Bounding the strength of a Stochastic GW Background in LIGO’s S3 Data
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Transcript of Bounding the strength of a Stochastic GW Background in LIGO’s S3 Data
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GWDAW-9, Annecy - 12/16/04
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Bounding the strength of a Stochastic GW Background
in LIGO’s S3 Data
Sukanta Bose (Washington State University, Pullman)
for the LIGO Scientific Collaboration
LIGO DCC No. LIGO-G050536-00-D
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SGWB: Properties
Individual detector
strain: Zero mean
Covariance:
SGWB power
spectrum:
What are we bounding?
0~
fhA
'2
1'
~~* ffffSfhfh ABgwBA
fd
fdf gw
gw ln
1
critical
32
20
10
3
f
fHfS gw
gw
[Christensen, PRD46 (1992)]
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The Search Statistic
Cross-correlation
(CC) statistic:
Theoretical mean
of CC statistic:
Theoretical variance:
Optimal filter:
fQffSdfT
ABgw
~
2
2
2 ~
4fQfPfPdf
TBA
)()()()(
~3 fPfPf
ff
fPfP
ffSfQ
BA
ABgw
BA
ABgw
QhhKttQththdtdtY BA
T
T
BA ,,''2/
2/
[Allen-Romano, PRD59, 102001 (1999)]
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The Search Statistic (contd.)
The optimal cross-correlation(CC) estimator is:
And the (inverse of the)optimal theoretical variance is: The measured Omega is:
i = 1 2 3 …
t
60sec
ii
iii Y
Y2
2
opt
i
i22
opt TYh opt2
1000
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S3: Reference sensitivities The figure shows the
typical equivalent-strain noise-densities of the 3 LIGO detectors during S3. Also shown is the strain density corresponding to a stochastic background with
40 10
2410
50 100 500 Frequency (Hz)
1810
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Optimallycombine Y
i ,
i2
Compute CC statistic Yi
Downsample, HP filter, Freq-mask & calibrate
Compute optimal filter Qi
and theoretical variance i2
Estimate PSDs (using prev & next segs)
Window & FFT
Detector 2 -60 sec data segments
Detector 1 -60 sec data segments
Downsample, HP filter,Freq-mask & calibrate
Estimate PSDs (using prev & next segs)
Window & FFT
Post-processing
Softwareinjections
Analysis pipeline
2 21 1 2 2{ , , , ,... } Y Y ˆ 1.28gw gw
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Choice of frequency cut-offs
Frequency bandwidth chosen from 70 - 220 Hz (H1-H2)
Overlap reduction functionsSensitivity vs Max cut-off for H1-H2 (S3)
0 50 100 150 200 250 300
Frequency (Hz)
50 100 150 200 250 300 350 400 450 500 Max. cut-off frequency (Hz)
[Flanagan, PRD48, 2389 (1993)]
0
1
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S3: H1-H2 Frequency mask
60 80 100 120 140 160 180 200 220
Frequency (Hz)
110 112 114 116 118 120 122 124 126 128 13010
-6
10-5
10-4
10-3
10-2
10-1
100
Co
her
ence
Frequency (Hz)
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Sigma-cut of data intervals Sigma-integrand is proportional to
1/(P1*P2) P1, P2 estimated using data outside
of 60s interval being analyzed, to avoid bias in cross-correlation
Not good PSD estimators when the noise is non-stationary over this time period
Compare this PSD to that computed with data in the interval; reject interval if they don’t agree
PI
t
60s
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Sigma-cut of data intervals Sigma-integrand is proportional to
1/(P1*P2) P1, P2 estimated using data outside
of 60s interval being analyzed, to avoid bias in cross-correlation
Not good PSD estimators when the noise is non-stationary over this time period
Compare this PSD to that computed with data in the interval; reject interval if they don’t agree
PI
t
60s
60 0
5
00
(S2) cutsoutlier with H1L1 of Histogram 00
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S2 H1-L1 analysis: Distribution of the
theoretical
Distribution of the theoretical S2
/
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S3 H1-H2 analysis: Distribution of the
theoretical
Distribution of the theoretical S3
Abs
S3 data was more non-stationary.
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H1-L1 analysis: Long-duration features in CC-statistics (S2)
Time (in days)
CC
-sta
tis
tic
5 15 25 35 45
S2 data was treated as “playground” for S3, esp., to check for long-duration trends.
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H1-L1 analysis: Lombe-Scargle Power Spectrum of CC statistics (S2)
Injected line at 1/f = 1 hour
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Frequency (in mHz)
Po
wer
1 day 10 min
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H1-L1 analysis: Distribution of the Power of the CC-statistics (S2)
0 2 4 6 8 10 12
Power
N
1
1000
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H1-L1 analysis: CC statistic trend (S2)
PRELIMINARY
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H1-L1 analysis(S2): Kolmogorov-Smirnov test
The K-S value of 0.483 implies that the distribution is close to normal. i
i Relative freq.
Relativefrequency
0.483 test
Smirnov-Kolmogorov
2/exp Curve 2
x
011.0
23249
2
yx
xy
N
0 9 10-5
0
-5
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S3 results: H1-H2Error-estimate (+3 plotted for the H1-H2 pair as a function of run time.
410
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S3 results: H1-L1Error-estimate (+3 plotted for the H1-L1 pair as a function of run time.
310
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LIGO results history on gw
h100
2
LIGO run H-L H1-H2 Freq range Observation Time
S1*
< 23 +/- 4.6
(H2-L1)
Cross-correlated instr. noise
found40-314 Hz
64 hours
(08/23/02 – 09/09/02)
S2< 0.018
+0.007- 0.003
(H1-L1)
Cross-correlated instr. noise
found 50-300 Hz
387 hours
(02/14/03 – 04/14/03)
S3 ??
Can account for instrument noise
in bounding 50-250 Hz (H1-L1)
70-220 Hz (H1-H2)
~350 hrs (H1-L1)
~550 hrs (H1-H2)
(10/31/03 – 01/09/04)
*[The LIGO Collaboration, PRD 69, 122004, (2004)]
PRELIMINARY
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Summary The current best IFO-IFO upper-limit (published) is from S1: 23 (+/-4.6)
» S2 bettered it to 0.018 (+0.007- 0.003) (PRELIMINARY)» The S3 studies are set to improve that
H1-H2 is the most sensitive pair, but it also suffers from cross-correlated terrestrial noise. H1-H2 coherence found weak in most frequency bands, except ~120Hz and ~180Hz; steps taken to excise these bands from analysis (in addition to frequency masking of certain lines).
The observed properties of the search statistics for the H1-H2 and H1-L1 pairs, after correcting for biases and known systematics, were found to closely fit the expected ones.
It now remains to run the search pipeline on the S3 science data to obtain upper-limits / confidence belts for a constant
Beyond current analysis:» Search for (f) ~ n(f/f0)n
» Targeted searches
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H1-L1 analysis: Long-duration features in CC-statistics (S2)
S2 data was treated as“playground”for S3, esp., to check forlong-durationtrends.