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Continuous Waves Searches ULs B. Allen, S.Anderson, S.Berukoff, P.Brady, D.Chin, R.Coldwell, T.Creighton, C.Cutler, R.Drever, R.Dupuis, S.Finn, D.Gustafson, J.Hough,M.Landry, G. Mendell, C.Messenger, S.Mohanty, S.Mukherjee, M.A. Papa, B.Owen, K.Riles, B.Schutz, X. Siemens, A.Sintes, A.Vecchio, H.Ward, A. Wiseman, G.Woan, M. Zucker www.lsc-group.phys.uwm.edu/pulgroup Science activities Plan of the talk • first priority item: UL on signals from known sources: • two search engines: - frequency domain - time domain (presented by G. Woan , Glasgow) • preliminary S1 investigations • other tasks that might meet the end-of-the year deadline: • UL on Sco-X1 (Birmingham) • Unbiased searches (Michigan) Frequency domain search: general overview • core LAL routines: LALDemod, LALBarycentering, LALComputeSky (AEI) • given template parameters and data in frequency domain (SFTs) it computes the F statistic • can be used for targeted searches, for area searches and as a module of a hierarchical search • efficient algorithm to run in a (cheap) distributed computing environment • there exists an LDAS DSO that runs on (LDAS-produced) SFTs, performs the search and write to the data-base table (G. Mendell) • what is F and how we compute it • how we go from F to an UL F statistic from gr-qc/9804014, P. Jaranowski, A. Krolak, B.F. Schutz Signals model and parameters isolated rotating pulsar emitting GWs at f0=2fr due to its configuration not being perfectly axysimmetric • (a,d) position in the sky • f0 emission frequency (at SSB) • {fs} spin-down parameters, • h0 amplitude of signal (as received on Earth) •i inclination angle •y polarization angle • f0 initial phase How does the signal that we receive depend on these parameters? h(t) F (t ) h (t ) F (t ) h (t ) F (t ) F (t ) beam-pattern functions and depend on the relative orientation of the detector w.r.t. the wave. They depend on y and on the amplitude modulation functions a(t) and b(t) that depend on the relative instantaneous position between source and detector. h ( t ) 12 h 0 (1 cos 2 i ) cos 2 (t) h ( t ) 12 h 0 cos i sin 2 (t) (t) Φ0 Φ (t ) the two independent wave polarizations. the phase of the received signal depends on the initial phase and on the frequency evolution of the signal. The latter depends on the spin-down parameters and on the Doppler modulation, thus on the frequency of the signal and on the relative instantaneous relative velocity between source and detector. What is the F statistic ? Given the values of the position in the sky, f0 and the spin-down parameters, the F statistic is a number: the maximum of the likelihood ratio. It is the likelihood ratio computed for the maximum likelihood estimators of (h0,i,f0,y), i.e. the values that maximize F, given the data set. We do not need to explicitly search over h0,i,f0,y. Given the data, F is the likelihood ratio for the quadruplet of values that best fits the data. This maximization is done analytically. As we will see, if a signal were present, the actual value of F would depend, of course, on the actual value of the signal’s h0,i,f0,y parameters. Why can we compute F efficiently ? • the core of the calculation consists in computing | xi e f i | i i with fi being the phase of the signal (template) and xi being the data. Let’s neglect for now the amplitude modulation. It’s irrelevant for the point I am making now. Input data in frequency domain TOTAL DATA SET: Sampling rate ~ 2kHz Our data: NM samples Tobs ~ 4 months Size: ~ 80 GB MN . . . . . . . . . . . . 1 t Band 0-1 kHz at 1e-7 Hz resolution. Can also be divided in M chunks, each one having N samples. Each of these chunks can be FFT-ed, producing a set of short time-baseline FFTs (SFTS): FFT the chunks ..... ..... ..... ..... ~ x 1k 1 N ~ x 2k 1 N ~ x 3k 1 N . . . . ~ xMk 1 N 1 2 EACH CHUNK: 3 . . . . M Duration of a chunk ~ 1 h In 4 months ~ 3000 chunks Each chunk ~28 MB Band 0-1 kHz at 3E-4 Hz resolution. The core of the calculation looks like this: M 1 a Sa ( f 0) N e ~ x a Da a S (f ) k k Hz k3 k ~ e x a Da k 8 Hz k 0 Hz a ya i 2 D k ...... k1 f0 1k k Hz ya i 1 M kM f0 D 2k f0 D3k f0 DMk Dak is the Dirichlet kernel centered at the instantaneous freq. at the mid time of every SFT (k*). Including the amplitude modulation F f 0 ai x e a f i ( f 0) i ( f 0) ai and bi are the amplitude modulation functions that only depend on the template sky-position. i i f f F 0 bi x e b i i i i To construct F one has to combine Fa and Fb like this: 2 2 A|F ( f 0)| B |F ( f 0)| 2C |F a ( f 0) F b ( f 0)| F ( f 0) D a b In every SFT the ai and bi can be considered constant. Thus we only have to multiply each SFT datum by the same number (different for each SFT) we do not need to modify our demod. scheme How is F distributed ? • in the case of noise only, F is expected to be a c2 random variable with 4 degrees of freedom. • if there’s a signal, F is expected to follow a non-central c2 distribution with non centrality parameter proportional to inner product of signal with itself. By studying how values of F(f0) computed for f0s different from those of the targeted signal are distributed, we can verify that our data follows the expected noise-only distribution. If