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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