1 An example or real data analysis: the VIRGO-bars search for bursts Andrea Viceré for VIRGO – Auriga - Rog

Andrea Viceré - IHP Paris, November 13th Motivations and outline Background: During Virgo Commissioning Run 7 (Sept. 2005), also INFN resonant bars, AURIGA, EXPLORER and NAUTILUS were taking data. A limited amount of data (24 hours) was exchanged, to permit the development of network analysis methods. Goals: understand the potentialities of a network of different detectors. Develop new techniques coping with their heterogeneous nature. Bring together the two DA communities and their experiences. Pave the way towards greater integration in the future. Methodology: Search for coincident events, no assumptions on the waveforms. Bring in physically motivated assumptions when evaluating the detection efficiency of the network. Use the assumptions to optimize the cuts on the events of each individual detectors, without compromising the detection efficiency Deduce upper limits from the fact that coincidences do not exceed background expectations.

Andrea Viceré - IHP Paris, November 13th The VIRGO-bars network in one slide 24 hours of data taken during Virgo C7 run: start at UTC time , (14 Sep :11 27s) Heterogeneous Network: spectral sensitivity directional response Patterns for circularly polarized signals

Andrea Viceré - IHP Paris, November 13th More details on the study Goal: assess interpreted confidence intervals on the flux of gravitational waves. The interpretation comes from software injections which are used to compute the efficiency of detection for a source population We have restricted the study to a class of signals, the Damped Sinusoids, and to one general direction in the sky, the Galactic Center. Main methodology: coincidence search on trigger lists made by each detector. The coincident counts, divided by efficiency and observation time, become observed rates (or upper limits on rates). Optimization of thresholds: for each template and each target amplitude, the best compromise between efficiency and false alarm rate is searched, using variable threshold for each detector. The efficiency acts not just passively at the end of the analysis to calibrate the results, but also actively during an optimization phase. Blind analysis: not to bias results by feedbacks on methods from looking at results, a “secret” time offset was added to detector times.

Andrea Viceré - IHP Paris, November 13th Signals and astrophysical motivation f gw and  are frequency and damping times h rss is the scale factor (we will define it precisely later)  and  are geometrical factors (polarization and “source plane” inclination Such signals could be produced by a ringdown of a system excited in a l=m=2 mode BH-BH ring-down. Andersson N. and Kokkotas K., Mon. Not. Roy. Astron. Soc. 299 (1998) Kokkotas K.D. and Schmidt B.G., (1999) f-mode of neutron stars. In this case the f-mode could produce a wave with variable frequency and damping time; to keep this into account we did not use matched filtering. Ferrari V. et al., Mon. Not. Roy. Astron. Soc. 342 (2003) 629

Andrea Viceré - IHP Paris, November 13th Event Trigger Generators and Observables AURIGA: WaveBursts (S. Klimenko et al, LIGO- T Z) adapted to AURIGA data. The cluster S/N (close to the optimal) was used as an indicator of the signal magnitude. NAUTILUS and EXPLORER: a single linear Wiener- Kolmogorov filter matched to the impulse response is applied to the output data. The impulse S/N was used as an indicator of the signal magnitude. VIRGO: PowerFilter is the chosen trigger generator. The logarithmic S/N was used as an indicator of the signal magnitude.

Andrea Viceré - IHP Paris, November 13th Assessing the background of accidentals To assess the significance of rates, we need an estimate of the rate of accidentals. Ideally one would like to have events at each detector distributed as independent Poisson processes. The auto- correlogram of the events at each detector should be flat. Instead, because of non-gaussianity, oscillations occur, for instance in Virgo which is under commissioning. However, the cross-correlogram is flat! So the coincidences can be regarded as a Poisson process.

Andrea Viceré - IHP Paris, November 13th A better view in the frequency domain

Andrea Viceré - IHP Paris, November 13th Software injections details Damped Sinusoids: elliptic polarization distributed signals assumed to come from the Galactic Center Several damping times and central frequencies to span our parameter space. 11 templates For each class, we generated randomly: injection times polarization angle  inclination angle  N=8640 (1/10 s) h rss =1e e-18 Hz -1/2

Andrea Viceré - IHP Paris, November 13th Which physical parameters? Take just the example of Quasi Normal Modes of Black Holes, and assume that an l=m=2 mode dominates the signal. Mass and ratio j = J/M 2 are correlated with frequency and damping time. So, we are looking also at  values which are incompatible with these modes

Andrea Viceré - IHP Paris, November 13th hrss= 1e-19 Hz -1/2 AURIGA N=1413 VIRGO N=24241 NAUTILUS N=8628 EXPLORER N=5614

Andrea Viceré - IHP Paris, November 13th Efficiency DS: f0=914 Hz tau=1ms DS: f0=930 Hz tau=30ms DS: f0=866 Hz tau=10ms Single detector efficiencies For VIRGO, ~ 7 hrs out of 24 have been excluded by epoch vetoes => Asymptotic 70%