it does not, we will take the measured distribution as the actual one. Presumably we will obtain a value of F from our search that lies well within the noise curve we want to set an UL on h0. Preliminary results on F F from 4 H2 S1 2048 s SFTs - IFO always locked - GPS 714887312 –714895504. distribution of F and fit to c2 with 4 deg. of freedom 0.128 Hz band around 1282.86 Hz. f0 fake pure gaussian noise passed through search code by G. Mendell and B.Cameron data ANALYSIS PIPELINE F* the result of our analysis a,d,f0,{fs} According to our analysis, with YYY confidence, this pulsar is not emitting GWs (of the type that we are looking for) with h0 greater than XXX. There are many ways to set upper limits Most conservative upper limit - a curve not a number ! F* the result of our analysis. If there’s a signal, the distribution depends of the signal’s i,y,f0 and h0 parameters. pw (F| h0=1) 20% pw (F| h0=2) Let us choose the triplet i,y,f0 that yields the smallest non-centrality parameter: iw,yw,f0w . Now the distribution only depends on h0: pw (F| h0). 80% F* It is a sample of a random variable drawn from a certain distribution. Which distribution ? For every values of h0 we can integrate the corresponding pw (F| h0) curve between F* and infinity, this yields P(h0), our final UL result: with probability (confidence) P we can say that if the data contained a signal with amplitude h0 (or greater), we would have measured a value of F higher than the one that we have measured (F*). We could set more “liberal” UL by eliminating the dependency from the i,y,f0 parameters differently: • we could construct the p(F|h0) by integrating p(F | h0, i,y,f0 ) over the values of i,y,f0 , with suitable weights, i.e.prior. This method, in general, will produce a stronger (lower) UL than the previous one presented. • It has also been suggested that we construct, just to provide “a feel” for the range of variability of our UL, also a “least conservative UL”. To compute this UL one follows the same recipe as for the most conservative UL, with the exception that the values of i,y,f0 are the ones for which the non-centrality parameter is maximum. • all of the methods proposed up to here are essentially Frequentist ULs. The time-domain analysis will use a Baysian approach. In the case of known pulsars we do not expect that the two approaches will provide significantly different results. Where do we stand now ? • constructing SFT data and looking at it • validating software pipeline to compute F • developing software for extensive Monte Carlo on real data to assess detection efficiency and estimate pdfs (might not be necessary, but certainly it is necessary to check that that’s the case) • want to meet end of the year UL deadline Power in 4 H2 S1 2048 s SFTs (IFO always locked; GPS 714887312 - 714895504) 2 Hz band centered on 1283.86 Hz The mean power goes increases by 13% increase during this interval by G. Mendell Power in 4 H2 S1 64 s SFTs (IFO always locked; GPS 714887296 - 714887552) 2 Hz band centered on 1283.86 Hz The mean power appears to increase by a factor of 12 during 2048 seconds Low frequency power “leaking out” -> high-pass filtering or windowing needed with such short time-baselines by G. Mendell Gaps in lock: GPS 714162320 - 714170512 Power in 4 H2 S1 2048 s SFTs padded gaps in lock Histograms by G. Mendell GEO - 10 SFTs - 60 s - 10 Hz band @ 1283 Hz |SFT|^2/<|SFT|^2> |SFT|^2/<|SFT|^2> with windowing before FFTing LIGO H1 - 1800 SFTs - 64 s 1 - Hz band @ 1283 Hz starting at 714151671 |SFT|^2/<|SFT|^2> <|SFT|^2> and s(|SFT|^2) |SFT|^2/<|SFT|^2> having removed the mean B. Allen’s SFTs More S1 investigations Unbiased CW Searches. Strategy (Michigan): • Measure power in selected bins of averaged periodograms • Bins defined by source parameters (f, RA, d) • Estimate noise level & statistics from neighboring bins • Set upper limit on quasi-sinusoidal signal on top of empirically determined noise • Scale upper limit by antenna pattern correction They have started to study the average spectral power distribution of the 2048s LDAS-produced SFTs Probably not optimum, but fine for exploration Upper limits will be based on excess power summed incoherently over subsets of bins in many SFT’s Study Range: 659-661 Hz (early H1) Early H1 – Average over 97 SFT’s with >90% livetime Bottom histogram now excludes region between bars Same but with 5 mHz binning (“typical” Δf searchband) D. Chin, K. Riles Study Range: 659-661 Hz (late H1) Late H1 – Average over 146 SFT’s with >90% livetime Same but with 5 mHz binning (“typical” Δf searchband) Bottom histogram now excludes region between bars Cleaner than early H1 !! D. Chin, K. Riles Preliminary conclusions from unbiased search team that in spirit apply to all S1 investigation: • At least some of the data looks “okay” – Gaussian, locally white noise not a terrible approximation • But non-Gaussianity and non-stationarity quite apparent and must be confronted • But devil is in the details: – Optimization of SFT’s (ΔT, windowing) – Determining detection efficiency: – Correction for calibration drifts Expect analysis to evolve in iterative refinements but on a time scale that will allow us to meet our end-of-the-year deadlines, as promised.