises laser freq. fluctuations, (optical bench),proof mass, optical path A LISA block diagram (very high level!) fluctuations of laser 3 at emission (t - L2) fluctuations of laser 1* (reference) at reception (t) Doppler shift measured for reception at spacecraft 1 and emission at spacecraft 3(laser travels along arm 2) shot noise at sc 1 proof-mass 1* noise Doppler yij inter-spacecraft relative frequency fluctuations Doppler zij intra-spacecraft relative frequency fluctuations Doppler shift measured between optical benches on spacecraft 1 proof-mass 1 noise fluctuations of lasers 1 and 1* GWDAW 2003: Michele Vallisneri on Synthetic LISA

  7. GW sources for plane waves, work from k, h+(t), hx(t) at SSB Doppler yij TDI observables LISA geometry inter-spacecraft relative frequency fluctuations time-delayed combinations of yij and zij laser-noise and optical-bench-noise free 3 independent observables spacecraft positions photon propagation armlengths Doppler zij intra-spacecraft relative frequency fluctuations LISA noises laser freq. fluctuations, (optical bench),proof mass, optical path A LISA block diagram (very high level!) theoryrand+digital filter Nyquist f: pfDt = p/2 theoryrand+digital filter Nyquist f: pfDt = p/2 • LISA noises: 18 time series (6 proof mass + 6 optical path + 6 laser) • Assume Gaussian, f-2, f2, white • Generate in the time domain by applying digital filters to uncorrelated white noise produced at fixed sampling time,then interpolate • For laser noise, use combination of Markov chain (exp(-Dt/l) correlation) and low-pass digital filter GWDAW 2003: Michele Vallisneri on Synthetic LISA

  8. GW sources for plane waves, work from k, h+(t), hx(t) at SSB Doppler yij TDI observables LISA geometry LISA geometry inter-spacecraft relative frequency fluctuations time-delayed combinations of yij and zij laser-noise and optical-bench-noise free 3 independent observables spacecraft positions photon propagation armlengths spacecraft positions photon propagation armlengths Doppler zij intra-spacecraft relative frequency fluctuations LISA noises laser freq. fluctuations, (optical bench),proof mass, optical path A LISA block diagram (very high level!) • Motion complicates GW signals (1): • by changing orientation of LISA plane(power spread through ~9 bins) • by Doppler-shifting incoming GW signals (due to relative motion, dominates for f>10-3 Hz; bandwidth ~(WR/c)f) One Solar orbit/yr; LISA triangle spins through 360°/orbit Armlengths deviate from equilateral triangle at ~ 2% Armlengths are time and direction dependent • Motion improves sensitivity to GW (1): • to source position and polarization • makes it homogeneous in the sky • Motion hinders noise suppression (1,2,3): • need accurate knowledge of armlengths • high-order time delays needed GWDAW 2003: Michele Vallisneri on Synthetic LISA

  9. The Synthetic LISA package Class LISA Defines the LISA time-evolving geometry (positions of spacecraft, armlengths) OriginalLISA: static configuration with fixed (arbitrary) armlengths ModifiedLISA: stationary configuration, rotating with T=1yr; different cw and ccw armlengths CircularRotating: spacecraft on circular, inclined orbits; cw/ccw, time-evolving,causal armlengths EccentricInclined: spacecraft on eccentric, inclined orbits; cw/ccw, time-evolving,causal armlengths NoisyLISA (use with any LISA): adds white noise to armlengths used for TDI delays ... Class Wave Defines the position and time evolution of a GW source SimpleBinary: GW from a physical monochromatic binary SimpleMonochromatic: simpler parametrization InterpolateMemory: interpolate user-provided buffers for h+, hx ... Implements the LISA block structure as a collection of C++ classes Class TDI(LISA,Wave) Return time series of noise and GW TDI observables (builds causal yij’s; includes 1st- and 2nd-generation observables) TDInoise: demonstrates laser-noise subtraction TDIsignal: causal, validated vs. LISA Simulator TDIfast: cached for multiple sources (Edlund) GWDAW 2003: Michele Vallisneri on Synthetic LISA

