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Heterodyne detection with LISA

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Heterodyne detection with LISA

for gravitational waves parameters estimation

Nicolas Douillet

- (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

- 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.

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.

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

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)

With

Amplitude modulation (envelope)

Shape depends on source location: (, )

and

Change ofreference frame for

and pattern beam functions.

: polarization angle

Spacecraft n° in LISA triangle.

4 sidebands

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

(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 surface between template and experimental envelope

- 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

Sources

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(-,+ )

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

envelope symmetries on the parametersand.

Examples:

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

space (varied topologies resolution problem).

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

Source localization

Simulated data from LISA data analysis community

Max error: polar source ( = /2)

Max sensitivity: source direction to LISA plan

( ~ /6)

= f()

= f()

True value

Estimations (180 runs on the noise)

~

X

Average relative errors for /3

(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)

- 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