2. Alicia M. Sintes Universitat de les Illes Balears Pisa,1 June 2007 Data analysis searches for continuous gravitational waves

3. VESF-school, Pisa, June 2007, A.M. Sintes 2 Content Gravitational waves from spinning neutron stars. –Emission mechanisms –Signal model Data analysis for continuous gravitational waves –Bayesian analysis –Frequentist approach Brief overview of the searches –Directed pulsar search –Wide-parameter search (all sky) Coherent methods Einstein@Home Hierarchical strategies Semi-coherent methods Summary of results and perspectives

4. VESF-school, Pisa, June 2007, A.M. Sintes 3 Rotating neutron stars (NASA/CXC/SAO) NASA Neutron stars can form from the remnant of stellar collapse Typical size of 10km, and are about 1.4 solar masses Some of these stars are observed as pulsars Gravitational waves from neutron stars could tell us about the equation of state of dense nuclear matter Our galaxy might contain ~10 9 NS, of which ~10 5 are expected to be active pulsars. Up to know ~1700 pulsars have been identified search for observed neutron stars all sky search (computing challenge)

5. VESF-school, Pisa, June 2007, A.M. Sintes 4 Neutron Stars Sources Great interest in detecting radiation: physics of such stars is poorly understood. –After 40 years we still don’t know what makes pulsars pulse. –Interior properties not understood: equation of state, superfluidity, superconductivity, solid core, source of magnetic field. –May not even be neutron stars: could be made of strange matter!

6. VESF-school, Pisa, June 2007, A.M. Sintes 5 “Continuous” gravitational waves from neutron stars Various physical mechanisms could operate in a NS to produce interesting levels of GW emission As the signal-strength is generally expected to be weak, long integrations times (several days to years) are required in order for the signal to be detectable in the noise. Therefore this GW emission has to last for a long time, which characterizes the class of ‘continuous-wave’ signals. NS might also be interesting sources of burst-like GW emission, f-mode or p-mode oscillations excited by a glitch crustal torsion modes

7. VESF-school, Pisa, June 2007, A.M. Sintes 6 Bumpy Neutron Star Emission mechanisms for continuous GW from spinning NS in the LIGO-VIRGO frequency band –Non-axisymmetric distortions –Unstable oscillation modes in the fluid part of the star –Free precession Wobbling Neutron Star Low Mass X-Ray Binaries Magnetic mountains R-modes in accreting stars

8. VESF-school, Pisa, June 2007, A.M. Sintes 7 Bumpy Neutron Star 1) Non-axisymmetric distortions A non-axisymmetric neutron star at distance a d, rotating with frequency ν around the I zz axis emits monochromatic GWs of frequency f = 2ν with an amplitude The strain amplitude h 0 refers to a GW from an optimally oriented source with respect to the detector The equatorial ellipticity is highly uncertain,  ~10 -7. In the most speculative model can reach up to 10 -4. Accreating neutron stars in binary systems can also have large crust deformations Strong internal magnetic fields could produce deformations of up to  ~10 -6. These deformations would result in GW emission at f = ν and f = 2ν. Magnetic mountains

9. VESF-school, Pisa, June 2007, A.M. Sintes 8 Wobbling Neutron Star 2) Non-axisymmetric instabilities At birth or during accretion, rapidly rotating NS can be subject to various non-axisymmetric instabilities, which would lead to GW emission, The r-mode instability has been proposed as a source of GWs (with frequency f = 4ν/3) from newborn NS and from rapidly accreting NS. R-modes in accreting stars 3) Free precession results in emission at (approximately) the rotation rate ν and twice the rotation rate, i.e. f =ν+ν prec and f = 2ν.

