Synthetic LISA simulating timedelay interferometry in a model LISA

1
Synthetic LISAsimulating time-delay
interferometryin a model LISA
lisa.jpl.nasa.gov

• (presenting) Michele Vallisneri
• (in absentia) John W. Armstrong
• LISA Science Office, Jet Propulsion Laboratory
• 12/17/2003

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

3
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
4
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
5
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)
geom. projection factor
wavefront retard. pi are spacecraft pos.
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A LISA block diagram (very high level!)
fluctuations of laser 1 (reference) at reception
(t)
fluctuations of laser 3 at emission (t - L2)
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
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A LISA block diagram (very high level!)
theoryranddigital filter
Nyquist f pfDt p/2
theoryranddigital 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

8
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 fgt10-3 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

9
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 T1yr
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 ...
Class TDI(LISA,Wave) Return time series of noise
and GW TDI observables (builds causal yijs
includes 1st- and 2nd-generation
observables) TDInoise demonstrates laser-noise
subtraction TDIsignal causal, validated vs. LISA
Simulator TDIfast cached for multiple sources
(Edlund)
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The Synthetic LISA package

• ...things to do with it right now!

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 T1yr
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 ...
Check the sensitivity of alternate LISA
configurations
Class TDI(LISA,Wave) Return time series of noise
and GW TDI observables (builds causal yijs
includes 1st- and 2nd-generation
observables) TDInoise demonstrates laser-noise
subtraction TDIsignal causal, validated vs. LISA
Simulator TDIfast cached for multiple sources
(Edlund)
11
The Synthetic LISA package

• ...things to do with it right now!

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 T1yr
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 ...

• Demonstrate laser-noise sub.
• 1st-generation TDI
• modified TDI
• 2nd-generation TDI
• degradation of subtraction for imperfect
  knowledge of arms
• with armlocking

Class TDI(LISA,Wave) Return time series of noise
and GW TDI observables (builds causal yijs
includes 1st- and 2nd-generation
observables) TDInoise demonstrates laser-noise
subtraction TDIsignal causal, validated vs. LISA
Simulator TDIfast cached for multiple sources
(Edlund)
12
The Synthetic LISA package

• ...things to do with it right now!

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 T1yr
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 ...
Produce synthetic time series to test
data-analysis algorithms
Class TDI(LISA,Wave) Return time series of noise
and GW TDI observables (builds causal yijs
includes 1st- and 2nd-generation
observables) TDInoise demonstrates laser-noise
subtraction TDIsignal causal, validated vs. LISA
Simulator TDIfast cached for multiple sources
(Edlund)
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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 unequalarmlis
a lisaswig.ModifiedLISA(15.0,16.0,17.0) unequ
alarmnoise lisaswig.TDInoise(unequalarmlisa, 1
.0,2.5e-48,1.0,1.8e-37,1.0,1.1e-26,1.0) lisaswi
g.printtdi("noise-X.txt",unequalarmnoise,1048576,1
.0,"X")
Create a LISA (geometry) objectuse 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)
Proof mass Sn ? f2 (Hz-1)
Opt. path Sn ? f-2 (Hz-1)
TDI variablesto print
Print X TDI noise to disk!
File name
samples requested,sampling time
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Example unequal-arm 1st-gen. noises
10-25
Note laser noise subtraction!
... lisaswig.printtdi("noise-a.txt",unequalarmnois
e,1048576,1.0,"a") lisaswig.printtdi("noise-z.txt
",unequalarmnoise,1048576,1.0,"z") lisaswig.print
tdi("noise-E.txt",unequalarmnoise,1048576,1.0,"E")

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Example noisyLISA subtraction
originallisa lisaswig.OriginalLISA(16.6782,16.67
82,16.6782) noisylisa lisaswig.NoisyLISA(origin
allisa,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
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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) m
ybinary lisaswig.SimpleBinary(frequency,initial
phase,inclination,amplitude,
ecliptic latitude,ecliptic longitude,polarization
angle) mysignal lisaswig.TDIsignal(mylisa,mybin
ary) lisaswig.printtdi("signal-X.txt",mysignal,sec
ondsperyear/16.0,16.0,"X")
LISA array parameters
samples requested,sampling time
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Comparison with LISA Simulator
Synthetic LISALISA Simulator
TDI X (no noise), T 1 yr f 1.94 mHzinc
1.60ecliptic lat. ? 0, long. 0
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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
Matlabcompute S/Ns
(Python)
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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 LISAs 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)

20
Synthetic LISAsimulating time-delay
interferometryin a model LISA
lisa.jpl.nasa.gov

• Michele Vallisneri
• Jet Propulsion Laboratory
• 12/12/2003

21
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 motiondominates for fgt10-3 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

22
A LISA block diagram (very high level!)
theoryranddigital 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(ti1)
  ay(tn) n(ti)
• Resulting spectrum Sy(f) Sn(f)/4 sin2(pf?t)
  for ??1 (non-unit ? cuts DC)

23
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(ti1)
  n(ti1) - n(ti)
• Resulting spectrum Sy(f) 4 sin2(pf?t) Sn(f)

24
A LISA block diagram (very high level!)
theory randdigital 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

25
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

26
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

27
The Synthetic LISA package

• ...things to do with it right now!

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 T1yr
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 ...
Generate syntheticgalactic-WD confusionbackgroun
ds
Class TDI(LISA,Wave) Return time series of noise
and GW TDI observables (builds causal yijs
includes 1st- and 2nd-generation
observables) TDInoise demonstrates laser-noise
subtraction TDIsignal causal, validated vs. LISA
Simulator TDIfast cached for multiple GW
sources (Jeff)
28
Example equal-arm 1st-gen. TDI noises
equalarmlisa lisaswig.OriginalLISA(16.6782,16.67
82,16.6782) equalarmnoise lisaswig.TDInoise(eq
ualarmlisa, 1.0,2.5e-48,1.0,1.8e-37,1.0,1.1e-26,
1.0) lisaswig.printtdi("noise-X.txt",equalarmnoi
se,1048576,1.0,"X")
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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")
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Example modified-TDI subtraction
Use different LISA for noise and TDI delays
modifiedlisa lisaswig.ModifiedLISA(16.6782,16.67
82,16.6782) modifiednoise lisaswig.TDInoise(equ
alarmlisa,modifiedlisa,
1.0,2.5e-48,1.0,1.8e-37,1.0,1.1e-26,1.0e-6) lisasw
ig.printtdi("noise-Xm.txt",modifiednoise,samples,1
.0,"X") correctednoise lisaswig.TDInoise(modif
iedlisa, 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
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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)