Heterodyne detection with lisa
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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

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Heterodyne detection with LISA

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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

  • (5): Conclusion & future work


LISA motion during one Earth period


LISA configuration

- Heliocentric orbits,

free falling spacecraft.

  • LISA center of mass

  • Follows Earth, delayed

  • from a 20° angle.

  • 60° angle between LISA

  • plan and the ecliptic plan.

- LISA arm’s length: 5. 109 m to detect gravitational waves with frequency in: 10-4   10-1 Hz

- LISA periodic motion -> information on the direction of the wave.


Motivations for LISA

Existing ground based detectors such as VIRGO and LIGO are « deaf » in low

frequencies (  < 10 Hz).

Limited sensitivity due to « seismic wall » (LF vibrations transmitted by the

Newtonian fields gradient)

  • A space based detector

    allows to get rid of this

    constraint.

  • Possibility to detect

    very low frequency

    gravitional waves.


Monochromatic waves

Sources: signals coming from coalescing binaries

long before inspiral step. Frequency  considered

as a constant.

h+ / h: amplitude following + / x polarization

+ / : directional functions

Gravitational wave causes

perturbations in the metric

tensor.

Effect (amplified) of a

Gravitational wave on a ring

of particles:

+ polarization

x polarization


Model for a monochromatic wave(1)

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

LISA response to the incoming GW:

T : LISA period (1 year)


Model for a monochromatic wave (2)

With

Amplitude modulation (envelope)

Shape depends on source location: (, )

and


Pattern beam functions (1)

Change ofreference frame for

and pattern beam functions.

: polarization angle

Spacecraft n° in LISA triangle.


4 sidebands

Pattern beam functions (2)‘+’ polarization


Pattern beam functions (3)‘x’ polarization


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:

Frequency precision is reached with a nested search.

0


Envelope heterodyne detection (2)

(3): Shift spectrum (offset zero-frequency) by heterodyning at , then low-pass filtering

(Filter above )

8 lateral bands: [0; 7] (empirical) -> compromise between accepted noise

level and maximum frequency needed to rebuild the envelope ( = 1/ T)

(2): Envelope reconstruction

Fourier sum


Correlation optimization (1)

Correlation surface between template and experimental envelope


Correlation optimization (2)

  • Principle: correlation maximization between signal envelope end envelope

  • template (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.


Signals and noises


Spectrum and instrumental noises


Sources

Sources mix

Possible to distinguish between n

sources since their fundamental

frequencies are spaced enough

(sidebands don’t cover each other):




Envelope detection (1)






Envelope detection (2)


LISA main symmetry

E(-, + ) = E(, )

Correlation symmetry

Corr(, ) = Corr(-,+ )

Symmetries & ambiguities


Symmetries (1)

Some parameters remains difficult to estimate due to the high number of the

envelope symmetries on the parametersand.

Examples:


Symmetries (2)

Ie -> risks of being stuck on correlation secondary maxima in N dimensions

space (varied topologies resolution problem).


How to remove sky location uncertainty (1)

Choice between (,) and ( -, +) depends on the sign of the product

If  is the colatitude (ie   [0;  ]), and when t=0

From the source signal, we compute the quantity

hence the sign of  and 


How to remove sky location uncertainty (2): Source -> LISA, Doppler effect


How to remove sky location uncertainty (3): Source -> LISA, Doppler effect


Source localization

Simulated data from LISA data analysis community


Max error: polar source ( = /2)

Max sensitivity: source direction  to LISA plan

( ~ /6)

Statistics on sky location angles (,)

 = f()

 = f()


Noise robustness tests (static source)

True value

Estimations (180 runs on the noise)


~

X

Typical errors on estimated parameters

Average relative errors for   /3


Compare two parameters estimation techniques: 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).

● muchlower 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)


Conclusion 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).


Thank you for listening


GW modelling effect on LISA


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