10. VESF-school, Pisa, June 2007, A.M. Sintes 9 Loudest expected signal From isolated neutron stars (Blandford’s argument): Although there is great uncertainty in the physics of the GW emission mechanisms and the strength of individual sources, one can argue for a statistical upper limit on the expected strongest GW signals from the galactic population of neutron stars, which is almost independent of individual source physics. h 0 ~4  10 -24 Is the optimistic upper limit, that there is a 50% chance that the strongest signal has at least that amplitude in the LIGO-VIRGO band, assuming NS are uniformly distributed in the galactic disc and a constant overall galactic bith-rate. Spindown limit for known pulsars If the GW emission is powered by the rotational energy of the NS, then one obtains:  ≤  sd, h 0 ≤ h sd

11. VESF-school, Pisa, June 2007, A.M. Sintes 10 The ‘periodic’ GW signal from a neutron star: Nearly-monochromatic continuous signal –spin precession at ~f rot –excited oscillatory modes such as the r-mode at 4/3* f rot –non-axisymmetric distortion of crystalline structure, at 2f rot The signal from a NS (Signal-to-noise) 2 ~

12. VESF-school, Pisa, June 2007, A.M. Sintes 11 The expected signal at the detector A gravitational wave signal we detect from a NS will be: –Frequency modulated by relative motion of detector and source –Amplitude modulated by the motion of the non- uniform antenna sensitivity pattern of the detector R

13. VESF-school, Pisa, June 2007, A.M. Sintes 12 Signal received from an isolated NS F + and F  are the strain antenna patterns. They depend on the orientation of the detector and source and on the polarization of the waves. the phase of the received signal depends on the initial phase, the frequency evolution of the signal and on the instantaneous relative velocity between source and detector. T(t) is the time of arrival of a signal at the solar system barycenter, t the time at the detector.

14. VESF-school, Pisa, June 2007, A.M. Sintes 13 Signal model: isolated non-precessing NS h 0 - amplitude of the gravitational wave signal  - angle between the pulsar spin axis and line of sight - equatorial ellipticity In the case of an isolated tri-axial neutron star emitting at twice its rotational frequency

15. VESF-school, Pisa, June 2007, A.M. Sintes 14 Signals in noise. The meaning of probability The strain x(t) measured by a detector is mainly dominated by noise n(t), such that even in the presence of a ignal h(t) we have x(t)=n(t)+h(t) Data analysis methods are not just simple recipes. We want tools capable of —dealing with very faint sources —handling very large data sets —diagnosing systematic errors —avoiding unnecessary assumptions —estimating parameters and testing models There are different ways of proceeding depending on the paradigm of statistics used. There are three popular interpretations of the word: Probability as a measure of our degree of belief in a statement Probability as a measure of limiting relative frequency of outcome of a set of identical experiments Probability as the fraction of favourable (equally likely) possibilities We will call these the Bayesian, Frequentist and Combinatorial interpretations.

16. VESF-school, Pisa, June 2007, A.M. Sintes 15 Algebra of (Bayesian) probability If there are two statements X and Y, then joint probability where the vertical line denotes the conditional statement “X given Y is true” – The Product Rule …because p(X,Y)=p(Y,X), we get which is called Bayes’ Theorem Thomas Bayes (1702 – 1761 AD) Likelihood Prior Evidence Posterior We can usually calculate all these terms Bayes’ theorem is the appropriate rule for updating our degree of belief when we have new data

17. VESF-school, Pisa, June 2007, A.M. Sintes 16 Algebra of (Bayesian) probability We can also deduce the marginal probabilities. If X and Y are propositions that can take on values drawn from and then this gives use the probability of X when we don’t care about Y. In these circumstances, Y is known as a nuisance parameter. All these relationships can be smoothly extended from discrete probabilities to probability densities, e.g. where “p(y)dy” is the probability that y lies in the range y to y+dy. =1

18. VESF-school, Pisa, June 2007, A.M. Sintes 17 Likelihood Prior Posterior What we know now Influence of our observations What we knew before In the Bayesian approach, we can test our model Bayesian parameter estimation As our data improve (e.g. our sample increases), the posterior pdf narrows and becomes less sensitive to our choice of prior. The posterior conveys our (evolving) degree of belief in different values of , in the light of our data If we want to express our result as a single number we could perhaps adopt the mean, median, or mode We can use the variance of the posterior pdf to assign an uncertainty for  It is very straightforward to define confidence intervals

