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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for gravitational waves parameters estimation
- Heliocentric orbits,
free falling spacecraft.
- 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)
allows to get rid of this
very low frequency
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
Effect (amplified) of a
Gravitational wave on a ring
(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)
Amplitude modulation (envelope)
Shape depends on source location: (, )
Change ofreference frame for
and pattern beam functions.
: polarization angle
Spacecraft n° in LISA triangle.
(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.
(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
Correlation surface between template and experimental envelope
Envelope detection (1)
Envelope detection (2)
Some parameters remains difficult to estimate due to the high number of the
envelope symmetries on the parametersand.
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
Simulated data from LISA data analysis community
Estimations (180 runs on the noise)
(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)