  10. The Synthetic LISA package Class LISA Defines the LISA time-evolving geometry (positions of spacecraft, armlengths) OriginalLISA: static configuration with fixed (arbitrary) armlengths ModifiedLISA: stationary configuration, rotating with T=1yr; different cw and ccw armlengths CircularRotating: spacecraft on circular, inclined orbits; cw/ccw, time-evolving,causal armlengths EccentricInclined: spacecraft on eccentric, inclined orbits; cw/ccw, time-evolving,causal armlengths NoisyLISA (use with any LISA): adds white noise to armlengths used for TDI delays ... Class Wave Defines the position and time evolution of a GW source SimpleBinary: GW from a physical monochromatic binary SimpleMonochromatic: simpler parametrization InterpolateMemory: interpolate user-provided buffers for h+, hx ... ...things to do with it right now! Check the sensitivity of alternate LISA configurations Class TDI(LISA,Wave) Return time series of noise and GW TDI observables (builds causal yij’s; includes 1st- and 2nd-generation observables) TDInoise: demonstrates laser-noise subtraction TDIsignal: causal, validated vs. LISA Simulator TDIfast: cached for multiple sources (Edlund) GWDAW 2003: Michele Vallisneri on Synthetic LISA

  11. The Synthetic LISA package Class LISA Defines the LISA time-evolving geometry (positions of spacecraft, armlengths) OriginalLISA: static configuration with fixed (arbitrary) armlengths ModifiedLISA: stationary configuration, rotating with T=1yr; different cw and ccw armlengths CircularRotating: spacecraft on circular, inclined orbits; cw/ccw, time-evolving,causal armlengths EccentricInclined: spacecraft on eccentric, inclined orbits; cw/ccw, time-evolving,causal armlengths NoisyLISA (use with any LISA): adds white noise to armlengths used for TDI delays ... Class Wave Defines the position and time evolution of a GW source SimpleBinary: GW from a physical monochromatic binary SimpleMonochromatic: simpler parametrization InterpolateMemory: interpolate user-provided buffers for h+, hx ... • Demonstrate laser-noise sub.: • 1st-generation TDI • modified TDI • 2nd-generation TDI • degradation of subtraction for imperfect knowledge of arms • with armlocking ...things to do with it right now! Class TDI(LISA,Wave) Return time series of noise and GW TDI observables (builds causal yij’s; includes 1st- and 2nd-generation observables) TDInoise: demonstrates laser-noise subtraction TDIsignal: causal, validated vs. LISA Simulator TDIfast: cached for multiple sources (Edlund) GWDAW 2003: Michele Vallisneri on Synthetic LISA

  12. The Synthetic LISA package Class LISA Defines the LISA time-evolving geometry (positions of spacecraft, armlengths) OriginalLISA: static configuration with fixed (arbitrary) armlengths ModifiedLISA: stationary configuration, rotating with T=1yr; different cw and ccw armlengths CircularRotating: spacecraft on circular, inclined orbits; cw/ccw, time-evolving,causal armlengths EccentricInclined: spacecraft on eccentric, inclined orbits; cw/ccw, time-evolving,causal armlengths NoisyLISA (use with any LISA): adds white noise to armlengths used for TDI delays ... Class Wave Defines the position and time evolution of a GW source SimpleBinary: GW from a physical monochromatic binary SimpleMonochromatic: simpler parametrization InterpolateMemory: interpolate user-provided buffers for h+, hx ... ...things to do with it right now! Produce synthetic time series to test data-analysis algorithms Class TDI(LISA,Wave) Return time series of noise and GW TDI observables (builds causal yij’s; includes 1st- and 2nd-generation observables) TDInoise: demonstrates laser-noise subtraction TDIsignal: causal, validated vs. LISA Simulator TDIfast: cached for multiple sources (Edlund) GWDAW 2003: Michele Vallisneri on Synthetic LISA

  13. Using Synthetic LISA The preferred interface to Synthetic LISA is through a simple script in the language Python. This is a Python script! Import the Synthetic LISA library(lisaswig.py, _lisaswig.so) so we can use it #!/usr/bin/python import lisaswig; unequalarmlisa = lisaswig.ModifiedLISA(15.0,16.0,17.0); unequalarmnoise = lisaswig.TDInoise(unequalarmlisa, 1.0,2.5e-48,1.0,1.8e-37,1.0,1.1e-26,1.0); lisaswig.printtdi("noise-X.txt",unequalarmnoise,1048576,1.0,"X"); Create a LISA (geometry) object;use static LISA, with equal arms Armlengths (s) Create a TDI object based on our chosen LISA Laser correlation (s) Noise sampling time (s) Laser Sn (Hz-1) Opt. path Sn  f-2 (Hz-1) Proof mass Sn  f2 (Hz-1) TDI variablesto print Print X TDI noise to disk! File name # samples requested,sampling time GWDAW 2003: Michele Vallisneri on Synthetic LISA

