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

H.J. Bulten - LSC-Virgo PSS June 9, 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 June 9, 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 maybe other sources with extra variance in frequency (e.g. systems with very high spin-down)

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

H.J. Bulten - LSC-Virgo PSS June 9, 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 June 9, solitary neutron stars alternative: F-statistics approach (Ligo, e.g. 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 June 9, Binary : Kepler orbitals ellipse We want to analyze: –frequency shifts up to 0.3%, frequency changes df/dt up to s -2 –this includes : orbital periods from 2 hours – infinite masses companion star up to 15 solar masses eccentricities up to about mHz shift in 1 second, at f=1000Hz

H.J. Bulten - LSC-Virgo PSS June 9, frequency shifts

H.J. Bulten - LSC-Virgo PSS June 9, frequency derivative

H.J. Bulten - LSC-Virgo PSS June 9, frequency shifts

H.J. Bulten - LSC-Virgo PSS June 9, coherence phase signal: signal should remain in-phase,e.g. maximally about 60 deg. out of phase anywhere during observation time – frequency within ½ bin - 1/(2Tobs)

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

H.J. Bulten - LSC-Virgo PSS June 9, 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 June 9, 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 June 9, 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 June 9, 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 Fourier- transformed data

H.J. Bulten - LSC-Virgo PSS June 9, 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 FFT products) –about a day of CPU-time on a single CPU, current desktop

H.J. Bulten - LSC-Virgo PSS June 9, 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 June 9, 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 June 9, 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 length 1024 seconds, 2048 samples/sec. 30 FFT sets (about 5 hours)

H.J. Bulten - LSC-Virgo PSS June 9, 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 June 9, Overlap of filters with signal maximum correlation with signal for all filters applied between 0 and 1000 Hz (81.5 million FFT products) (most are <0.001)

H.J. Bulten - LSC-Virgo PSS June 9, signal-to-noise cut, 4 

H.J. Bulten - LSC-Virgo PSS June 9, Power spectral density PSD signal+noise time (interleaved FFT sets)

H.J. Bulten - LSC-Virgo PSS June 9, PSD, signal only time (interleaved FFT sets)

H.J. Bulten - LSC-Virgo PSS June 9, PSD: 30 FFTs added

H.J. Bulten - LSC-Virgo PSS June 9, PSD of 30 FFTs added

H.J. Bulten - LSC-Virgo PSS June 9, PSD, signal only time (interleaved FFT sets)

H.J. Bulten - LSC-Virgo PSS June 9, Search results 30 FFTs, about 5h of data analyzed between 100 and 500 Hz –2405 different filters –in total about 30 h CPU-time on my desktop PC (dual core ~ 3GHz) Applied threshold: 4 sigma about 1.3 billion filter multiplications, hits in simulated dataset (10 pulsars+noise) analysis on files with signal (10 pulsars) only: hits

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

H.J. Bulten - LSC-Virgo PSS June 9, correlations with signal-only

H.J. Bulten - LSC-Virgo PSS June 9, Search results, signal+noise

H.J. Bulten - LSC-Virgo PSS June 9, Search results, signal only

H.J. Bulten - LSC-Virgo PSS June 9, Comparison: cut on power Cut: 4 sigma on power FFT –number

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

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

H.J. Bulten - LSC-Virgo PSS June 9, PSD: signal only signal highest PSD data still spread out over about 30 bins here

H.J. Bulten - LSC-Virgo PSS June 9, Summary we propose to search for gravitational waves by applying filters that describe the phase of the signal ad hoc (large parameter reduction), up to cubic time dependence –method should be good for short FFT base –less parameters than F-statistics approach –longer FFT base possible than Hough approach hits 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 June 9, Summary first step: for high thresholds these filters are very effective to remove noise –for a time base of 1000 seconds, a 4-sigma threshold may be set. Full power is recovered, even though it may be spread out over 50 bins or more. after first step, amplitude and frequency of the signal can be parameterized as a function of time. –time/frequency dependence is known, filter is known. –candidates can be followed up from 1 FFT to the next : a hit predicts possible follow-ups

H.J. Bulten - LSC-Virgo PSS June 9, Concluding remarks Second step hierarchical approach needs to be developed yet –how to go to a longer time base parameterize frequency as a function of time? use amplitude? use hits in consecutive FFTs as confirmation/rejection? –how to go to the astrophysical source in case of GW detection although we would be glad with the detection anyways... Next actions –documenting –further developing framework, run on grid