heterodyne detection with lisa
DESCRIPTION
Heterodyne detection with LISA. for gravitational waves parameters estimation. Nicolas Douillet. Outline. (1) : LISA (Laser Interferometer Space Antenna (2) : Model for a monochromatic wave (3) : Heterodyne detection principle (4) : Some results on simulated data analysis - PowerPoint PPT PresentationTRANSCRIPT
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Heterodyne detection with LISAHeterodyne detection with LISA
for gravitational waves parameters for gravitational waves parameters estimationestimation
Nicolas Douillet
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OutlineOutline
• (1): LISA (Laser Interferometer SpaceAntenna
• (2): Model for a monochromatic wave
• (3): Heterodyne detection principle
• (4): Some results on simulated data analysis
• (5): Conclusion & future work
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LISA motion during one Earth periodLISA motion during one Earth period
23
( )1 2 3S , S , S
LISA geometry rotation symmetry in ecliptic longitude ( ) between
two consecutive spacecraft orbits
2 1 3 2
2 2;
3 3S S S S
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- LISA arm’s length: 5. 109 m to detect gravitational waves with frequency in: 10-4 10-1 Hz
- Heliocentric orbits,free falling spacecraft.
- LISA center of massFollows Earth, delayedfrom a 20° angle.
- 60° angle between LISAplan and the ecliptic plan.
- LISA periodic motion -> information on the direction of the wave.
LISA configurationLISA configuration
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Motivations for LISAMotivations for LISA
A space based detectorallows to get rid of thisconstraint.
Possibility to detectvery low frequencygravitional waves.
Existing ground based detectors such as VIRGO and LIGO are « deaf » in lowfrequencies ( < 10 Hz).
Limited sensitivity due to « seismic wall » (LF vibrations transmitted by theNewtonian fields gradient)
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Monochromatic wavesMonochromatic waves
H h h
Sources: signals coming from coalescing binarieslong before inspiral step. Frequency consideredas a constant.
+ polarization x polarization
h+ / h : amplitude following + / x polarization
+ / : directional functions
Gravitational wave causesperturbations in the metrictensor.
Effect (amplified) of aGravitational wave on a ringof particles:
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Model for a monochromatic wave(1)Model for a monochromatic wave(1)
cos 2 sin cos
sin 2 sin cos
s t F t h t R
F t h t R
1, , , , , , , i i N
s s t h s
LISA response to the incoming GW: 2 t
T
Unknown parameters:
(Hz): source frequency (rad): ecliptic latitude (rad): ecliptic longitude (rad): polarization angle (rad): orbital inclination angle h (-): wave amplitude (rad): initial source phase
T : LISA period (1 year)
4 110 10Hz Hz
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Model for a monochromatic wave (2)Model for a monochromatic wave (2)
arctanh F
th F
cos 2 sin cosR
s t E t t tc
1/ 22 2
E t h F h F
Amplitude modulation (envelope)Shape depends on source location: (, )
With
21 cos ; 2 cosh h h h and
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Pattern beam functions (1)Pattern beam functions (1)
, , ,
, , ,
1cos 2 sin 2
2
1sin 2 cos 2
2
n n n
n n n
F t D t D t
F t D t D t
Change of reference frame for and pattern beam functions. ,nD ,nD
1 3n
Spacecraft n° in LISA triangle.
: polarization angle
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Pattern beam functions (2)Pattern beam functions (2)‘+’ polarization‘+’ polarization
2,
3[ 36sin sin 2 2 1
64 3
3 cos 2 cos 2 9sin 2 1 sin 4 2 13 3
sin 2 cos 4 2 1 9cos 2 13 3
4 3 sin 2 sin 3 2 1 3sin 2 1 ]3 3
nD t n
n n
n n
n n
4 sidebands2
: t
T
LISA orbital phase
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Pattern beam functions (3)Pattern beam functions (3)‘x’ polarization‘x’ polarization
1[ 3 cos 9cos 2 1 2
16 3
cos 4 2 1 23
6sin cos 3 2 13
3cos 2 1 ]3
nD t n
n
n
n
2:
t
T
LISA orbital phase
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Envelope heterodyne detection (1)Envelope heterodyne detection (1)Principle:
(1): Fundamental frequency (0) search
Detect the maximum in the spectrum of the product between source signal (s) and a template signal (m) which frequency lays in the range:
0
0 0[ ; ]
max S M
Frequency precision is reached with a nested search.
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17 2 1
0
kj n
N
k
E n Y k e
(3): Shift spectrum (offset zero-frequency) by heterodyning at , then low-pass filtering
8 lateral bands: [0; 7] (empirical) -> compromise between accepted noiselevel and maximum frequency needed to rebuild the envelope ( = 1/ T)
Envelope heterodyne detection (2)Envelope heterodyne detection (2)
0
020 0
i ts t e S S S
Fourier sum
Y S H
(2): Envelope reconstruction
(Filter above )0
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Correlation optimization (1)Correlation optimization (1)
Correlation surface between template and experimental envelope
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Correlation optimizationCorrelation optimization (2) (2)
0lim , 0j
Corr E E
0 00
1 0 02 2
,N
i i
i i i
E E E ECorr E E
E E E E
(1) Principle: correlation maximization between signal envelope end envelopetemplate (or mean squares minimization).