  14. 10-25 Example: unequal-arm 1st-gen. noises Note laser noise subtraction! ... lisaswig.printtdi("noise-a.txt",unequalarmnoise,1048576,1.0,"a"); lisaswig.printtdi("noise-z.txt",unequalarmnoise,1048576,1.0,"z"); lisaswig.printtdi("noise-E.txt",unequalarmnoise,1048576,1.0,"E"); GWDAW 2003: Michele Vallisneri on Synthetic LISA

  15. Example: noisyLISA subtraction originallisa = lisaswig.OriginalLISA(16.6782,16.6782,16.6782) noisylisa = lisaswig.NoisyLISA(originallisa,1.0,measurement noise) originalnoise = lisaswig.TDInoise(originallisa, 1.0,2.5e-48,1.0,1.8e-37,1.0,1.1e-26,0.1) noisynoise = lisaswig.TDInoise(noisylisa,originallisa, 1.0,2.5e-48,1.0,1.8e-37,1.0,1.1e-26,0.1) measurement noise Sn (s2 Hz-1) Use different LISA for noise and TDI delays GWDAW 2003: Michele Vallisneri on Synthetic LISA

  16. Example: monochromatic binary f = 2 mHzT = 1 yr ecliptic lat. = p/2ecliptic long. = 0 lat. = p/5long. = p/3 mylisa = lisaswig.CircularRotating(0.0,0.0,1.0) mybinary = lisaswig.SimpleBinary(frequency,initial phase,inclination,amplitude, ecliptic latitude,ecliptic longitude,polarization angle) mysignal = lisaswig.TDIsignal(mylisa,mybinary) lisaswig.printtdi("signal-X.txt",mysignal,secondsperyear/16.0,16.0,"X") LISA array parameters # samples requested,sampling time GWDAW 2003: Michele Vallisneri on Synthetic LISA

  17. Comparison with LISA Simulator Synthetic LISALISA Simulator TDI X (no noise), T = 1 yr f = 1.94 mHzinc = 1.60ecliptic lat.  0, long. = 0 GWDAW 2003: Michele Vallisneri on Synthetic LISA

  18. Case study: S/Nsfor extreme-mass ratio inspirals Hughes-Glampedakis-Kennefick integrator(C++): output h+, hx Synthetic LISA: generate A, E, T, X GW & noise time series Matlab:compute S/Ns (Python) GWDAW 2003: Michele Vallisneri on Synthetic LISA

  19. Summary! • Synthetic LISA is the package I would have wanted to download and use, had I not written it • Synthetic LISA simulates LISA fundamental noises and GW responseat the level of science/technical requirements • Synthetic LISA includes a full model of the LISA science process (2nd-generation TDI, laser-noise subtraction) • Synthetic LISA’s modular design allows easy interfacing to extended modeling and data-analysis applications • Synthetic LISA is user-friendly and extensible (C++, Python, other scripting languages) • Synthetic LISA is planned for open-source release in Jan/Feb (NASA permitting) GWDAW 2003: Michele Vallisneri on Synthetic LISA

  20. Synthetic LISAsimulating time-delay interferometryin a model LISA Michele Vallisneri Jet Propulsion Laboratory 12/12/2003 lisa.jpl.nasa.gov

  21. GW sources for plane waves, work from k, h+(t), hx(t) at SSB Doppler yij TDI observables LISA geometry LISA geometry inter-spacecraft relative frequency fluctuations time-delayed combinations of yij and zij laser-noise and optical-bench-noise free 3 independent observables spacecraft positions photon propagation armlengths spacecraft positions photon propagation armlengths Doppler zij intra-spacecraft relative frequency fluctuations LISA noises laser freq. fluctuations, (optical bench),proof mass, optical path A LISA block diagram (very high level!) • One Solar orbit/yr, equilateral-triangle configuration kept to ~2% • The triangle spins through 360°/orbit • Motion complicates signals: • by changing orientation of LISA plane (power spread through ~9 bins) • by Doppler-shifting incoming GW signals (due to relative motion;dominates for f>10-3 Hz; bandwidth ~(WR/c)f) • Motion improves sensitivity: • to source position and polarization • homogeneous in the sky • Full model must include: • Time dependence of arms • Aberration GWDAW 2003: Michele Vallisneri on Synthetic LISA

