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Data Analysis Techniques for Gravitational Wave Observations

Data Analysis Techniques for Gravitational Wave Observations. S. V. Dhurandhar. I U C A A Pune, India. Great strides taken by experimentalists in improving sensitivity of GW detectors. Technology driven to its limits. Gravitational Wave Data Analysis

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Data Analysis Techniques for Gravitational Wave Observations

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  1. Data Analysis Techniques for Gravitational Wave Observations S. V. Dhurandhar I U C A A Pune, India

  2. Great strides taken by experimentalists in improving sensitivity of GW detectors Technology driven to its limits Gravitational Wave Data Analysis Important component of GW observation

  3. Signals with parametrizable waveforms • Deterministic Binary inspirals –modelled on the Hulse-Taylor binary pulsar Continuous wave sources • Stochastic Stochastic background • Unmodeled sources • Bursts and transients h ~ 10- 23 to 10-27

  4. Source Strengths Binary inspiral : Periodic: Stochastic background:

  5. Detector Sensitivity for the S2 run *http://www.ligo.caltech.edu/~lazz/distribution/Data

  6. Data Analysis Techniques Techniques depend on the type of source • Binary Inspirals: Matched filtering • Continuous wave signals: Fourier transforms after applying Doppler/spin-down corrections • Stochastic background: Optimally weighted cross-correlated data from independent detectors • Unmodeled sources: Bursts Time-frequency methods: Excess power statistics

  7. Inspiraling compact binary Waveform well modelled: - PN approximations (Damour, Blanchet, Iyer) - Resummation techniques: Pade, Effective one body – extend the validity of the PN formalism (Damour, Iyer, Sathyaprakash, Buonanno, Jaranowski, Schafer) Waveform: h Noise: Sh (f) Thematched filter : Stationary noise: Optimal filter in Gaussian noise: Detection probability is maximised for a given false alarm rate

  8. Matched filtering the inspiraling binary signal

  9. Detection Strategy Signal depends on many parameters Parameters: Amplitude, ta , fa , m1 , m2 , spins Strategy: Maximum likelihood method • Spinless case: • Amplitude: Use normalised templates • ta : FFT • Initial phase fa : Quadratures – only 2 templates needed for 0 and p/2 • masses chirp times: t0 , t3 bank required • For each template the maximised statistic is compared • with a threshold set by the false alarm rate. (SVD and Sathyaprakash)

  10. Thresholding , false alarm & detection Detection probability

  11. Parameter Space Parameter space for the mass range 1 – 30 solar masses

  12. Hexagonal tiling of the parameter space LIGO I psd Minimal match: 0 .97 Number of templates: ~ 104 Online speed: ~ 3 GFlops

  13. Inspiral Search (contd) • Reduced lower mass limit .2 Msun , fs ~ 10 Hz , then online speed ~ 300 Gflops • Hierarchical search required • - 2 step search: 2 banks - coarse & fine (Mohanty & SVD) • Step I : coarser bank – fewer templates, low threshold - high false alarm rate • Step II: follow-up the false alarms by a fine search • - Extended hierarchical search: over ta and masses • (Sengupta, SVD, Lazzarini) (Tanaka & Tagoshi)

  14. Hierarchical search frees up CPU for searching over more parameters LIGO I psd - mass range 1 to 30 solar masses 92% power at fc = 256 Hz Factor of 4 in FFT cost

  15. Relative size of templates in the 2 stages of hierarchy Total gain factor 60 over the flat search

  16. Multi-detector search for GW signals GEO: 0.6km VIRGO: 3km LIGO-LHO: 2km, 4km TAMA: 0.3km LIGO-LLO: 4km AIGO: (?)km

  17. Inspiral search with a network of detectors • Coincidence analysis: • – event lists, windows in parameter space(S. Bose) • Coherent search: -phase informationused (Pai, Bose, SVD) (S. Finn) • *Full data from all detectors necessary to carry out the data analysis • * A single network statistic constructed to be compared with a threshold • * Analytical maximisation over amplitude, initial phase, orientation of binary orbit • * FFT over the time-of-arrival • * direction search: time-delay window • * Filter bank over the intrinsic parameters: masses – metric depends on extrinsic parameters • Computational costs soar up in searching over time-delays ( ~ x 103 for LIGO-VIRGO)

  18. L S2 S1 Spin • Orbital-plane precesses –spin-orbit coupling -modulates the waveform (Blanchet, Damour, Iyer, Will, Wiseman, Jaranowski, Schafer) • Too many parameters – high computational cost (Apostolatus) • Detection template families – detection only(Buonnano, Chen, Vallisneri) • - few physical parameters, model well the modulation (FF > .97) • - automatic search over several (extrinsic) parameters – no template bank • For searching single-spin binaries: 7 M < m1 < 12 M , 1 M < m2 < 3 M • Templates in just 3 parameters: S1 , m1 and m2 • 76000 templates needed at .97 match (average) - LIGO I sensitivity

  19. Periodic Sources Target sources: Slowly varying instantaneous frequency eg. Rapidly rotating neutron starsh ~ 10-25 , 10-26 Integration time: months, years - motion of detector phase modulates the signal Doppler modulation: depends on direction of GW : Df = (n . v) f0/c 1 kHz wave gets spread into a million Fourier bins in 1 year observation time Intrinsic: spin down

