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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 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 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
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.
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 parametersand. 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).