  22. GW sources for plane waves, work from k, h+(t), hx(t) at SSB Doppler yij TDI observables LISA geometry inter-spacecraft relative frequency fluctuations time-delayed combinations of yij and zij laser-noise and optical-bench-noise free 3 independent observables spacecraft positions photon propagation armlengths Doppler zij intra-spacecraft relative frequency fluctuations LISA noises laser freq. fluctuations, (optical bench),proof mass, optical path A LISA block diagram (very high level!) theoryrand+digital filter Nyquist f: pfDt = p/2 • Proof-mass f/f noise: six time series • Assume Gaussian and red; baseline Sn 2.5 10-48 f-2 Hz-1 • Generate white noise n(ti) (independent Gaussian variates) at sampling interval t • Filter through digital integrator: y(ti+1) = ay(tn) + n(ti) • Resulting spectrum Sy(f) = Sn(f)/[4 sin2(pft)] for 1 (non-unit  cuts DC) GWDAW 2003: Michele Vallisneri on Synthetic LISA

  23. theoryrand+digital filter GW sources for plane waves, work from k, h+(t), hx(t) at SSB Doppler yij TDI observables Nyquist f: pfDt = p/2 LISA geometry inter-spacecraft relative frequency fluctuations time-delayed combinations of yij and zij laser-noise and optical-bench-noise free 3 independent observables spacecraft positions photon propagation armlengths Doppler zij intra-spacecraft relative frequency fluctuations LISA noises laser freq. fluctuations, (optical bench),proof mass, optical path A LISA block diagram (very high level!) • Optical path f/f noise: six time series • Assume Gaussian and blue; baseline Sn 1.8 10-37 f2 Hz-1 • Generate white noise n(ti) (independent Gaussian variates) at sampling interval t • Filter through digital differentiator: y(ti+1) = n(ti+1) - n(ti) • Resulting spectrum Sy(f) = 4 sin2(pft) Sn(f) GWDAW 2003: Michele Vallisneri on Synthetic LISA

  24. GW sources for plane waves, work from k, h+(t), hx(t) at SSB Doppler yij TDI observables LISA geometry inter-spacecraft relative frequency fluctuations time-delayed combinations of yij and zij laser-noise and optical-bench-noise free 3 independent observables spacecraft positions photon propagation armlengths Doppler zij intra-spacecraft relative frequency fluctuations LISA noises laser freq. fluctuations, (optical bench),proof mass, optical path A LISA block diagram (very high level!) theory rand+digital filter(sampling time = 1 s) • Noise interpolation: • The TDI observables operate on noise values at times specified to 30 ns • If noise is band limited, the exact time structure can be reconstructed by Fourier series resummation (but this requires the entire data train!) • Simple linear interpolation between samples introduces some structure above the effective Nyquist frequency (of noise generation) • Moral: generate noise (and sample TDI) comfortably above frequency of interest GWDAW 2003: Michele Vallisneri on Synthetic LISA

  25. GW sources for plane waves, work from k, h+(t), hx(t) at SSB Doppler yij TDI observables LISA geometry inter-spacecraft relative frequency fluctuations time-delayed combinations of yij and zij laser-noise and optical-bench-noise free 3 independent observables spacecraft positions photon propagation armlengths Doppler zij intra-spacecraft relative frequency fluctuations LISA noises laser freq. fluctuations, (optical bench),proof mass, optical path A LISA block diagram (very high level!) • Laser f/f noise: six time series • Assume Gaussian and white, band-limited between 1 Hz and 10 Hz, Sn 1.1 10-26 Hz-1 • To understand TDI laser-frequency-noise subtraction, it is crucial to model correctly the short-time correlation structure of the noise: residual n(t) ≈ n(t + L est. error.) - n(t) • Generating white noise at fixed sampling interval and then interpolating overestimates this correlation (imposing lax requirements on armlength-measurement error) • It is also possible to generate exp(-Dt/l) correlated noise at arbitrary times using an unequal-timestep Markov process (Ornstein-Uhlenbeck process); this underestimates the real laser-noise correlation (imposing exacting requirements on armlength-measurement error) • A good balance can probably be found by producing noise with a Markov chain, followed by a digital filter GWDAW 2003: Michele Vallisneri on Synthetic LISA

