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

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

H.J. Bulten - LSC-Virgo PSS 9 Jun 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 solitary neutron stars alternative: F-statistics approach (Ligo, Jaranowski et all PRD58, ) –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, 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 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 s -2 1 mHz shift in 1 second, at f=1000Hz

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

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

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

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

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

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

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

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:

H.J. Bulten - LSC-Virgo PSS 9 Jun 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 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 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 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 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 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 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 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 < , 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 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 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 signal-to-noise

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

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

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

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

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

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

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

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

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

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

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

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