19. VESF-school, Pisa, June 2007, A.M. Sintes 18 p(  | data, I )  95% of area under pdf 11 22 We are 95% sure that lies between and Note: the confidence interval is not unique, but we can define the shortest C.I. 11 22  Bayesian confidence intervals Bayesian parameter estimation

20. VESF-school, Pisa, June 2007, A.M. Sintes 19 Recall that in frequentist (orthodox) statistics, probability is limiting relative frequency of outcome, so : only random variables can have frequentist probabilities as only these show variation with repeated measurement. So we can’t talk about the probability of a model parameter, or of a hypothesis. E.g., a measurement of a mass is a random variable, but the mass itself is not. So no orthodox probabilistic statement can be interpreted as directly referring to the parameter in question! For example, orthodox confidence intervals do not indicate the range in which we are confident the parameter value lies. That’s what Bayesian intervals do. Frequentist framework

21. VESF-school, Pisa, June 2007, A.M. Sintes 20 The method goes like this: –To test a hypothesis H 1 consider another hypothesis, called the null hypothesis, H 0, the truth of which would deny H 1. Then argue against H 0 … –Use the data you have gathered to compute a test statistic  obs which has a calculable pdf if H 0 is true. This can be calculated analytically or by Monte Carlo methods. –Look where your observed value of the statistic lies in the pdf, and reject H 0 based on how far in the wings of the distribution you have fallen. Frequentist hypothesis testing – significance tests p(  |H 0 ) Reject H 0 if your result lies in here 

22. VESF-school, Pisa, June 2007, A.M. Sintes 21 H 0 is rejected at the x% level if x% of the probability lies to the right of the observed value of the statistic (or is ‘worse’ in some other sense): and makes no reference to how improbable the value is under any alternative hypothesis (not even H 1 !). We will chose a critical region of  *, so that if  obs >  * we reject H 0 and therefore accept H 1. Frequentist hypothesis testing – significance tests p(  |H 0 )   obs X% of the area p(  |H 0 )   p(  |H 1 )

23. VESF-school, Pisa, June 2007, A.M. Sintes 22 –A Type I error occurs when we reject the null hypothesis when it is true (false alarm) –A Type II error occurs when we accept the null hypothesis when it is false (false dismissal) both of which we should strive to minimise. According to the Neyman-Pearson lemma, this optimal test is the so-called likelihood ratio: For Gaussian noise, one finds which is the well-known expression for the matched-filtering amplitude. If some of the parameters of the signal h(t;A, ) are unknown, one has to find the maximum of lnΛ as a function of the unknown parameters Frequentist hypothesis testing  Type II errorType I error

24. VESF-school, Pisa, June 2007, A.M. Sintes 23 Calibrated output: LIGO noise history Integration times S1 - L1 5.7 days, H1 8.7 days, H2 8.9 days S2 - L1 14.3 days, H1 37.9 days, H2 28.8 days S3 - L1 13.4 days, H1 45.5 days, H2 42.1 days S4 - L1 17.1 days, H1 19.4 days, H2 22.5 days S5 (so far...) >1 year

25. VESF-school, Pisa, June 2007, A.M. Sintes 24 Calibrated output: GEO noise history

26. VESF-school, Pisa, June 2007, A.M. Sintes 25 Continuous wave searches Signal parameters: position (may be known), inclination angle, [orbital parameters in case of a NS in a binary system], polarization, amplitude, frequency (may be known), frequency derivative(s) (may be known), initial phase. Most sensitive method: coherently correlate the data with the expected signal (template) and inverse weights with the noise. If the signal were monochromatic this would be equivalent to a FT. –Templates: we assume various sets of unknown parameters and correlate the data against these different wave-forms. –Good news: we do not have to search explicitly over polarization, inclination, initial phase and amplitude. Because of the antenna pattern, we are sensitive to all the sky. Our data stream has signals from all over the sky all at once. However: low signal-to-noise is expected. Hence confusion from many sources overlapping on each other is not a concern. Input data to our analyses: –A calibrated data stream which with a better than 10% accuracy, is a measure of the GW excitation of the detector. Sampling rate 16kHz (LIGO-GEO, 20kHz VIRGO), but since the high sensitivity range is 40-1500 Hz we can downsample at 3 kHz.

