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
Heterodyne detection with LISA
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
Signals and noises
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(, )
Corr(, ) = Corr(-,+ )
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
Max error: polar source ( = /2)
Max sensitivity: source direction to LISA plan
( ~ /6)
Estimations (180 runs on the noise)
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)
Thank you for listening
GW modelling effect on LISA