INTERPRETATION of IGEC RESULTS Lucio Baggio, Giovanni Andrea Prodi - PowerPoint PPT Presentation

INTERPRETATION of IGEC RESULTS Lucio Baggio, Giovanni Andrea Prodi University of Trento and INFN Italy or unfolding gw source parameters starting point: • IGEC 1997-2000 results ( P.Astone et al., PRD 68 (2003) 022001 ) with reference to: • LIGO S1 burst gw results ( B.Abbott et al., gr-qc/0312056 )

COMPARISON at a GLANCE IGEC 1997-2000 LIGO S1 • systematic search over many • playground data to tune the amplitude thresholds: search: many data selections one data selection many data points one data point • bound of maximum false • montecarlo for some specific dismissal probability of detection: source models : conservative efficiency is efficiency is measured vs gw estimated for δ -like waveform amplitude for sample waveforms results are upper limits on rate of results are upper limits on rate of detected burst gws above incoming burst gws: threshold: rate vs true amplitudes rate vs search threshold cumulative Lacking the “unfolding” to gw Source model: source parameters sample waveforms incoming at fixed (“uninterpreted” results) amplitude + directional corrections…

UPPER LIMIT on the RATE of BURST GW from the GALACTIC CENTER DIRECTION Poisson dashed region excluded rate of with probability � 90% detected gw overcoverage [ year –1 ] search threshold • signal template = � -like gw from the Galactic Center direction signal amplitude H S = FT[h S ] at � � 2 � 900 Hz ⋅ − ↔ 21 H ~ 2 10 / Hz 0 02 . M converted in burst gw atGalacticCe nter S ฀

UPPER LIMIT on the RATE of BURST GW from the GALACTIC CENTER DIRECTION (2) dashed region excluded Poisson with probability � 90% rate of overcoverage detected gw [ year –1 ] 1.8 yr -1 search threshold • no coincidences found, limited by the observation time • limited by accidental coincidences • observation time cuts off: sensitivity cut

UPPER LIMIT on the RATE of BURST GW from the GALACTIC CENTER DIRECTION (3) Poisson rate of detected gw [ year –1 ] search threshold • analysis includes all the measured signal amplitudes � search threshold � result is cumulative for H M � H t • systematic search vs threshold H t � many trials ( 20 /decade ) almost independent results

Case of gw flux of constant amplitude: δ -like signal from GC Poisson correct each result for the rate of detection efficiency as a detected function of gw amplitude H S δ gw [ year –1 ] convert in terms of parameters of the source search model threshold at H S ≥ H t efficiency ≥ 0.25 due to 2-fold observations at threshold Poisson rate of at H S ≥ 2 H t incoming efficiency = 1 gw true δ enough above the threshold [ year –1 ] H S amplitude

Case of gw flux of constant amplitude: δ -like signal from GC (2) • complete conservative efficiency estimation for the single data point • … on all data points • convert from H S = FT[h S ] at � � 2 � 900 Hz to template amplitude parameter e.g. for a sine-gaussian(850 Hz;Q=9) h rss = 10 Hz 0.5 H S Poisson rate of incoming gw [ year –1 ] true δ amplitude H S

Remarks • IGEC time coincidence search provides a systematic search as a function of common threshold a directional search strategy ⇒ is able to deal with • detectors with different sensitivities (level & bandwidths) search with templates search resctricted on the common sensitivity bandwidth • detectors with different antenna patterns and locations if gw polarization is modeled or simply linear • IGEC method is able to assess the false detection probability Of course, relevant improvements are possible: - provide measurements of detection efficiency Monte Carlo injection of selected templates - feed a further stage of coherent analysis - effective control of false detections of surveys

HOW to UNFOLD IGEC RESULTS in terms of GW FLUX at the EARTH • Take a model for the distribution of events impinging on the detector H S � H t (dashed line) • Estimate the distribution of measured coincidences H M � H t (cont.line) • Compare with IGEC results to set confidence intervals on gw flux parameters 1,000 1,000 coverage 100 100 0.60 0.60 rate rate 0.80 0.80 ( year –1 ) ( year –1 ) 0.90 0.90 0.95 0.95 10 10 1 1 1E-21 1E-21 1E-20 1E-20 search threshold search threshold 1E-19 1E-19 H t ( Hz -1 ) ( Hz -1 )

Case of gw flux of constant amplitude: δ -like signal from GC (3) • the resulting interpreted upper limit H S = FT[h S ] at � � 2 � 900 Hz • convert from to template amplitude parameter h rss = 10 Hz 0.5 H S e.g. for a sine-gaussian(850 Hz;Q=9) Poisson rate of detected gw [ year –1 ] search threshold

Case of gw flux of constant amplitude: comparison to LIGO • the resulting interpreted upper limit H S = FT[h S ] at � � 2 � 900 Hz • convert from to template amplitude parameter h rss = 10 Hz 0.5 H S e.g. for a sine-gaussian(850 Hz;Q=9) Poisson rate of detected gw [ year –1 ] h rss