27. VESF-school, Pisa, June 2007, A.M. Sintes 26 Four neutron star populations and searches Known pulsars Position & frequency evolution known (including derivatives, timing noise, glitches, orbit) Unknown neutron stars Nothing known, search over position, frequency & its derivatives Accreting neutron stars in low-mass x-ray binaries Position known, sometimes orbit & frequency Known, isolated, non-pulsing neutron stars Position known, search over frequency & derivatives What searches? –Targeted searches for signals from known pulsars –Blind searches of previously unknown objects –Coherent methods (require accurate prediction of the phase evolution of the signal) –Semi-coherent methods (require prediction of the frequency evolution of the signal) What drives the choice? The computational expense of the search

28. VESF-school, Pisa, June 2007, A.M. Sintes 27 Coherent detection methods Frequency domain Conceived as a module in a hierarchical search Matched filtering techniques. Aimed at computing a detection statistic. These methods have been implemented in the frequency domain (although this is not necessary) and are very computationally efficient. Best suited for large parameter space searches (when signal characteristics are uncertain) Frequentist approach used to cast upper limits. Time domain process signal to remove frequency variations due to Earth’s motion around Sun and spindown Standard Bayesian analysis, as fast numerically but provides natural parameter estimation Best suited to target known objects, even if phase evolution is complicated Efficiently handless missing data Upper limits interpretation: Bayesian approach There are essentially two types of coherent searches that are performed

29. VESF-school, Pisa, June 2007, A.M. Sintes 28 Summary of directed pulsar searches Within the LIGO sensitive band ( gw > 50 Hz) there are currently 163 known pulsars We have rotational and positional parameter information for 97 of these S1 (LIGO and GEO: separate analyses) –Upper limit set for GWs from J1939+2134 (h 0 <1.4 x 10 -22 ) –Phys. Rev. D 69, 082004 (2004) S2 science run (LIGO: 3 interferometers coherently, TDS) –End-to-end validation with 2 hardware injections –Upper limits set for GWs from 28 known isolated pulsars –Phys. Rev. Lett. 94, 181103 (2005) S3 & S4 science runs (LIGO and GEO: up to 4 interferometers coherently, TDS) –Additional hardware injections in both GEO and LIGO –Add known binary pulsars to targeted search –Full results with total of 93 (33 isolated, 60 binary) pulsars S5 science run (ongoing, TDS) –32 known isolated, 41 in binaries, 29 in globular clusters

30. VESF-school, Pisa, June 2007, A.M. Sintes 29 Frequency domain method The outcome of a target search is a number F* that represents the optimal detection statistic for this search. 2F* is a random variable: For Gaussian stationary noise, follows a  2 distribution with 4 degrees of freedom with a non-centrality parameter (h|h). Fixing ,  and  0, for every h 0, we can obtain a pdf curve: p(2F|h 0 ) The frequentist approach says the data will contain a signal with amplitude  h 0, with confidence C, if in repeated experiments, some fraction of trials C would yield a value of the detection statistics  F* Use signal injection Monte Carlos to measure Probability Distribution Function (PDF) of F

31. VESF-school, Pisa, June 2007, A.M. Sintes 30 Measured PDFs for the F statistic with fake injected worst-case signals at nearby frequencies 2F* = 1.5: Chance probability 83% 2F* 2F* = 3.6: Chance probability 46% 2F* = 6.0: chance probability 20%2F* = 3.4: chance probability 49% h 0 = 1.9E-21 95% h 0 = 2.7E-22 h 0 = 5.4E-22h 0 = 4.0E-22 S1 Note: hundreds of thousands of injections were needed to get such nice clean statistics!

