1. 1SBPI 16/06/2009 Heterodyne detection with LISA for gravitational waves parameters estimation Nicolas Douillet

2. 2SBPI 16/06/2009 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

3. 3SBPI 16/06/2009 LISA motion during one Earth period

4. 4SBPI 16/06/2009 - LISA arm’s length: 5. 10 9 m to detect gravitational waves with frequency in: 10 -4   10 -1 Hz - 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 periodic motion -> information on the direction of the wave. LISA configuration

5. 5SBPI 16/06/2009 Motivations for LISA  A space based detector allows to get rid of this constraint.  Possibility to detect very low frequency gravitional waves. 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)

6. 6SBPI 16/06/2009 Monochromatic waves Sources: signals coming from coalescing binaries long before inspiral step. Frequency  considered as a constant. + polarization x polarization 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:

7. 7SBPI 16/06/2009 Model for a monochromatic wave(1) LISA response to the incoming GW: 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 T : LISA period (1 year)

8. 8SBPI 16/06/2009 Model for a monochromatic wave (2) Amplitude modulation (envelope) Shape depends on source location: ( ,  ) With and

9. 9SBPI 16/06/2009 Pattern beam functions (1) Change of reference frame for and pattern beam functions. Spacecraft n° in LISA triangle.  : polarization angle

10. 10SBPI 16/06/2009 Pattern beam functions (2) ‘+’ polarization 4 sidebands

11. 11SBPI 16/06/2009 Pattern beam functions (3) ‘x’ polarization

12. 12SBPI 16/06/2009 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: 00 Frequency precision is reached with a nested search.

13. 13SBPI 16/06/2009 (3): Shift spectrum (offset zero-frequency) by heterodyning at, then low-pass filtering 8 lateral bands: [0; 7  ] (empirical) -> compromise between accepted noise level and maximum frequency needed to rebuild the envelope (  = 1/ T) Envelope heterodyne detection (2) Fourier sum (2): Envelope reconstruction (Filter above )

14. 14SBPI 16/06/2009 Correlation optimization (1) Correlation surface between template and experimental envelope

15. 15SBPI 16/06/2009 Correlation optimization (2) (1)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.

16. 16SBPI 16/06/2009 Signals and noises

17. 17SBPI 16/06/2009 Spectrum and instrumental noises

18. 18SBPI 16/06/2009 Sources mix Possible to distinguish between n sources since their fundamental frequencies are spaced enough (sidebands don’t cover each other): Sources

19. 19SBPI 16/06/2009 Envelope detection (1) 

20. 20SBPI 16/06/2009 Envelope detection (2) 

21. 21SBPI 16/06/2009 Symmetries & ambiguities Correlation symmetry Corr( ,  ) = Corr(  - ,  +  ) LISA main symmetry E(  - ,  +  ) = E( ,  )

22. 22SBPI 16/06/2009 Symmetries (1) Some parameters remains difficult to estimate due to the high number of the envelope symmetries on the parameters  and . Examples:

23. 23SBPI 16/06/2009 Symmetries (2) Ie -> risks of being stuck on correlation secondary maxima in N dimensions space (varied topologies resolution problem).

24. 24SBPI 16/06/2009 How to remove sky location uncertainty (1) 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 

25. 25SBPI 16/06/2009 How to remove sky location uncertainty (2): Source -> LISA, Doppler effect

26. 26SBPI 16/06/2009 How to remove sky location uncertainty (3): Source -> LISA, Doppler effect

27. 27SBPI 16/06/2009 Source localization Simulated data from LISA data analysis community

28. 28SBPI 16/06/2009 Statistics on sky location angles ( ,  )  = f(  )  = f(  ) Max error: polar source (  =  /2 ) Max sensitivity: source direction  to LISA plan (  ~  /6 )

29. 29SBPI 16/06/2009 Noise robustness tests (static source) True value Estimations (180 runs on the noise)

30. 30SBPI 16/06/2009 Typical errors on estimated parameters Average relative errors for    /3  i / i Ecliptic latitude  5. 10 -2 Ecliptic longitude  1. 10 -3 Polarization  1.5 10 -1 Inclination angle  3. 10 -1 Frequency  8.5 10 -6 Amplitude h 0.5 – 1 ~ X

31. 31SBPI 16/06/2009 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). ● much lower 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)

32. 32SBPI 16/06/2009 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 (N 5 + N 2 instead of N 7 for example).

33. 33SBPI 16/06/2009 Thank you for listening

34. 34SBPI 16/06/2009 GW modelling effect on LISA