  20. 1013 patches in the sky Computational cost in searching for periodic sources Parameters: f0, q, f, spin down parameters Targeted search: known pulsar: window in parameter space, heterodyne `All sky all frequency search ‘ - ACHALLENGE f0 is also a parameter Number of Doppler corrections (patches in the sky): spin-down parameters not included Brady et al (1998) Parameter space large: typical Tobs ~ 107 secs – weak source Effective GW telescope size ~ 2 AU, thus resolution = l / D ~ .2 arc sec

  21. Hierarchical Searches • Alternate between coherent & incoherent stages • Hough transform (Schutz, Papa, Frasca) • - short term Fourier Transforms • - Look for patterns in peaks in the time-frequency plane which • correspond to parameter values • -histogram in parameter space – do full time coherent search around the peak • Stack and slide search (Brady & T. Creighton) • - Given fixed computing power look for an efficient search algorithm • - Divide the data into N stacks, compute power spectra, slide and then sum • Results: gain 2-4 in sensitivity + 20-60% hierarchical , 99% confidence • Classes of pulsars: fmax = 1 kHz, t = 40 yr; fmax = 200 Hz, t = 1000 yr

  22. Stochastic Background Cannot distinguish instrumental noise from signal with one detector Cross-correlate the output of two detectors: Q: filter (Allen & Romano) (E. Flanagan)

  23. Stochastic Background Overlap reduction function g(f): Non-coincident & non-aligned detectors SNR : functional of g(f), WGW (f), P1(f), P2(f) LIGO detector pair, Tobs = 4 months, PF = 5%, PD = 95% Initial: WGW ~ 10-5 - 10-6 Advanced: WGW ~ 10-10 - 10-11

  24. Unmodeled sources Burst sources: Supernovae, Hypernovae, Binary mergers, Ring-downs of binary blackholes Excess power statistics: Sum the power in the time-frequency window Anderson, Brady, J.Creighton, Flanagan E is distributed:c2 if no signal and noise Gaussian non-central c2 if signal is present Q: How to distinguish non-gaussianity from the signal? (statistic can detect non-gaussianity) Network of detectors: autocorrelation v/s cross-correlation Slope statistic:

  25. Coherent detection of bursts with a network of detectors (J. Sylvestre) • Linearly combine the data with time-delays and antenna pattern functions for a given source direction: • Polarisation plane:Signal lies in theplane spanned by h+ (t) and hx (t) Y: data from a single synthetic detector andP = || Y ||2 P = z + h and r2 = z / E(h) and maximise r2 Only 2 parameters needed in addition to source direction: length ratio, angle Direction to the source can be found: LHV network ~ 1o – 10 o Source model required !

  26. Dealing with real data • Algorithms, codes working - yielding sensible results • Real detector noise is neither stationary nor Gaussian • - algorithms have been developed for G & S noise • - need to adapt the algorithms to the real world • Vetos: • - Excess noise level veto • - Instrumental vetos • For inspirals • - time frequency veto(Bruce Allen et al)

  27. Based on the fact that irrespective of the masses: Divide the frequency domain into p subbands so that the signal has equal power in each subband k and compute the c2 as : where rk is the SNR in subband k (normalised templates) Compare the value of c2 with a threshold for deciding detection Veto for inspirals (Allen et al) Better vetos: follow the ambiguity function

  28. Clustering of triggers for real events

  29. Clustering of triggers for real events • Condensing the `cloud of events’ – graph theory?

  30. Setting upper limits • Although at this early stage no detection can be announced we can place upper limits for example on the inspiral event rate • S1 data from the LIGO detectors gives A rate > than above means there is more than 90% probability that one inspiral event will be observed with SNR > highest SNR observed in S1 data. (gr-qc/0308069)

  31. Setting upper-limits (contd.) • Upper limits can be set for other types of sources: • Stochastic - WGW • Continuous wave sources - h for a given source < 23 for S1 data L1- H2 Source: PSR J1939+2134 (fastest known rotating neutron star) located 3.6 kpc from Earth - fGW ~ 1283.86 Hz Best upper limit from S1 data (L1) ~ 10-22

  32. Data Analysis as diagonistic tool • Detector characterisation: • Understanding of instrumental couplings to GW channel • Calibration • Line removal techniques – adaptive methods

  33. LISA : ESA & NASA project Space based detector for detecting low frequency GW

  34. LISA sensitivity curve

  35. Laser Interferometric Space Antenna (LISA) • LISA is an unequal arm interferometer in a triangular configuration • LISA will observe low frequency GW in the band-width of 10-5 Hz- 1 Hz. Six Doppler data streams Unequal arms: Laser frequency noise uncancelled Suitably delayed data streams form data combinations cancelling laser frequency noise (Tinto, Estabrook, Armstrong) Polynomial vectors in time-delay operators (SVD, Vinet, Nayak, Pai) Coherent detection

  36. LISA data analysis • Polynomial vectors in 3 time-delay operators • -Module of syzygies • 4 generators: a , b , g , z • - linear combinations generate the module • There are optimal combinations which perform better than the Michelson – LISA curve • The z combination can be used to `switch off ‘ GW • - calibration Current effort: generalise to moving LISA, changing • arm-lengths etc. (Tinto, SVD, Vinet, Nayak)

  37. Summary • Data analysis important aspect of GW observation • Different types of sources need different data analysis strategies • Algorithms must be computationally efficient – sophisticated analysis is required • Algorithms, codes now being tested on real data • LISA data analysis: combining data streams for • optimal performance

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