Search for periodic sources C. Palomba

1. Search for periodic sources C. Palomba • Signal duration much larger than typical observation time Possibility to reduce the false alarm probability to negligible values • Signal often (but not always) predictable and • depending on the kind of source • depending on a (large) number of (poorly known) parameters

2. Different approach in the analysis depending if we know the source parameters (targeted search) or not (blind search) • Blind searches are computationally bound • Different kinds of periodic sources can be considered: • isolated NS • NS in binary systems • accreting NS (r-modes excitation) • We will focuse attention on the first type and discuss what changes in the other cases

3. Signal characterization - 1 • Doppler frequency modulation, due to the detector motion and to the source motion

4. Spin-down or spin-up • Intrinsic frequency modulation, due to a companion, an accretion disk or a wobble • Amplitude modulation, due to the detector radiation pattern and possibly to intrinsic effects (e.g. wobble) • Glitches

5. Signal at the detector detector beam pattern function wave polarizations phase evolution amplitudes

6. Data characterization • Stationarity • Gaussianity • Impulsive noise • Holes in the data • Correct data timing

7. Detection of periodic signals • If we had a monochromatic signal, the most natural strategy would be that of looking for significant peaks in the spectrum of data • Due to the frequency modulation and spin-down the signal power is spread in a large number of frequency bins • If the signal frequency evolution is known we can correct for the modulation (targeted search) • Otherwise we need to perform a ‘rough’ search, select some candidates and refine the analysis on them (blind search -> hierarchical methods)

8. Targeted vs blind searches • Targeted: • possibility to apply optimal methods • computationally not expensive • upper limits • Blind: • oriented to detection • computationally expensive • non optimal methods

9. Targeted search for isolated NS - 1 • Assume sky position, emission frequency and spin-down are known • Amplitude, source inclination, polarization angle, initial phase typically are unknown • Allow to use optimal DA methods • Nominal sensitivity • For f.a.p=0.01 and f.d.p.=.1 the sensitivity, averaging over source position and inclination and wave polarization, is

10. Targeted search for isolated NS - 2 • Time-domain methods • Re-sampling procedure: use a sampling frequency proportional to the varying received frequency

11. Targeted search for isolated NS - 3 • Heterodyning procedure (Abbott et al., PRL94 181103, 2005): multiply the signal by • Allow to take into account complex phase evolution, e.g. using data from radio telescopes, in a straightforward way • Frequency domain methods • F-statistics (Jaranowski et al., PRD58 063001, 1998; Abbott et al. PRD69 082004, 2004) : based on the maximization of the likelihood function

12. Targeted search for isolated NS - 4 • Can be used as coherent step in a hierarchical procedure • Analytical signal (Astone et al., PRD65 022001, 2001) : start from a set of short FFTs, compute the analytical signal, correct for the frequency variations, compute the new spectrum • Can be used as coherent step in a hierarchical procedure

13. Blind search - 1 • Assume source position, frequency and spin-down are not known (or only loosely constrained) • Cannot be performed with optimal methods due to the huge number of points in the source parameter space Unreachable computing power

14. Blind search - 2 • Hierarchical methods have been developed which strongly reduce the needed computing power at the price of a small sensitivity loss (Rome, Potsdam…) • Typically based on alternation between coherent steps and incoherent ones. • Two kinds of incoherent steps are popular: • stack-slide (Radon transform) • Hough transform • Both methods start from a collection of ‘short’ FFTs: their length is such that a signal would be confined within a frequency bin

15. Radon transform • Compute periodograms from short FFTs • Shift (slide) periodograms according to the frequency evolution f t 3. Sum (stack) the periodograms

16. Hough transform - 1 • Parameter estimator of patterns in digital images • Developed in the ’60 by P. Hough to analyze particle tracks in bubble chamber images • Example: find parameters of a straight line y=mx+q y q (xi, yi) q=yi-mxi x m

17. Frequency Time Hough transform - 2 • In our case the HT connects the time-frequency plane to the source parameter space • On the periodograms select peaks above a threshold

18. Hough transform - 3 • For each point in the peak-map we have a circle in the sky

19. Hough transform - 4

20. Hough transform - 5 • Slightly less sensitive than Radon (~12% in amplitude) Hough vs. Radon Sensitivity ratio Threshold for peak selection • More robust against non-stationarities and disturbances • Computing power reduced by ~10