32. VESF-school, Pisa, June 2007, A.M. Sintes 31 F-statistics We can express h(t) in terms of amplitude A {A +, A , ,  0 } and Doppler parameters

33. VESF-school, Pisa, June 2007, A.M. Sintes 32 F-Statistics Analytically maximize the likelihood over A

34. VESF-school, Pisa, June 2007, A.M. Sintes 33 Time domain target search Time-domain data are successively heterodyned to reduce the sample rate and take account of pulsar slowdown and Doppler shift, –Coarse stage (fixed frequency) 16384  4 samples/sec –Fine stage (Doppler & spin-down correction)  1 samples/min  B k –Low-pass filter these data in each step. The data is down-sampled via averaging, yielding one value B k of the complex time series, every 60 seconds Noise level is estimated from the variance of the data over each minute to account for non-stationarity.   k Standard Bayesian parameter fitting problem, using time-domain model for signal -- a function of the unknown source parameters h 0, ,  and  0

35. VESF-school, Pisa, June 2007, A.M. Sintes 34 We take a Bayesian approach, and determine the joint posterior distribution of the probability of our unknown parameters, using uniform priors on h 0,cos ,  and  0 over their accessible values, i.e. The likelihood  exp(-  2 /2), where To get the posterior PDF for h 0, marginalizing with respect to the nuisance parameters cos ,  and  0 given the data B k Time domain: Bayesian approach posterior prior likelihood

36. VESF-school, Pisa, June 2007, A.M. Sintes 35 Posterior PDFs for CW time domain analyses Simulated injection at 2.2 x10 -21 p shaded area = 95% of total area p

37. VESF-school, Pisa, June 2007, A.M. Sintes 36 Progression of targeted pulsars upper limits Crab pulsar Amplitudes of < 10 -25 and ellipticities <10 -6 for many objects Our most stringent ellipticities (10 -7 ) are starting to reach into the range of neutron star structures for some neutron- proton-electron models

38. VESF-school, Pisa, June 2007, A.M. Sintes 37 APS Meeting, Jacksonville, April 2007 Preliminary S5 results Used parameters provided by Pulsar Group, Jodrell Bank Joint 95% upper limits from first ~13 months of S5 using H1, H2 and L1 (97 pulsars) preliminary Lowest h 0 upper limit: PSR J1435-6100 ( n gw = 214.0 Hz, r = 3.3 kpc) h 0_min = 4.2x10 -26 Lowest ellipticity upper limit: PSR J2124-3358 ( n gw = 405.6Hz, r = 0.25 kpc)  = 9.6x10 -8 Due to pulsar glitches the Crab pulsar result uses data up to the glitch on 23 Aug 2006, and the PSRJ0537-6910 result uses only three months of data between two glitches on 5th May and 4th Aug 2006

39. VESF-school, Pisa, June 2007, A.M. Sintes 38 APS Meeting, Jacksonville, April 2007 preliminary Black curve represents one full year of data for all three interferometers running at design sensitivity Blue stars represent pulsars for which we are reasonably confident of having phase coherence with the signal model Green stars represent pulsars for which there is uncertainty about phase coherence

40. VESF-school, Pisa, June 2007, A.M. Sintes 39 Blind searches and coherent detection methods Coherent methods are the most sensitive methods (amplitude SNR increases with sqrt of observation time) but they are the most computationally expensive, why? –Our templates are constructed based on different values of the signal parameters (e.g. position, frequency and spindown) –The parameter resolution increases with longer observations –Sensitivity also increases with longer observations –As one increases the sensitivity of the search, one also increases the number of templates one needs to use.

41. VESF-school, Pisa, June 2007, A.M. Sintes 40 Number of templates [Brady et al., Phys.Rev.D57 (1998)2101] The number of templates grows dramatically with the coherent integration time baseline and the computational requirements become prohibitive

42. VESF-school, Pisa, June 2007, A.M. Sintes 41 S2 run: Coherent search for unknown isolated sources and Sco-X1 Entire sky search Fully coherent matched filtering 160 to 728.8 Hz df/dt < 4 x 10 -10 Hz/s 10 hours of S2 data; computationally intensive 95% confidence upper limit on the GW strain amplitude range from 6.6x10 -23 to 1.0x10 -21 across the frequency band Scorpius X-1 Fully coherent matched filtering 464 to 484 Hz, 604 to 624 Hz df/dt < 1 x 10 -9 Hz/s 6 hours of S2 data 95% confidence upper limit on the GW strain amplitude range from 1.7x10 -22 to 1.3x10 -21 across the two 20 Hz wide frequency bands See gr-qc/0605028

