SBPI 16/06/2009 1

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