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Ensemble tests and sensitivity calculations

Overview of ensemble creation, event selection, and statistical analysis for sensitivity estimation in background sources simulation using MaGe. Includes compiling ensembles, event selection criteria, and Bayesian analysis for signal and background estimation.

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Ensemble tests and sensitivity calculations

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  1. Ensemble tests and sensitivity calculations Kevin Kröninger, MPI für Physik GERDA Collaboration Meeting, Tübingen, 11/09 – 11/11/2005

  2. Overview Monte Carlo simulation of int. background sources (MaGe) Creation of ensembles according to activities Cut-based event selection Statistical analysis: Definition of discovery ↔ Limit estimation procedure Kevin Kröninger, MPI München GERDA Collaboration Meeting Tübingen, 11/09 – 11/11/2005

  3. Monte Carlo Simulation • Use MaGe for the simulation of background sources (internal) • → see Xiang‘s talk and background note GSTR-05-019 • Setup is ‚ideal‘ Phase II: • 21 segmented detectors (3 z / 6 φ segments) • Total of 44.2 kg germanium • Material according to Phase II design (holder, etc.) • Energy resolution 5 keV FWHM • No primordial or muon induced neutrons included • External background from infrastructure neglected Kevin Kröninger, MPI München GERDA Collaboration Meeting Tübingen, 11/09 – 11/11/2005

  4. Ensembles I • Compile list of materials used in set-up and corresponding activities • Calculate mean number of events for each background source and part • Compile ensemble: a set of events that mimic data after run-time T • Actual number of events in ensemble are Poisson fluctuated • Store time, e.g. halflife of Ge-68 taken into account (exponential decay) A : activity per mass m : mass T : run-time <N> = A · m · T Kevin Kröninger, MPI München GERDA Collaboration Meeting Tübingen, 11/09 – 11/11/2005

  5. Ensembles II Phase II Holder: Copper POOL Co-60 10 μBq/kg POOL Th-232 19 μBq/kg POOL U-238 16 μBq/kg POOL K-40 88 μBq/kg x mass : (31 x 21) g x time : 1 year = 205 events 390 events 328 events 1807 events ENSEMBLE Kevin Kröninger, MPI München GERDA Collaboration Meeting Tübingen, 11/09 – 11/11/2005

  6. Ensembles III 1 year run-time 2.3·10-3 counts/kg/keV/y Kevin Kröninger, MPI München GERDA Collaboration Meeting Tübingen, 11/09 – 11/11/2005

  7. Ensembles IV • Compiling ensembles is CPU time intensive • Use toy ensembles: • Spectra created with flat background + Gaussian peak signal • Tested flat background hypothesis with 2500 kg·years • Vary background and signal contribution Kevin Kröninger, MPI München GERDA Collaboration Meeting Tübingen, 11/09 – 11/11/2005

  8. Event Selection I • Similiar to selection done for background estimate: • Anti-coincidence between segments • Energy window ±80 keV around Qββ • X-ray veto against decay of Ge-68 • No pulse shape analysis used yet • For details on the background contributions see Note GSTR-05-019 Kevin Kröninger, MPI München GERDA Collaboration Meeting Tübingen, 11/09 – 11/11/2005

  9. Statistical Analysis I • Estimate two parameters: signal (A) and background (B) • Question: What is ? • Assume flat background and Gaussian peak at Qββ with width ~ resolution • Divide energy spectrum in 1 keV bins with events ni • Expectation in ith bin Kevin Kröninger, MPI München GERDA Collaboration Meeting Tübingen, 11/09 – 11/11/2005

  10. Statistical Analysis II • Apply Bayes‘ Theorem: • with Poissonian fluctuations in each bin • For details see note GSTR-05-020 Kevin Kröninger, MPI München GERDA Collaboration Meeting Tübingen, 11/09 – 11/11/2005

  11. Statistical Analysis III p(A, B|{ni}) Background B [keV-1] Signal A Kevin Kröninger, MPI München GERDA Collaboration Meeting Tübingen, 11/09 – 11/11/2005

  12. Statistical Analysis IV mode A* mode B* Signal A Background B [keV-1] • Marginalize w.r.t. signal (A) and background (B) Kevin Kröninger, MPI München GERDA Collaboration Meeting Tübingen, 11/09 – 11/11/2005

  13. Statistical Analysis V • Definition of discovery: • Discovery potential: fraction of ensembles with discovery (Freq. prob.) • Limit estimation: integrate p(A|{ni}) to 95% probability • Test different scenarios: • Background index between 0 and 10-2 counts/kg/keV/y • Halflife between (0.8 ·1025 – 5.0 ·1026) years • Run-time between 1 and 10 years A* : most probable value 6·103 corresponds to 5 σ Kevin Kröninger, MPI München GERDA Collaboration Meeting Tübingen, 11/09 – 11/11/2005

  14. Statistical Analysis VI MC simulation (best estimate) 1 year run-time Fraction of discovering ensembles Kevin Kröninger, MPI München GERDA Collaboration Meeting Tübingen, 11/09 – 11/11/2005

  15. Statistical Analysis VII MC simulation (best estimate) 1 year run-time Fraction of discovering ensembles Kevin Kröninger, MPI München GERDA Collaboration Meeting Tübingen, 11/09 – 11/11/2005

  16. Statistical Analysis IX 1 year run-time Kevin Kröninger, MPI München GERDA Collaboration Meeting Tübingen, 11/09 – 11/11/2005

  17. Statistical Analysis X 2 σ environment of recent claim no background 10-4 counts/kg/keV/y 10-3 counts/kg/keV/y 10-2 counts/kg/keV/y Kevin Kröninger, MPI München GERDA Collaboration Meeting Tübingen, 11/09 – 11/11/2005

  18. Conclusion • Ensemble test have been done with fake data sets • Statistical analysis yields following results: • Probability of observing 1.6·1025 years >95% after 1 year • … after 5 years ~5.0·1025 years • Exclusion limit after 1 years ~5.0·1025 years • … after 5 years ~ 2.0·1026 years • Results stable against resolution up to 10 keV FWHM • Results stable against miscalibration up to 2 keV • Need to be better than ~ 10-2 counts/kg/keV/y • Details can be found in note GSTR-05-020 10-3 counts/kg/keV/y Kevin Kröninger, MPI München GERDA Collaboration Meeting Tübingen, 11/09 – 11/11/2005

  19. Event Selection II 1 year run-time before event selection • Signal efficiency ~90% • Resolution added • After event selection ~6% of event left after event selection Kevin Kröninger, MPI München GERDA Collaboration Meeting Tübingen, 11/09 – 11/11/2005

  20. Statistical Analysis VIII Fraction of discovering ensembles Kevin Kröninger, MPI München GERDA Collaboration Meeting Tübingen, 11/09 – 11/11/2005

  21. Resolution Study 1 year run-time, 2.3·10-3 counts/kg/keV/y, T1/2 = 1.6·1025 years Fraction of discovering ensembles Kevin Kröninger, MPI München GERDA Collaboration Meeting Tübingen, 11/09 – 11/11/2005

  22. Miscalibration Study 1 year run-time, 2.3·10-3 counts/kg/keV/y, T1/2 = 1.6·1025 years Fraction of discovering ensembles Kevin Kröninger, MPI München GERDA Collaboration Meeting Tübingen, 11/09 – 11/11/2005

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