43. VESF-school, Pisa, June 2007, A.M. Sintes 42 Computational Engine Searchs offline at: Medusa, Nemo clusters (UWM) Merlin, Morgane cluster (AEI) Tsunami (B’ham) Others

44. VESF-school, Pisa, June 2007, A.M. Sintes 43 Einstein@home http://einstein.phys.uwm.edu/ Like SETI@home, but for LIGO/GEO data American Physical Society (APS) publicized as part of World Year of Physics (WYP) 2005 activities Use infrastructure/help from SETI@home developers for the distributed computing parts (BOINC) Goal: pulsar searches using ~1 million clients. Support for Windows, Mac OSX, Linux clients From our own clusters we can get ~ thousands of CPUs. From Einstein@home hope to get order(s) of magnitude more at low cost Currently : ~140,000 active users corresponding to about 80Tflops

45. VESF-school, Pisa, June 2007, A.M. Sintes 44 Public distributed computing project to look for isolated pulsars in LIGO/GEO data ~ 80 TFlops 24/7 Initially making use of coherent F-statistic method S3 - no spindown Using segment lengths of 10 hours No evidence of strong pulsar signals Outliers are consistent with instrumental artifacts or bad bands. None of the low significance remaining candidates showed up in follow-up on S4 data. S4 - one spindown parameter, up to f/fdot ~ 10,000 yr Using segment lengths of 30 hours Analysis took ~ 6 months Post-processing complete. Currently under review. S5 (R1)- similar to S4 Faster more efficient application Currently in post-processing stage Coherent searches on Einstein@home

46. VESF-school, Pisa, June 2007, A.M. Sintes 45 http://www.boincsynergy.com/stats/ User/Credit History

47. VESF-school, Pisa, June 2007, A.M. Sintes 46 Current performance Einstein@Home is currently getting 84 Tflops http://www.boincstats.com/

48. VESF-school, Pisa, June 2007, A.M. Sintes 47 All-Sky surveys for unknown gravity-wave emitting pulsars It is necessary to search for every signal template distinguishable in parameter space. Number of parameter points required for a coherent T=10 7 s search [Brady et al., Phys.Rev.D57 (1998)2101]: Number of templates grows dramatically with the integration time. To search this many parameter space coherently, with the optimum sensitivity that can be achieved by matched filtering, is computationally prohibitive. Class f (Hz)  (Yrs) NsNs DirectedAll-sky Slow-old<200 >10 3 13.7x10 6 1.1x10 10 Fast-old<10 3 >10 3 11.2x10 8 1.3x10 16 Slow-young <200 >4038.5x10 12 1.7x10 18 Fast-young<10 3 >4031.4x10 15 8x10 21

49. VESF-school, Pisa, June 2007, A.M. Sintes 48 Coherent wide-parameter searches The second effect of the large number of templates N p is to reduce the sensitivity compared to a targeted search with the same observation time and false-alarm probability: increasing the number of templates increases the number of expected false-alarm candidates at fixed detection threshold. Therefore the detection-threshold needs to be raised to maintain the same false-alarm rate, thereby decreasing the sensitivity. Note that increasing the number of equal-sensitivity detectors N improves the SNR in the same way as increasing the integration time T obs. However, increasing the number of detectors N does — contrary to the observation time T obs — not increase the required number of templates N p, which makes this the computationally cheapest way to improve the SNR of coherent wide-parameter searches.