(2) Method: gradient convergence and quasi-Newton optimization methods.
(3) Conditions: already lay on the convex area which contains the maximum.
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Signals and noisesSignals and noises
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Spectrum and instrumental noisesSpectrum and instrumental noises
1 1 1,nS N 2 2 2,nS N
0
2
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Sources mixSources mix
Possible to distinguish between n
sources since their fundamental
frequencies are spaced enough
(sidebands don’t cover each other):
15j i
Sources
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Envelope detection (1)Envelope detection (1)
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Envelope detection (2)Envelope detection (2)
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Symmetries & ambiguities Symmetries & ambiguities
Correlation symmetry
Corr(, ) = Corr(-,+ )
LISA main symmetry
E(-, + ) = E(, )
,S
,S
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Symmetries (1)Symmetries (1)
Some parameters remains difficult to estimate due to the high number of theenvelope symmetries on the parameters and .
Examples:
, , , , , ,
, , , , , ,
; ; ; 0;24 4
; ; ;4 4
i i i i i i
i i i i i i
F F F F F F
F F F F F F
1/ 22 2
E t h F h F
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Symmetries (2)Symmetries (2)
, , , ,
, , , ,
; ; 0;22 2
; ;2 2
i i i i
i i i i
F F F F
F F F F
Ie -> risks of being stuck on correlation secondary maxima in N dimensions space (varied topologies resolution problem).
; ; ; 0;2
;
h h h h h h
h h h h h h
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How to remove sky location uncertainty (1)How to remove sky location uncertainty (1)
2sin cos
Rk r t
c
Choice between (,) and ( -, +) depends on the sign of the product
If is the colatitude (ie [0; ] ), and when t=0
sgn sgn cosk r t
From the source signal, we compute the quantity
sgn tana
h Ft Arg s t Arc
h F
hence the sign of and
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How to remove sky location uncertainty (2):How to remove sky location uncertainty (2): Source -> LISA, Doppler effect Source -> LISA, Doppler effect
max
min
sin cos tan
1 2sin sin ; tan 0
2
21 sin 1 ;
2
21 sin 1 ;
h F tRt t arc
c h F t
h F tRf t arc
t cT t h F t
R vf
cT c
R vf
cT c
2
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How to remove sky location uncertainty (3):How to remove sky location uncertainty (3): Source -> LISA, Doppler effect Source -> LISA, Doppler effect
2 2
2 2
2: LISA tangential speed
2 : Ecliptic length
: Light year
1 4sin cos
2
Moreover, if 0 (colatitude),
0 si - : frequency seen by LISA increases.2 2
0 s
Rv
TR
cT
f t R
t t cT
f t
t
f t
t
3
i : frequency seen by LISA decreases.2 2
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Source localizationSource localization
Simulated data from LISA data analysis community
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Statistics on sky location angles Statistics on sky location angles ((,,))
= f() = f()
Max error: polar source ( = /2) 0 / 2, / 3S
Max sensitivity: source direction to LISA plan( ~ /6)
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Noise robustness tests (static source)Noise robustness tests (static source)
True value
Estimations (180 runs on the noise)
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Typical errors on estimated parametersTypical errors on estimated parameters
Average relative errors for /3
i/i
Ecliptic latitude 5. 10-2
Ecliptic longitude 1. 10-3
Polarization 1.5 10-1
Inclination angle 3. 10-1
Frequency 8.5 10-6
Amplitude h 0.5 – 1
~
X
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Compare two parameters estimation Compare two parameters estimation techniques: template bank vs MCMCtechniques: template bank vs MCMC
(1): Matching templates (template bank and scan parameters space till reaching correlation maximum -> systematic method)
- Advantages: ● easy/friendly programmable ● quite good robustness
- Limitations: ● N dimensions parameters space. (memory space and computation time expensive)
● difficulties to adapt and apply this method for more complex waveforms
(2): MCMC methods, max likelihood ratio: motivations(statistics & probability based methods)
- Advantages: ● No exhaustive scan of the parameters space (dim N). ● much lower computing cost and smaller memory space- Limitations: ● Careful handling: high number parameters to tune in the
algorithm (choice of probability density functions of the parameters)
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Conclusion and future workConclusion and future work
- Encouraging results of this method (heterodyne detection) on monochromatic waves. Could still to be improved however.
- Continue to develop image processing techniques for trajectories segmentation (chirp & EMRI) in time-frequency plan. (level sets, ‘active contours’ methods import from medical imaging and shape optimization)
- Combining this methods (graphic first estimation of parameters) with Monte-Carlo Markov Chains algorithms (numeric finest estimation) allows in a way to ‘‘ log-divide’’ the dimensions of the parameters space (N5 + N2 instead of N7 for example).
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Thank you for listeningThank you for listening
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GW modelling effect on GW modelling effect on LISALISA