1. Bounding the strength of a Stochastic GW Background in LIGO’s S3 Data
Sukanta Bose
(Washington State University, Pullman)
for the LIGO Scientific Collaboration
Include references
LIGO DCC No. LIGO-G D
GWDAW-9, Annecy - 12/16/04
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2. SGWB: Properties Individual detector strain: Zero mean Covariance:
SGWB power
spectrum:
What are we bounding?
[Christensen, PRD46 (1992)]
Essentially, the first 2 lines describe it
A is the detector index;
2nd line: SGWB is generally colored, stationary, and gaussian.
SGWB is isotropic % unpolarized
Will consider only the Omega = constant here.
GWDAW-9, Annecy - 12/16/04
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3. The Search Statistic Cross-correlation (CC) statistic:
Theoretical mean
of CC statistic:
Theoretical variance:
Optimal filter:
Q determined up to a normalization constant.
[Allen-Romano, PRD59,
(1999)]
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4. The Search Statistic (contd.)
60sec
The optimal cross-correlation
(CC) estimator is:
i = 1 2 3 … t
And the (inverse of the)
optimal theoretical variance is:
i is the (60-sec) segment number
Sigma may vary from one i to another due to non-stationarity; the optimal way of combining them (coherently) is given here.
The measured Omega is:
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5. 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
Log-log plot. What is plotted as dashed line is: tilde{h} (f) ~ \sqrt{Sgw} ~ \sqrt{Omega0 / f^3}
Frequency (Hz)
GWDAW-9, Annecy - 12/16/04
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6. Analysis pipeline Post-processing Detector 1 - 60 sec data segments
Downsample, HP filter,
Freq-mask & calibrate
Software
injections
Downsample, HP filter,
Freq-mask & calibrate
Estimate PSDs
(using prev & next segs)
Estimate PSDs
(using prev & next segs)
Compute optimal filter Qi
and theoretical variance i2
Window & FFT
Window & FFT
One-sided: 1.28sigma; two-sided: 1.65sigma
Compute CC statistic Yi
Post-processing
Optimally
combine Yi , i2
GWDAW-9, Annecy - 12/16/04
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7. Choice of frequency cut-offs
Overlap reduction functions
Sensitivity vs Max cut-off for H1-H2 (S3)
Frequency (Hz)
MaxSens = SNR(50-F Hz) / SNR( Hz); MinSens = SNR(F-500 Hz) / SNR( Hz)
Fixed freq to a variable frequency
[Flanagan, PRD48, 2389 (1993)]
Frequency bandwidth chosen from Hz (H1-H2)
Max. cut-off frequency (Hz)
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8. S3: H1-H2 Frequency mask
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9. 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
This is a new veto in S3. Illustrated on S2 data, which was “playground” for S3.
GWDAW-9, Annecy - 12/16/04
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10. 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
This is a new veto in S3. Illustrated on S2 data, which was “playground” for S3.
GWDAW-9, Annecy - 12/16/04
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11. Distribution of the theoretical s (S2)
S2 H1-L1 analysis: Distribution of the
theoretical s
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12. Distribution of the theoretical s (S3)
S3 H1-H2 analysis:
Distribution of the
theoretical s
S3 data was more
non-stationary.
GWDAW-9, Annecy - 12/16/04
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13. H1-L1 analysis: Long-duration features in CC-statistics (S2)
S2 data was treated as “playground” for S3, esp.,
to check for long-duration trends.
CC-statistic
This study done to check for TRENDS.
In the SGWB analysis, S2 data was treated as a “playground” for the S3.
CENTRAL plot: PSD consistent with WHITE noise; left-end ~1e-5 = 1 day; right-end 1.6e-3 = 10 minutes = 600sec
Bottom plot: PSD is consistent with a Rayleigh distribution in power.
Time (in days)
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14. H1-L1 analysis: Lombe-Scargle Power Spectrum of CC statistics (S2)
Injected line at 1/f = 1 hour
Power
This study done to check for TRENDS.
In the SGWB analysis, S2 data was treated as a “playground” for the S3.
CENTRAL plot: PSD consistent with WHITE noise; left-end ~1e-5 = 1 day; right-end 1.6e-3 = 10 minutes = 600sec
Bottom plot: PSD is consistent with a Rayleigh distribution in power.
Frequency (in mHz)
1 day
10 min
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15. H1-L1 analysis: Distribution of the Power of the CC-statistics (S2)
1000 N 1
This study done to check for TRENDS.
In the SGWB analysis, S2 data was treated as a “playground” for the S3.
CENTRAL plot: PSD consistent with WHITE noise; left-end ~1e-5 = 1 day; right-end 1.6e-3 = 10 minutes = 600sec
Bottom plot: PSD is consistent with a Rayleigh distribution in power.
Power
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16. H1-L1 analysis: CC statistic trend (S2)
Horizontal axis: in seconds, the whole S2 run of ~57 days = 5,000,000sec
PRELIMINARY
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17. H1-L1 analysis(S2): Kolmogorov-Smirnov test
Relative
frequency -5
The K-S value of implies that the distribution is close to normal.
These plots are made withOUT the sigma-outlier cut. With the cut, the results are still qualitatively similar to the above (except that the numbers change a bit). This is expected because for S2 the data was much more stationary at 1min timescales than S3. -5 9 1
Relative freq.
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18. S3 results: H1-H2
Error-estimate (+3s) plotted for the H1-H2 pair as a function of run time.
S3: improvement in sensitivity of run.
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19. S3 results: H1-L1
Error-estimate (+3s) plotted for the H1-L1 pair as a function of run time.
S3: improvement in sensitivity of run.
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20. LIGO results history on gw h1002
LIGO run
H-L
H1-H2
Freq range
Observation Time
S1*
< 23 +/- 4.6
(H2-L1)
Cross-correlated instr. noise
found Hz
64 hours
(08/23/02 – 09/09/02) S2
< 0.018
(H1-L1) Hz
387 hours
(02/14/03 – 04/14/03) S3 ??
Can account for instrument noise in bounding W
Hz (H1-L1)
Hz (H1-H2)
~350 hrs (H1-L1)
~550 hrs (H1-H2)
(10/31/03 – 01/09/04)
PRELIMINARY
For S2: the positive error is larger in magnitude than the negative error owing to a bias we found in the SW and HW injection pt. Est, that lowered its value. This was later found to be caused by the fact that we were using the PSD of the same interval on which the CC statistic was being computed -> so we graduated to doing sliding PSDs.
*[The LIGO Collaboration, PRD 69, , (2004)]
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21. Summary
The current best IFO-IFO upper-limit (published) is from S1: W < 23 (+/-4.6)
S2 bettered it to ( ) (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 W.
Beyond current analysis:
Search for (f) ~ n(f/f0)n
Targeted searches
Targeted searches: See Albert’s talk in this session
GWDAW-9, Annecy - 12/16/04
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22. H1-L1 analysis: Long-duration features in CC-statistics (S2)
S2 data was
treated as
“playground”
for S3, esp.,
to check for
long-duration
trends.
This study done to check for TRENDS.
In the SGWB analysis, S2 data was treated as a “playground” for the S3.
CENTRAL plot: PSD consistent with WHITE noise; left-end ~1e-5 = 1 day; right-end 1.6e-3 = 10 minutes = 600sec
Bottom plot: PSD is consistent with a Rayleigh distribution in power.
GWDAW-9, Annecy - 12/16/04
LSC/SB