All-sky search for gravitational waves from neutron stars in binary systems

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

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

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

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

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

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

14. binary neutron stars • different set of filters: parameterizethe phase as a function of time! • assume that within Tobs, 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 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 H.J. Bulten - LSC-Virgo PSS 9 Jun 2008

16. FFT 1 FFT 2 filter, lag=0 filter, scan to lag = T/2 Filter search data, split in overlapping periods 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 H.J. Bulten - LSC-Virgo PSS 9 Jun 2008

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

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

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

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

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

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

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

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

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

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

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

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

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