Andrea Viceré - IHP Paris, November 13th Time coincidence We are not using matched filtering Time errors are therefore dominated by systematic biases. The narrower the bandwidth, the greater the signal is distorted Example : AURIGA – VIRGO coincidences. The double peak is due to the multi-modal time error of the Virgo Power Filter The coincidence “window”, T w = 40 ms f0=914 Hz tau=1ms f0=866 Hz tau=10ms f0=930 Hz tau=30ms

Andrea Viceré - IHP Paris, November 13th Optimization of the thresholds (1) To shrink interpreted confidence interval we choose to optimize the 2-fold coincidence searches => Better Upper Limits For each configuration/template/amplitude, the magnitude thresholds for the 2 detectors are tuned => large trial factor. We keep this into account when calculating the statistics. The criterion is to maximize the ratio efficiency over the fluctuation of the accidental coincidences. The efficiency is calculated on the data sets containing the MDC injections The average background of accidental coincidence is estimated by means of +/- 400 time shifts (~ +/- 7 min). Coincidences are Poisson point processes: fluctuation is sqrt(counts). The magnitude thresholds are optimized every 30 min

Andrea Viceré - IHP Paris, November 13th Optimization (2): 914 Hz, 1ms, 1e-19 Hz -1/2 AURIGA VIRGO

Andrea Viceré - IHP Paris, November 13th Statistical Analysis (1) Blind Analysis: we do not open the box of zero-lag until all tunable parameters are fixed, and the methodology to be used is chosen. Large trial factor => multiple tests performed, increase of the false claim probability to reduce the trial factor, for each template/amplitude, we analyze only on the best couples of detectors (72). The effective global probability is empirically estimated over the 400 time shifted data sets => the single trial confidence is tuned in order to reach a total false claim of 99% Single trial confidence Global confidence

Andrea Viceré - IHP Paris, November 13th Statistical Analysis (2) After approval by the Collaborations, we exchanged the secret time offsets and we “opened the box” and… The confidence intervals were set according to the confidence belt already used by IGEC1 (see L. Baggio and G.A. Prodi, “Setting confidence intervals in coincidence search analysis" in Statistical problems in particle physics, astrophysics and cosmology, R.Mount, L.Lyonsand and R.Reitmeyer editors, Stanford (2003) 238) When the null hypothesis test is fulfilled, than the confidence interval is simply an Upper Limit Note: a rejection of the null is a claim for an excess correlation in the observatory at the true time, not taken into account in the measured noise background at different time lags. Whether these correlations are true GW or just correlated noise signals is not known. A Virgo-note was produced to discuss the methodology: VIR-NOT-FIR

Andrea Viceré - IHP Paris, November 13th Results for the 2-fold coincidence searches No excess of Coincidences found. Null hypothesis survives... Upper Limits at 95% coverage

Andrea Viceré - IHP Paris, November 13th experimental data physical unknown confidence interval coverage For each outcome x one should be able to determine a confidence interval I x For each possible , the measures which lead to a confidence interval consistent with the true value have probability C(  ), i.e. 1-C(  ) is the false dismissal probability Confidence Belt & Coverage

Andrea Viceré - IHP Paris, November 13th Interpretation of the found limits We go back to the signal model The hrss is just a spectral scale of the signall The definition of the energy flux distribution over angle and frequency With the signal model, the total radiated energy is easily computed as An h rss = Hz -1/2 would correspond to ~ M o radiated at 10kpc

Andrea Viceré - IHP Paris, November 13th Next Steps The 2-fold coincidences have a high level of accidental background, single detection not possible=> 3-fold coincidence searches. Goal: to be able to issue a claim at 99.5% confidence on a single observed triple coincidence. In the next weeks, we plan to try the 3-fold coincidence search. The methodology and all the key parameters have been decided before “opening the box” of the double coincidence searches. Optimization of thresholds: for each template and some amplitudes (i.e. 1e-18, 5e-19 and 1e-19 Hz -1/2. ), the best compromise between efficiency and FAR is searched, using variable thresholds for each detector with ½ hour bins in order to reach the target level of background. The zero-delay will be analyzed with the optimization for the minimal signal amplitude which allows at least a level of efficiency of 40%. Configurations of detectors/template, which do not reach such minimal level for any of the chosen amplitudes, will be discarded. Given the chosen 99.5% of confidence level, to be compared with the 99% (~1% spread) for the 2-fold coincidence searches, performing the 3-fold coincidence searches will slightly affect the global confidence.

Andrea Viceré - IHP Paris, November 13th Conclusions Because of the limitations on the observation time, the Virgo-bars study does not yield stringent limits It is however a good example of the different ingredients of the analysis Efficient event search: to see as much as possible with “open eyes” in the data. Careful statistical analysis. Take into account that If you look long enough, you see anything you want. Power of the coincidence method. As well known from IGEC and LIGO experience, the network brings difficult statistics to more manageable ones Need of good theoretical glasses. We may not need waveforms to catch all classes of signals. But we need them to assess the significance and constrain physical parameters.