50. VESF-school, Pisa, June 2007, A.M. Sintes 49 Hierarchical strategies Set upper-limit Pre-processing Divide the data set in N chunks raw data GEO/LIGO Construct set of short FT (t SFT ) Coherent search ( ,f i ) in a frequency band Template placing Incoherent search Hough transform (  f 0, f i ) Peak selection in t-f plane Candidates selection Candidates selection

51. VESF-school, Pisa, June 2007, A.M. Sintes 50 Hierarchical method for ‘blind’ searches h-reconstructed data Data quality SFDB Average spectrum estimation peak map Hough transf. candidates peak map Hough transf. candidatescoincidences coherent step events Data quality SFDB Average spectrum estimation The procedure involves two or more data sets belonging to a single or more detectors

52. VESF-school, Pisa, June 2007, A.M. Sintes 51 Incoherent power-sum methods The idea is to perform a search over the total observation time using an incoherent (sub-optimal) method: Three methods have been developed to search for cumulative excess power from a hypothetical periodic gravitational wave signal by examining successive spectral estimates: –Stack-slide (Radon transform) –Hough transform –Power-flux method They are all based on breaking up the data into segments, FFT each, producing Short (30 min) Fourier Transforms (SFTs) from h(t), as a coherent step (although other coherent integrations can be used if one increasing the length of the segments), and then track the frequency drifts due to Doppler modulations and df/dt as the incoherent step. Frequency Time

53. VESF-school, Pisa, June 2007, A.M. Sintes 52 Differences among the incoherent methods What is exactly summed? StackSlide – Normalized power (power divided by estimated noise)  Averaging gives expectation of 1.0 in absence of signal Hough – Weighted binary counts (0/1 = normalized power below/above SNR), with weighting based on antenna pattern and detector noise PowerFlux – Average strain power with weighting based on antenna pattern and detector noise  Signal estimator is direct excess strain noise (circular polarization and 4 linear polarization projections)

54. VESF-school, Pisa, June 2007, A.M. Sintes 53 Peak selection in the t-f plane Input data: Short Fourier Transforms (SFT) of time series For every SFT, select frequency bins i such exceeds some threshold  th time-frequency plane of zeros and ones p(  |h, S n ) follows a  2 distribution with 2 degrees of freedom: The false alarm and detection probabilities for a threshold  th  are

55. VESF-school, Pisa, June 2007, A.M. Sintes 54 Standard Hough statistics After performing the HT using N SFTs, the probability that the pixel { , , f 0, f i } has a number count n is given by a binomial distribution: The Hough false alarm and false dismissal probabilities for a threshold n th  Candidates selection For a given  H, the solution for n th is Optimal threshold for peak selection :  th ≈1.6 and  ≈0.20

56. VESF-school, Pisa, June 2007, A.M. Sintes 55 Noise only case

57. VESF-school, Pisa, June 2007, A.M. Sintes 56 Frequentist upper limit Perform the Hough transform for a set of points in parameter space ={ , ,f 0,f i }  S, given the data: HT: S  N   n( Determine the maximum number count n* n* = max (n(  S Determine the probability distribution p(n|h 0 ) for a range of h 0 The 95% frequentist upper limit h 0 95% is the value such that for repeated trials with a signal h 0  h 0 95%, we would obtain n  n* more than 95% of the time Compute p(n|h 0 ) via Monte Carlo signal injection, using  S, and  0  [0,2  ],   [-  /4,  /4], cos  [-1,1].

58. VESF-school, Pisa, June 2007, A.M. Sintes 57 0.1%30.5% 87.0%1 Number count distribution for signal injections p(n|h 0 ) ideally binomial for a target search, but: –Amplitude modulation of the signal for different SFTs –Different sensitivity for different sky locations –Random mismatch between signal & templates ‘smear’ out the binomial distributions L1: 200-201 Hz, n* =202 1000 injections

59. VESF-school, Pisa, June 2007, A.M. Sintes 58 Hough S2: UL Summary Feb.14-Apr.14,2003 DetectorL1H1H2 Frequency (Hz)200-201259-260258-259 h 0 95% 4.43x10 -23 4.88x10 -23 8.32x10 -23 S2 analysis covered 200-400Hz, over the whole sky, and 11 values of the first spindown (Δf = 5.55×10 – 4 Hz, Δf 1 = – 1.1×10 – 10 Hz s – 1 ) Templates: Number of sky point templates scales like (frequency) 2 –1.5×10 5 sky locations @ 300 Hz –1.9×10 9 @ 200-201 Hz –7.5×10 9 @ 399-400 Hz Three IFOs analyzed separately No signal detected Upper limits obtained for each 1 Hz band by signal injections: Population-based frequentist limits on h 0 averaging over sky location and pulsar orientation

