synthetic lisa simulating time-delay interferometry in a model lisa

Synthetic LISAsimulating time-delay interferometryin a model LISA

(presenting) Michele Vallisneri(in absentia) John W. Armstrong

LISA Science Office, Jet Propulsion Laboratory12/17/2003

lisa.jpl.nasa.gov

GWDAW 2003: Michele Vallisneri on Synthetic LISA 212/17/2003

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


A LISA block diagram (very high level!)

LISA geometryspacecraft positions

photon propagation armlengths

LISA noiseslaser freq. fluctuations, (optical bench),proof mass, optical path

GW sources

for plane waves, work from k, h+(t), hx(t) at SSB

time-delayed combinations of yij and zij

laser-noise and optical-bench-noise free

3 independent observables

TDI observables

Doppler yij

Doppler zij

inter-spacecraft relative frequency fluctuations

intra-spacecraft relative frequency fluctuations






GW sources





TDI observables




Doppler yij

Doppler zij



TDI observables









TDI observables

Doppler yij

Doppler zij



Doppler yij

Doppler zij



GW sources


geom. projection factor

photon propagation vectorGW TT tensor

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)





GW sources





TDI observables

Doppler yij

Doppler zij



Doppler yij

Doppler zij




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





GW sources





TDI observables

Doppler yij

Doppler zij




theoryrand+digital filter

Nyquist f: f

t = /2

theoryrand+digital

filter

Nyquist f: f

t =

/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(-t/) correlation) and low-pass digital filter






GW sources





TDI observables

Doppler yij

Doppler zij





1.One Solar orbit/yr; LISA triangle spins through 360°/orbit

2.Armlengths deviate from equilateral triangle at ~ 2%

3.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 ~(R/c)f)


The Synthetic LISA packageImplements the LISA block structure as a collection of C++ classesClass 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 WaveDefines 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 LISADefines the LISA time-evolving geometry (positions of spacecraft, armlengths)






...

The Synthetic LISA package...things to do with it right now!

Check the sensitivity of alternate LISA configurations










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...


Demonstrate laser-noise sub.:• 1st-generation TDI• modified TDI• 2nd-generation TDI• degradation of subtraction

for imperfect knowledge of arms• with armlocking










...







...


Produce synthetic time series to test data-analysis algorithms


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 variablesto 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: unequal-arm 1st-gen. noises

...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");

Note laser noise subtraction!

10-

25


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

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. = /2ecliptic long. = 0

lat. = /5long. = /3

f = 2 mHzT = 1 yr

LISA array parameters

# samples requested,

sampling time


Comparison with LISA Simulator

Synthetic LISALISA Simulator

TDI X (no noise), T = 1 yr

f = 1.94 mHzinc = 1.60ecliptic lat. 0, long. = 0


Case study: S/Nsfor 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!

• 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

Michele VallisneriJet Propulsion Laboratory

12/12/2003

lisa.jpl.nasa.gov






GW sources





TDI observables

Doppler yij

Doppler zij





• 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 ~(R/c)f)

• Motion improves sensitivity:• to source position and polarization• homogeneous in the sky

• Full model must include:• Time dependence of arms• Aberration





GW sources





TDI observables

Doppler yij

Doppler zij




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) = y(tn) + n(ti)• Resulting spectrum Sy(f) = Sn(f)/[4 sin2(ft)] for 1 (non-unit

cuts DC)

theoryrand+digital

filter Nyquist f: f

t =

/2





GW sources





TDI observables

Doppler yij

Doppler zij




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(ft) Sn(f)

theoryrand+digital filter N

yquist f: f

t =

/2





GW sources





TDI observables

Doppler yij

Doppler zij




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

theoryrand+digital filter(sampling time = 1 s)





GW sources





TDI observables

Doppler yij

Doppler zij




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(-t/) 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






GW sources





TDI observables

Doppler yij

Doppler zij



GW sources


For the purpose of LISA detection, plane gravitational waves are completely specified by their ecliptic coordinates (,) 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 (,) defines the two GW polarizations





TDIfast: cached for multiple GW sources (Jeff)





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...


Generate syntheticgalactic-WD confusionbackgrounds


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");


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");


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

modifiedTDI obs


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)

synthetic lisa simulating time-delay interferometry in a model lisa

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