ensemble tests and sensitivity calculations

Ensemble tests and sensitivity calculations

Kevin Kröninger, MPI für Physik

GERDA Collaboration Meeting, Tübingen, 11/09 – 11/11/2005

Overview

Kevin Kröninger, MPI München GERDA Collaboration Meeting Tübingen, 11/09 – 11/11/2005

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

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

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)

<N> = A · m · T

A : activity per mass

m : mass

T : run-time

Ensembles II


POOL

Co-6010 μBq/kg

POOL

U-23816 μBq/kg

POOL

K-4088 μBq/kg

Phase II Holder: Copper

POOL

Th-23219 μBq/kg

ENSEMBLE

205 events 390 events 328 events 1807 events

x mass : (31 x 21) g x time : 1 year

=

Ensembles III


1 year run-time2.3·10-3 counts/kg/keV/y

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

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

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

inBAp |,

BeABANQE

i

i

2

2

1,

Statistical Analysis II


• Apply Bayes‘ Theorem:

with Poissonian fluctuations in each bin

For details see note GSTR-05-020

dAdBBApBAnp

BApBAnpnBAp

i

ii

,,|

,,||,

i

ii BAnp ,|

),(

!

),(,| BAN

i

ni

iii

i

en

BANBAnp

Statistical Analysis III


Ba

ckg

rou

nd

B [

ke

V-1]

Signal A

p(A, B|{ni})

Statistical Analysis IV


• Marginalize w.r.t. signal (A) and background (B)

dAnBApnBp

dBnBApnAp

iii

iii

|,|

|,|

mode A* mode B*

Signal A Background B [keV-1]

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

3*

106|0|

ii

iinAp

nAAp A* : most probable value

6·103 corresponds to 5 σ

Statistical Analysis VI


1 year run-timeF

ract

ion

of d

isco

verin

g en

sem

bles

MC simulation (best estimate)

Statistical Analysis VII


1 year run-timeF

ract

ion

of d

isco

verin

g en

sem

bles

MC simulation (best estimate)

Statistical Analysis IX


1 year run-time

Statistical Analysis X


10-4 counts/kg/keV/y



no background2 σ environment of recent claim

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 cou

nts

/kg

/keV

/y

Event Selection II


1 year run-timebefore event selection

after event selection

• Signal efficiency ~90%

• Resolution added

• After event selection ~6%

of event left

Statistical Analysis VIII


Fra

ctio

n of

dis

cove

ring

ense

mbl

es

Resolution Study


1 year run-time, 2.3·10-3 counts/kg/keV/y, T1/2 = 1.6·1025 yearsF

ract

ion

of d

isco

verin

g en

sem

bles

Miscalibration Study


Fra

ctio

n of

dis

cove

ring

ense

mbl

es1 year run-time, 2.3·10-3 counts/kg/keV/y, T1/2 = 1.6·1025 years

ensemble tests and sensitivity calculations

Documents

background estimate

external background

background bmode

background contributions

background b kev

background note gstr

overviewkevin krninger

ensembles iiikevin krninger