60. VESF-school, Pisa, June 2007, A.M. Sintes 59 Improvements for S4 We use a weighted Hough to give more weight to SFTs having greater SNR. Weights are proportional to the beam pattern functions and inversely proportional to the SFT noise floor. Number count n is not an integer anymore Using the weights does not lead to any loss in computational efficiency or robustness. It has also been generalized to the Multi-IFO case Mean number count is unchanged due to normalization of weights. Standard deviation always increases: Number count threshold for a given false alarm:

61. VESF-school, Pisa, June 2007, A.M. Sintes 60 Contribution of different IFOs Figure plots the relative noise weights from H1, H2 and L1

62. VESF-school, Pisa, June 2007, A.M. Sintes 61 Loudest events for every 0.25 Hz Multi interferometer case 50-1000 Hz Significance defined as s=(n max - )/  Preliminary

63. VESF-school, Pisa, June 2007, A.M. Sintes 62 The S4 Hough search As before, input data is a set of N 1800s SFTs (no demodulations) Weights allow us to use SFTs from all three IFOs together: 1004 SFTS from H1, 1063 from H2 and 899 from L1 Search frequency band 50-1000Hz 1 spin-down parameter. Spindown range [-2.2,0]×10 -9 Hz/s with a resolution of 2.2×10 -10 Hz/s All sky search All-sky upper limits set in 0.25 Hz bands Multi-IFO and single IFOs have been analyzed Best UL for L1: 5.9×10 -24 for H1: 5.0×10 -24 for Multi H1-H2-L1: 4.3×10 -24 Preliminary

64. VESF-school, Pisa, June 2007, A.M. Sintes 63 Improvements due to the weights Comparison of the All-sky 95% upper limits obtained by Monte- Carlo injections for the multi- IFO case. The average improvement by using weights in this band is 9.25% for the multi-IFO case, but only ~6% for the single IFO

65. VESF-school, Pisa, June 2007, A.M. Sintes 64 StackSlide/Hough/PowerFlux differences StackSlideHoughPowerFlux WindowingTukey Hann Noise estimationMedian-based floor tracking Time/frequency decomposition Line handlingCleaning Skyband exclusion Antenna pattern weighting NoYes Noise weightingNoYes Spindown step size2 x 10 -10 Hz/s Freq dependent Limit at every skypoint No Yes Upper limit typePopulation-based Strict frequentist

66. VESF-school, Pisa, June 2007, A.M. Sintes 65 H1 (Hanford 4-km) and Multi-IFO Results PRELIMINARY

67. VESF-school, Pisa, June 2007, A.M. Sintes 66 L1 (Livingston 4-km) and Multi-IFO Results PRELIMINARY

68. VESF-school, Pisa, June 2007, A.M. Sintes 67 S5 incoherent searches preliminary PowerFlux results Preliminary

69. VESF-school, Pisa, June 2007, A.M. Sintes 68 E@H-Hierarchical search Basic algorithm Started test-run (~3 months) –currently working on improving stability & reliability Will be followed by a full 1-year run Should permit a search that extends hundreds of pc into the Galaxy This should become the most sensitive blind CW search possible with current knowledge and technology

70. VESF-school, Pisa, June 2007, A.M. Sintes 69 Sensitivity estimate of different E@H Runs

71. VESF-school, Pisa, June 2007, A.M. Sintes 70 Searches for Continuous Waves, present, past and future

72. VESF-school, Pisa, June 2007, A.M. Sintes 71 Conclusions Analysis of LIGO data is in full swing, and results from LIGO searches from science runs 4, 5 are now appearing. –Significant improvements in interferometer sensitivity since S3. –In the process of accumulating 1 year of data (S5). –Known pulsar searches are beginning to place interesting upper limits in S5 –All sky searches are under way and exploring large area of parameter space Veto strategies and data conditioning are becoming more relevant in order to reduce the selection threshold and improve sensitivity