21. Hierarchicalmethodoutline h-reconstructed data SFDB SFDB peak map peak map hough transf. hough transf. candidates coincidences candidates coherent step events

22. Short FFT database - 1 • Time domain disturbances • identification of events through an adaptive threshold (on the CR); • background estimated from the AR mean of the absolute value and square of the samples; • events removal. (Astone et al., CQG22, S1197)

23. Short FFT database - 2 • Construction of the short FFT database • Maximum length • 4 SFDB for frequency bands [0,31.25Hz], [31.25Hz,125Hz], [125Hz,500Hz], [500Hz,2kHz] • Estimation of the average spectrum • based on AR estimation; • used for peaks selection and in the Hough transform;

24. C6 spectrum and average spectrum

25. Peak map • Construction based on the ratio R between the spectrum and its AR estimation; • A threshold is set on R and all the local maxima above it are selected as peaks; • The threshold is chosen in order to maximize the CR on the Hough map (see next) • With thr=2.5 we have that ~1/12 of the frequency bins are selected

27. Hough transform • For each search frequency takes a Doppler band around it and compute the hough map for all the possible spin-down values • Adaptive hough transform for non-stationarities (Palomba et al., CQG22, S1255, 2005) • Computationally heaviest part of the hierarchical analysis • Efficient implementation needed • Use of computing grids

28. Hough statistics • In the case of pure noise the number count in a Hough map follows a binomial distribution N: total number of spectra h0(q): peak selection probability (depending on the threshold q) • In presence of a signal (e.g. Krishnan et al., PRD70, 082001)

29. The choice of the threshold is done maximizing the critical ratio CR: • The optimal choice would be • is still nearly optimal and reduce the prob. of peak selection to ~1/12 • We select candidates putting a threshold on the number count • The threshold is chosen on the basis of the maximum number of candidates we can manage (e.g. )

30. Sensitivity - 1 • Loss factor for nominal sensitivity • Nominal sensitivity • Number of points in the parameter space • The number of ‘basic’ hough operations (increasing by 1 the number count in a pixel) is

31. Sensitivity - 2 • The needed computing power is • where is the ‘number of equivalent floating point operations’ needed for pixel increase and the analysis time is assumed to be half of the observation time • Computing power of the order of 1Tflops needed for • Larger CP available, reduce the spin-down age

32. Sensitivity - 3 • To compute the ‘effective’ sensitivity loss of the hierarchical method we have to take into account candidate selection • For the optimal method, the loss due to the selection of 10^9 candidates is (exponential statistics) • The Hough number count distribution is binomial • Using the gaussian approximation, the threshold as a function of the number of selected candidates is

33. Sensitivity - 4 hough map mean value hough map std. dev. peak selection prob. in the peak map • The 10^9 sensitivity reduction factor is ~2.2-2.8 depending on the frequency band • The ‘effective’ loss respect to the optimal method is 2 – 4 depending on the freq. band

34. Sensitivity - 5

35. Pulsars in binary systems - 1 • Orbital parameters must be taken into account (up to 5) • Orbital Doppler shift may give more stringent limits to the maximum length of FFTs (from Dhurandhar & Vecchio, PRD63, 122001)

36. Pulsars in binary systems - 2 • Optimal methods can be applied only if the system parameters are known or the uncertainties are small • Otherwise, hierarchical non-optimal methods are needed

37. Accreting NS (e.g. LMXB) • Same parameters as in the previous case • The frequency will change randomly due to fluctuations in the rate of matter accretion • In LMXB a clustering of frequencies is observed, though the exact rotation frequency is not known • Possibility to apply coherent methods over short time period (few hours) (Vecchio, GWDAW10)

38. Summary of results • Coherent analysis • Explorer: 0.72Hz, 1 sd, all-sky, h90=1e-22 • LIGO S2: 28 isolated NS, h95>1.7e-24 • LIGO S2: Sco-X1, h95=2e-22 • Incoherent analysis: • LIGO S2: all-sky, 1 sd, 200-400Hz, h95=4.4e-23

39. SPARE SLIDES

41. Two thresholds enter into the game: peaks selection candidates selection • False dismissal probability