Synthetic LISAsimulating time-delay interferometryin 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?:

Why Synthetic LISA? Simulate LISA fundamental noises at 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

A LISA block diagram (very high level!):

A LISA block diagram (very high level!) time-delayed combinations of yij and zij
laser-noise and optical-bench-noise free
3 independent observables TDI observables

A LISA block diagram (very high level!):

A LISA block diagram (very high level!) time-delayed combinations of yij and zij
laser-noise and optical-bench-noise free
3 independent observables TDI observables time-delayed combinations of yij and zij
laser-noise and optical-bench-noise free
3 independent observables TDI observables

A LISA block diagram (very high level!):

A LISA block diagram (very high level!) Doppler yij Doppler zij inter-spacecraft relative frequency fluctuations intra-spacecraft relative frequency fluctuations GW buffeting of spacecraft s at emission (t-Ll) GW buffeting of spacecraft r at reception (t) geom. projection factor wavefront retard.; pi are spacecraft pos. Doppler shift due to GWs (Wahlquist-Estabrook response) measured for reception at spacecraft r and emission at spacecraft s (laser travels along arm l)

A LISA block diagram (very high level!):

A LISA block diagram (very high level!) Doppler yij Doppler zij inter-spacecraft relative frequency fluctuations intra-spacecraft relative frequency fluctuations shot noise at sc 1 fluctuations of laser 1* (reference) at reception (t) fluctuations of laser 3 at emission (t - L2) proof-mass 1* noise Doppler shift measured for reception at spacecraft 1 and emission at spacecraft 3 (laser travels along arm 2) Doppler shift measured between optical benches on spacecraft 1 fluctuations of lasers 1 and 1* proof-mass 1 noise

A LISA block diagram (very high level!):

A LISA block diagram (very high level!) theory rand+digital filter Nyquist f: pfDt = p/2 theory rand+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

A LISA block diagram (very high level!):

A LISA block diagram (very high level!) One Solar orbit/yr; LISA triangle spins through 360°/orbit
Armlengths deviate from equilateral triangle at ~ 2%
Armlengths are time and direction dependent Motion hinders noise suppression (1,2,3):
need accurate knowledge of armlengths
high-order time delays needed Motion improves sensitivity to GW (1):
to source position and polarization
makes it homogeneous in the sky 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)

The Synthetic LISA package:

The Synthetic LISA package Implements the LISA block structure as a collection of C++ classes 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 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) 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
...

The Synthetic LISA package:

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) 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
... 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
... The Synthetic LISA package ...things to do with it right now! Check the sensitivity of alternate LISA configurations

The Synthetic LISA package:

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) 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
... 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
... The Synthetic LISA package ...things to do with it right now! Demonstrate laser-noise sub.:
1st-generation TDI
modified TDI
2nd-generation TDI
degradation of subtraction for imperfect knowledge of arms
with armlocking

The Synthetic LISA package:

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) 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
... 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
... The Synthetic LISA package ...things to do with it right now! Produce synthetic time series to test data-analysis algorithms

Using Synthetic LISA:

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 Create a LISA (geometry) object; use static LISA, with equal arms Armlengths (s) Create a TDI object based on our chosen LISA Noise sampling time (s) Proof mass Sn f2 (Hz-1) Opt. path Sn f-2 (Hz-1) Laser Sn (Hz-1) Laser correlation (s) Print X TDI noise to disk! File name # samples requested, sampling time TDI variables to print #!/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");

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

Example: monochromatic binary:

Example: monochromatic binary 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") ecliptic lat. = p/2 ecliptic long. = 0 lat. = p/5 long. = p/3 f = 2 mHz T = 1 yr LISA array parameters # samples requested, sampling time

Comparison with LISA Simulator:

Comparison with LISA Simulator Synthetic LISA LISA Simulator TDI X (no noise), T = 1 yr
f = 1.94 mHz inc = 1.60 ecliptic lat. 0, long. = 0

Case study: S/Nsfor extreme-mass ratio inspirals:

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

Summary!:

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 response at 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)

Synthetic LISAsimulating time-delay interferometryin a model LISA:

Synthetic LISA simulating time-delay interferometry in a model LISA Michele Vallisneri
Jet Propulsion Laboratory
12/12/2003 lisa.jpl.nasa.gov

A LISA block diagram (very high level!):

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

A LISA block diagram (very high level!):

A LISA block diagram (very high level!) 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(pft)] for 1 (non-unit cuts DC) theory rand+digital filter Nyquist f: pfDt = p/2

A LISA block diagram (very high level!):

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(pft) Sn(f)

A LISA block diagram (very high level!):

A LISA block diagram (very high level!) 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 theory
rand+digital filter (sampling time = 1 s)

A LISA block diagram (very high level!):

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

A LISA block diagram (very high level!):

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

The Synthetic LISA package:

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) 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
... 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
... The Synthetic LISA package ...things to do with it right now! Generate synthetic galactic-WD confusion backgrounds

Example: modified-TDI subtraction 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"); Use different LISA for noise and TDI delays modified TDI obs

Example: realistic LISA noises:

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)

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