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A Time-Domain Veto for Binary Inspirals Search Gianluca M. Guidi - - PowerPoint PPT Presentation
A Time-Domain Veto for Binary Inspirals Search Gianluca M. Guidi - - PowerPoint PPT Presentation
A Time-Domain Veto for Binary Inspirals Search Gianluca M. Guidi Universit di Urbino- INFN Firenze-Urbino LIGO Visitor - CALTECH A Time-Domain Veto for Binary Inspirals search The distinction between actual gravitational waves from binary
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The correlator and its statistic distribution
The Gaussian variables x(t) and y(t) are the correlation between the ITF output h(t) and the cosine and sine polarizations of the inspiral signal:
( )
* 2
( ) ( ) ( ) 2
ift c h
h f h f x t e df S f
π ∞ −∞
= ∫
- (
)
* 2
( ) ( ) ( ) 2
ift s h
h f h f y t e df S f
π ∞ −∞
= ∫
- The statistic normally considered is:
( ) ( ) ( )
2 2 2 2
x t y t t ρ σ + =
( )
* 2
( ) ( ) 2
c c h
h f h f df S f σ
∞ −∞
= ∫
- with:
In absence of signal ρ2(t) has a χ2 distribution.
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The correlator and its statistic distribution
( ) ( )
( )
( )
( )
2 2 2 2
( ) ( ) x t x t y t y t t ρ σ − + − =
- When a signal has been detected,
we can consider the statistic:
2 2 n
χ = ( )
* 2
( ) ( ) ( ) 2
ift c c h
h f h f e df S f
π ∞ −∞
= ∫
- is the mean of x(t) for cosine component
- f the template that has matched the signal.
x t Where
2( )
t ρ
Simulated data
2( )
t ρ
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Two Possible Time-Domain Vetoes
1) Analysis of the probalility that N uncorrelated values of exceed a fixed threshold before the time
- f
coalescence.
2
ρ
- 2
ρ
- 2
ρ
- 2) Analysis of the probability that the max
- ver a time T exceeds a fixed threshold
before the time of coalescence.
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1-Crossing Veto
We consider N uncorrelated times before the coalescence.
1 4
a
t f δ ≥
..
t
..
t
1 N
t
− N
t
coalescence
( )
( )
( )
2
2 2 2 2 2
Thr i
Thr i i n
P t d
ρ
ρ ρ χ ρ
=
< = ∫
- Probability that the statistic at ti
does not exceed :
( )
2 Thr i
ρ
- (
)
( )
2 2 1 N Thr tot i i i
P P t ρ ρ
=
= <
∏
- Probability that anyone of the N
values exceeds the thresholds:
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1-Crossing Veto ( )
( )
1 2 2 Thr N i i tot
P t P ρ ρ < =
- N
tot
P p =
( )
2
10.55
Thr i
ρ =
- (
)
( )
( )
1 2 2 10
0.95 0.995
Thr i i
P t ρ ρ < = =
- Ex: We fix N=10 and choose a
confidence level of 5% ( = 0.05)
P
2
ρ
- (
)
( )
2 2
, 1,
Thr i i
P t p i N ρ ρ < = ∀ =
- Probability that the thresholds are
exceeded at least one time:
1
tot
P P = −
If we suppose that:
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1-Crossing Veto
Monte Carlo with 7 < SNR < 20 – simulated LIGO noise. SNR Number of crossing
= 0.05
P
= 0.2
P
= 0.1
P
= 0.01
P
( )
2
10.55
Thr i
ρ =
- (
)
2
7.63
Thr i
ρ =
- (
)
2
9.12
Thr i
ρ =
- (
)
2
13.81
Thr i
ρ =
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Veto on the Maxima
2 max
ρ
- We choose an interval of time
before the coalescence and find the maximum of
2
ρ
- The distribution of the maxima follows a Fréchet distribution:
(1 )
1 ( )
F
x F F
x f x e
α
α σ
α σ σ
−
− + −
=
2
ρ
- 2
max
x ρ =
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Veto on the Maxima
We fix a cofidence level C and find:
( )
F
x
F x e
α
σ
−
−
=
Threshold for C = 5%
( )
1 2 max
1 ln 1
Thr F
C
α
ρ σ
−
= −
- α and σF are the location and
shape parameters which can be found from the mean and stdv
- f
( )
2 max
ln ρ
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Veto on the Maxima
max
x ρ =
10% 5% 1%
The threshold of 6.5 corresponds to a confidence level of C = 0.02%
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