Case of gw flux of constant amplitude: comparison with LIGO results • IGEC sets an almost independent result per each tried threshold H t • correct each result for the detection efficiency as a function of gw amplitude H S : at H S ≥ H t efficiency ≥ 0.25 due to 2-fold observations at threshold e.g. at H S ≥ 2 H t efficiency = 1 enough above the threshold Poisson rate of detected gw [ year –1 ] search threshold

DIRECTIONAL SEARCH: sensitivity modulation 10 amplitude (Hz -1 ·10 -21 ) 9 8 7 6 5 4 3 2 1 0 0 6 12 18 24 30 36 42 48 54 60 time (hours) 1.0 10 10 amplitude (Hz -1 ·10 -21 ) 0.9 9 9 amplitude 0.8 8 8 directional 0.7 7 7 sensitivity 0.6 6 6 ϑ 2 sin 0.5 5 5 GC 0.4 − ϑ 4 4 2 sin GC 0.3 3 3 0.2 2 2 0.1 1 1 0.0 0 0 0 0 6 6 12 12 18 18 24 24 30 30 36 36 42 42 48 48 54 54 60 60 0 6 12 18 24 30 36 42 48 54 60 time (hours)

Resampling statistics by time shifts amplitude (Hz -1 ·10 -21 ) 10 10 10 10 10 9 9 9 9 9 8 8 8 8 8 7 7 7 7 7 6 6 6 6 6 5 5 5 5 5 4 4 4 4 4 3 3 3 3 3 2 2 2 2 2 1 1 1 1 1 0 0 0 0 0 0 0 0 0 6 6 6 6 12 12 12 12 18 18 18 18 24 24 24 24 30 30 30 30 36 36 36 36 42 42 42 42 48 48 48 48 54 54 54 54 60 60 60 60 0 6 12 18 24 30 36 42 48 54 60 time (hours) We can approximately resample the stochastic process by time shift. in the shifted data the gw sources are off , along with any correlated noise Ergodicity holds at least up to timescales of the order of one hour . The samples are independent as long as the shift is longer than the maximum time window for coincidence search ( few seconds )

Setting confidence intervals IGEC approach is frequentistic in that it computes the confidence level or coverage as the probability that the confidence interval contains the true value unified in that it prescribes how to set a confidence interval automatically leading to a gw detection claim or an upper limit based on maximum likelyhood confidence intervals ( different from Feldman & Cousins ) false dismissal is under control (but detection efficiency is only lower- bounded) estimation of the probability of false detection (many attempts made to enhance the chances of detection)

TESTING the NULL HYPOTHESIS 18 16 14 12 10 N gw 8 6 4 2 0 1.0 10.0 100.0 search threshold [10 -21 /Hz] ⇒ testing the null hypothesis many trials ! all upper limits but one: overall false alarm probability 33% for 0.95 coverage NULL HYPOTHESIS WELL IN 56% for 0.90 coverage AGREEMENT WITH THE at least one detection in the set OBSERVATIONS in case NO GW are in the data

FALSE ALARM RATES false 10 alarm AL-AU dramatic improvement by 1 rate AL-AU-NA increasing the detector number: 0.1 -1 ] [ yr 3-fold or more would allow 0.01 to identify the gw candidate 1E-3 1E-4 1E-5 1E-6 2E-21 1E-20 -1 ] common search threshold [ Hz mean � timing mean [ ms ] rate of events [ yr -1 ]

UPPER LIMIT on the RATE of BURST GW from the GALACTIC CENTER DIRECTION (3) Poisson rate of detected gw [ year –1 ] search threshold • analysis includes all the measured signal amplitudes � search threshold � result is cumulative for H M � H t • systematic search vs threshold H t � many trials ( 20 /decade ) almost independent results

MULTIPLE DETECTOR ANALYSIS network is needed to estimate (and reduce) the false alarms time coincidence search among exchanged triggers time window is set according to timing uncertainties by requiring a conservative false dismissal 1 − ≤ σ + σ ↔ ≤ 2 2 t t k false dismissal by Tchebyscheff inequality i j i j 2 k false alarms ∝ k efficiency of detection maximize the chances of detection i.e. the ratio fluctuations of false alarms measure the false alarms: time shifts � resampling the stochastic processes so that: • gw sources are off (as well as any correlated noise) • statistical properties are preserved (max shift ~ 1 h ) • independent samples (min shift > largest time window ~ few s )

DIRECTIONAL SENSITIVITY The achieved sensitivity of bar detectors limits the observation range to sources in the Milky Way . The almost parallel orientation of the detectors guarantees a good coverage of the Galactic Center 1 0.8 0.6 ← ALLEGRO Sin 2 ( θ ) ← AURIGA -EXPLORER –NAUTILUS 0.4 ← NIOBE 0.2 0 0 4 8 12 16 20 24 amplitude directional sensitivity factor Time (h) vs sideral time (hours)