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Upper Limits from Loudest Events
Jolien CreightonPatrick Brady and Alan Wiseman
See Also:R. Cousins, Nucl. Instrum. Meth. A, 557 (1994)
S. Yellin, Phys. Rev. D, 66 032005 (2002)
2
Outline
Loudest Event Rate Upper Limit
Loudest Event Method vs. Fixed Threshold
Rate vs. Strain Exclusion Curves
Upper Limit on Strain
Accounting for a Background
3
Basic Idea
0%
25%
50%
75%
100%
Observed Events Population Efficiency
Frac
tion
Exc
eeding
Th
resh
old
Threshold SNR Level ρ
Loudest Event ρmax
Efficiency at Loudest Event ε
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Loudest EventRate Upper Limit
Assume loudest event is from a signal (conservative upper limit)
Probability that no events would have SNR greater than loudest event SNR ρmax:
90% confidence (frequentist) upper limit:
R90%=2.3T ε
P= e–RTε
P = 10% →
R: RateT: Observation timeε: Efficiency at ρmax
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Loudest vs. Threshold
3 4 5 6 7threshold SNR level
90
% r
ate
up
pe
r lim
it (
arb
itra
ry)
thresholdloudest
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Rate vs. Strain
Excluded(90% confidence)
Strain, h
Rate
Lim
it, R
90%
R90%=2.3
T ε(h)
Allowed
(Similar to S1 burst analysis)
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Goal: bound GW flux rather than rate
Assume only one signal is present
Probability that this event would not exceed loudest event SNR:
90% confidence (frequentist) upper limit:
Upper Limit on Strain
P = 1–ε(h)
h90% = ε–1(90%)P = 10% →
(Triggered Bursts, Pulsars)
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Background
Probability that no background event would exceed the loudest event SNR: PB
Probability that no event (background or signal) would exceed the loudest event SNR:
90% confidence (frequentist) upper limit:
P = PBe–RTε
R90%=2.3 – |ln PB|
T εP = 10% →
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Summary
Upper limits often sensitive to SNR threshold
Loudest event upper limits are often stronger than threshold-based limits
Background information can be used to strengthen limit (conservative)
Setting an upper limit does not preclude claiming a detection