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All-sky search for gravitational waves from neutron stars in binary systems strategy and algorithms H.J. Bulten.

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Presentation on theme: "All-sky search for gravitational waves from neutron stars in binary systems strategy and algorithms H.J. Bulten."— Presentation transcript:

1 All-sky search for gravitational waves from neutron stars in binary systems strategy and algorithms H.J. Bulten

2 H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 2 analysis of PSS from binaries thesis work of Sipho van der Putten Sipho van der Putten, R. Ebeling (siesta) staff involved: JFJ van den Brand, Th. Bauer, HJB, T.J. Ketel, S. Klous (grid) theory dept. : G. Koekoek and J.W. van Holten

3 H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 3 motivation: binary systems Virgo/Ligo: better sensitivity at higher frequency (>10 Hz) fixed quadrupole deformation: most high-frequency neutron stars are in binary systems –spin-up via gas transfer

4 H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 4 motivation Brady et al. PRD57,2101: old binary new? constant Power

5 H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 5 solitary neutron stars solitary neutron star: Doppler shifts from earth movement Hierarchical search possible, T~ 1h (Rome group, e.g. Astona, Frasca, Palomba CQG 2005.) signal-to-noise ~

6 H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 6 solitary neutron stars alternative: F-statistics approach (Ligo, Jaranowski et all PRD58, 063001) –produce templates that remain in phase over the template search time –parameters –solitary neutron stars: all-sky search –many templates needed, e.g. Brady et al. PRD61, 082001 coherent all-sky search of length of 0.5days would take 10,000 Tflops (fmax=1000 Hz) smaller spin-down, fmax=200 Hz: 5 days

7 H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 7 Binary : Kepler orbitals ellipse We want to analyze: –orbital periods from 2 hours – infinite –masses companion star up to 15 solar masses –eccentricities up to about 0.7 –frequency shifts up to 0.3%, frequency changes df/dt up to 10 -6 s -2 1 mHz shift in 1 second, at f=1000Hz

8 H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 8 frequency shifts

9 H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 9 frequency derivative

10 H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 10 frequency shifts

11 H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 11 coherence phase signal: signal should remain in-phase,e.g. maximally 90 deg. out of phase anywhere during observation time – frequency within ½ bin - 1/(2Tobs)

12 H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 12 binary neutron stars how many extra parameters? –e.g. orbital period >=2 hours, eccentricity <=0.6, mass companion <=15 solar masses, frequency <=1000 Hz –coherent: phase: distance to neutron star within 75 km w.r.t. template anywhere during the coherence time. –all power coherent within 1 FFT-bin: Tmax = 30s –FFT length 1 hour: signal spreads over 4000 bins. –Tobs = 1 hour: detectable difference in orbital period: ~70 ms a factor of 100,000 in parameter space to scan all orbital periods between 2 and 4 hours in a blind search

13 H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 13 binary neutron stars additional parameters: –even with Tobs = 1 hour, at least 100 billion times as many templates are required to keep the phase of the filter coherent for all possibilities within the boundaries: T_orbit => 2hour 0< eccentricity < 0.6 all orientations of semi-major and semi-minor axes all starting phases in orbital up to 1000 Hz g.w. frequencies full parameter scan is not feasible.

14 H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 14 binary neutron stars different set of filters: parameterize the phase as a function of time! –assume that within T obs, the frequency can be described by a second-order function of time –third-order effects are assumed to be negligible. scan for presence of signal by calculating the correlation with the template

15 H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 15 Correlation Correlation is given by presence of signal defined by overlap with filter. data is not periodic: make filter equal to zero for last N/2 samples and shift it maximally N/2 samples to the right FFT: interleave, to cover full dataset

16 H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 16 Filter search FFT 1 FFT 2 filter, lag=0 Filter: zero-padded for half length check correlations from t=0 to t= ½T (FFT1) check correlations from t= ½T to t=1T (FFT2) check correlations from t=1T to t= 1½T (FFT3) maximum overlap: amplitude and time known data, split in overlapping periods filter, scan to lag = T/2

17 H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 17 Filter search filter

18 H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 18 Example Filters

19

20 parameter space phase should be given by filter: –coherent times up to about T=500 seconds: for times <500 seconds, fourth-order corrections due to orbital movements are small –quadratic change of frequency: can be parameterized with about 120 parameters –linear change of frequency:

21 H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 21 Phase: parameters for coherent times up to 500 seconds, the frequency should be accurate within about 1mHz. –phase description of data: about 10 phases about 1 million values of f0 about 500 values of alpha=df/dt about 120 values of beta. –however: scan with FFT template: in time direction: can be determined templates can be re-used 600,000 templates reduce to about 5000

22 H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 22 shifting in time shifting a filter in time by a lag tau gives a filter with parameters: you do not have to apply filters with with

23 H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 23 shifting in frequency frequency changes are smaller than 1 Hz within the set of filters produce filters in a small frequency band, a complete set for 1 fixed value of f(t=0). –reduction of a factor of Fourier-transform them heterodyne data, or alternatively: compare the filter in frequency domain with the appropriate frequency band of the FFT of the data

24 H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 24 Scan Step in frequency: if the filter has small frequency dependence, you have to step 1 frequency bin. So a filter with a constant frequency is applied (Fmax/binwidth) times (e.g. 1 million times for an FFT of 1000 second) if the filter has large linear or quadratic dependence, you can step with a stepsize total scans needed to analyze 0 - 1000 Hz, 1000 seconds –about 10,000 filters suffice. –about 300 million correlations in total (300 million FFTs) –a few days of CPU-time on a single CPU, current desktop

25 H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 25 Hits a hit: overlap is larger than pre-defined threshold –PSD from FFT from complete set (needs to be optimized) sets noise threshold –normalize data in frequency domain to have mean amplitude of in each bin

26 H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 26 Procedure tests we tested with white noise, 4096 samples per second, 1024 seconds FFT: –filters can pick signal with 20 times smaller amplitude (time domain) out of the noise (Total power signal is 800 times smaller than that of noise) –overlap filter-signal is 1.0 if signal is equal to filter+noise: amplitude is reproduced correctly. –frequency is reproduced correctly (filter gives only hits in the right frequency band) –average overlap between filters is about 0.43 (at same frequency)

27 H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 27 First tests spectrum : Gaussian-distributed noise with mean zero and amplitude –one-sided PSD of signals: 10 binary neutron stars: –frequency between 200 and 250 Hz –random angles, deformations, etc –maximum amplitude < 10 -23, total power of 10 signals is 0.2 percent of the power in the noise FFT lenght 1024 seconds, 2048 samples/sec. 30 FFT sets (about 5 hours)

28 H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 28 Overlap of filters, only noise maximum correlation for all filters applied between 0 and 1000 Hz (81.5 million FFT products, 4096 lags per filter)

29 H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 29 Overlap of filters with signal maximum correlation with signal for all filters applied between 0 and 1000 Hz (81.5 million FFT products)

30 H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 30 signal-to-noise

31 H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 31 Power spectral density PSD signal+noise

32 H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 32 PSD, signal only

33 H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 33 PSD, signal only

34 H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 34 Search results 30 FFTs, about 5h of data analyzed between 100 and 500 Hz –2405 different filters about 1.3 billion filter multiplications, 28731 hits (10 pulsars+noise) pulsars only: 14972 hits

35 H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 35 Search results, all hits

36 H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 36 Search results

37 H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 37 Alternative: cut on power Cut: 4 sigma on power FFT –number

38 H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 38 Alternative: cut on power Cut: 4 sigma on power 7649 hits between 450 and 460 Hz

39 H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 39 highest PSD in data FFT –number

40 H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 40 PSD: signal only FFT –number signal highest PSD still data spread out over about 30 bins

41 H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 41

42 H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 42 Summary we propose an all-sky search for gravitational waves from neutron stars in binary systems a complete set of filters (complete to third order in frequency) is used to parameterize the signal. the correlation of the filters with the data yield –time of overlap – with better resolution than FFT-time –amplitude and frequency of signal –first and second derivative of the frequency as function of time

43 H.J. Bulten - LSC-Virgo PSS 9 Jun 2008 43 Summary after first step, amplitude and frequency of the signal can be parameterized as a function of time. –candidates can be followed from 1 FFT to the next Filters can be produced in a small frequency band –compared to different frequency bands in the data –stepsize in frequency determined by frequency dependence of filter amount of CPU time is manageable

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