  26. GW sources GW sources for plane waves, work from k, h+(t), hx(t) at SSB for plane waves, work from k, h+(t), hx(t) at SSB Doppler yij TDI observables LISA geometry inter-spacecraft relative frequency fluctuations time-delayed combinations of yij and zij laser-noise and optical-bench-noise free 3 independent observables spacecraft positions photon propagation armlengths Doppler zij intra-spacecraft relative frequency fluctuations LISA noises laser freq. fluctuations, (optical bench),proof mass, optical path A LISA block diagram (very high level!) • For the purpose of LISA detection, plane gravitational waves are completely specified by their ecliptic coordinates(l,b) and by their h+(t) and hx(t) time series at the solar system baricenter • Retardation to the LISA spacecraft is trivial given the plane-wave structure • A conventional rotation angle (l,b) defines the two GW polarizations GWDAW 2003: Michele Vallisneri on Synthetic LISA

  27. The Synthetic LISA package Class LISA Defines the LISA time-evolving geometry (positions of spacecraft, armlengths) OriginalLISA: static configuration with fixed (arbitrary) armlengths ModifiedLISA: stationary configuration, rotating with T=1yr; different cw and ccw armlengths CircularRotating: spacecraft on circular, inclined orbits; cw/ccw, time-evolving,causal armlengths EccentricInclined: spacecraft on eccentric, inclined orbits; cw/ccw, time-evolving,causal armlengths NoisyLISA (use with any LISA): adds white noise to armlengths used for TDI delays ... Class Wave Defines the position and time evolution of a GW source SimpleBinary: GW from a physical monochromatic binary SimpleMonochromatic: simpler parametrization InterpolateMemory: interpolate user-provided buffers for h+, hx ... ...things to do with it right now! Generate syntheticgalactic-WD confusionbackgrounds Class TDI(LISA,Wave) Return time series of noise and GW TDI observables (builds causal yij’s; includes 1st- and 2nd-generation observables) TDInoise: demonstrates laser-noise subtraction TDIsignal: causal, validated vs. LISA Simulator TDIfast: cached for multiple GW sources (Jeff) GWDAW 2003: Michele Vallisneri on Synthetic LISA

  28. Example: equal-arm 1st-gen. TDI noises equalarmlisa = lisaswig.OriginalLISA(16.6782,16.6782,16.6782); equalarmnoise = lisaswig.TDInoise(equalarmlisa, 1.0,2.5e-48,1.0,1.8e-37,1.0,1.1e-26,1.0); lisaswig.printtdi("noise-X.txt",equalarmnoise,1048576,1.0,"X"); GWDAW 2003: Michele Vallisneri on Synthetic LISA

  29. Example: equal-arm 1st-gen. TDI noises ... lisaswig.printtdi("noise-a.txt",equalarmnoise,1048576,1.0,"z"); lisaswig.printtdi("noise-z.txt",equalarmnoise,1048576,1.0,"z"); lisaswig.printtdi("noise-E.txt",equalarmnoise,1048576,1.0,"E"); GWDAW 2003: Michele Vallisneri on Synthetic LISA

  30. Example: modified-TDI subtraction Use different LISA for noise and TDI delays modifiedlisa = lisaswig.ModifiedLISA(16.6782,16.6782,16.6782) modifiednoise = lisaswig.TDInoise(equalarmlisa,modifiedlisa, 1.0,2.5e-48,1.0,1.8e-37,1.0,1.1e-26,1.0e-6) lisaswig.printtdi("noise-Xm.txt",modifiednoise,samples,1.0,"X"); correctednoise = lisaswig.TDInoise(modifiedlisa, 1.0,2.5e-48,1.0,1.8e-37,1.0,1.1e-26,1.0e-6) lisaswig.printtdi("noise-Xmc.txt",correctednoise,samples,1.0,"Xm"); modifiedTDI obs GWDAW 2003: Michele Vallisneri on Synthetic LISA

  31. Example: realistic LISA noises For 1 yr of integration, including galactic-WD confusion noise “short LISA” (L = 1.66x106 km) baseline LISA (L = 1.66x106 km) GWDAW 2003: Michele Vallisneri on Synthetic LISA

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