1. A Waveform Consistency Test for Binary Inspirals using LIGO data
Andres C. Rodriguez
Gabriela González
(Louisiana State University)
Peter Shawhan
(LIGO | Caltech)
Evan Oshsner
(University of Maryland)
LIGO Observatories
I’m going to talk today about a waveform consistency test for binary inspirals using data from the LIGO interferometer.
LSC Inspiral Analysis Working Group
LIGO-G Z
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LIGO Observatories
Hanford: two interferometers in same vacuum envelope (4km(H1), 2km(H2))
*For one of the sources of gravitational waves that could be detected by LIGO I, binary inspirals are one example.
Initial LIGO (40Hz-2 KHz)will be sensitive to gravitational waves emitted as far as the VIRGO cluster for a Msol neutron star inspiralling binary.
Matched filtering is the optimal detection strategy if detector noise is stationary & white
Livingston: one interferometer (4km(L1))
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3. Template based Matched Filtering
Data stream searched using matched filtering between
data & template waveform.
Transform data to frequency domain : s(f)
Calculate template in frequency domain: h(f)
s(t) = n(t) + h(t) ~ ~
n(t), where it is assumed to be stationery
The matched filter output is described by s(f): data , h*(f) : template,
with a weight inversely proportional to the power spectral density of the noise, sn(f): , estimated from nearby data
* Find maxima of over arrival time and phase of |z(t)| above some threshold
***************************
Noise power spectrum estimated from data in each chunk;
Lay out a bank of templates to catch any signal for which each component in the binary system has a mass between 1 Msol and 3 Msol
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4. Template based Matched Filtering
Signal to Noise ():
2:
Where we Characterize event by signal-to-noise ratio, ,
Sigma: variance
Check consistency of signal with expected chirp waveform
By checking that the candidate signals have the expected distribution of signal power as a function of frequency: chi^2
zl(t): template filter output for a given bin, z(L): data filter output
Divide template into p frequency bands which contribute equally to power, on average
Currently use p=16 frequency bands
Next is to define some thresholds:
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5. Template based Matched Filtering - Thresholds
Signal to Noise ():
2 :
» really:
So if  > * and r2 < r2* --> inspiral “trigger”
 > threshold (* )
< threshold (r2*)
For our search, we threshold on the output of the matched filter and demand that the SNR be above some value, SNR*
* We also use a chi^2 threshold, but since
Large signals to have higher c2 values, due to mismatch with discrete template bank
This was also introduced to finding the siulations
And introduce a weighting factor p: number of bins, delta: mismatch in bank - tunable parameter, SNR^2, to avoid losing real signals
* So, let’s look at an example simulation plus data event.
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6. Simulated Inspiral (Injection)*
SNR
r2* r2
SNR and r^2 time series for a:
This particular injection in H1 data was a Msolar template at 8 Mpc Optimally oriented to Hanford 4km .
The thresholds used for the test are denoted by the dashed green lines.
We expected for 1-4*2 Msol binary optimally oriented at have a SNR 8
time (s) to inferred coalescence
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7. time (s) to inferred coalescence
Injection vs. Trigger
Injection
Trigger
SNR r2
* On the left we have an inspiral trigger of the sample template. One obvious thing here is that there is a significant difference between the output of an injection with a trigger.
This trigger did pass the snr and r^2 theshold test, we see a case of excess noise
Looking further I see that there seems to be much fluctuation in r^2 series vs. signal to noise
This conveys the fact earlier that any slight mismatches of template with data over short duration and excess noise could lead to ringing of the matched filter.
time (s) to inferred coalescence
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8. time (s) to inferred coalescence
Injection vs. Trigger
Injection
Trigger r2
Chisq test does not do as well here, can we try to improve on this, as there are significant deviations prior to the trigger:
* Can we find a quantitative measure of the obvious difference between our simulated event and and data event, some sort of check that the detector noise around the time of the trigger was consistent with the stationary noise assumed by the matched filter.
time (s) to inferred coalescence
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The Test
Use the r2 time series as a method to search for excess noise
Impose a “looser” r2 threshold than the search employs
» Count the number of time samples above a given
threshold in a time interval before inferred coalescence
» Count the number of threshold crossings
Time interval “window” chosen to be 2 seconds
Do my triggers behave like software, hardware injections?
» yes ->> keep
» no ->> veto
Design 2 tests
A crossing is defined as an instance of the r^2(t) time series crossing over a given threshold r^2*, in an upward going direction.
One way to decide on thesholds is to look at the distributions:
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2 , r2 distribution
log10(counts)
* One way to decide thresholds is to look at the distribution of r^2.
Blue : rho^2 and r^2 time series for a segment of data: 128 seconds.
Red: gaussian detector noise simulation, variance of 32.
200 r2 70 2
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2 , r2 distribution
log10(counts)
In the case of the r^2 time series,We see already at around 6, there seems to be fluctuations in noise of the detector, there could be a good place to start. 40 r2 12 2
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Injection vs. Trigger
Crossings = 0
Time above Threshold = 0 sec
Crossings = 52
Time above Threshold = 0.9 sec r2
looser r2 threshold
So, let’s go back to my first example and see how this test does?
* Show only 1 threshold, rsq = 6
Crossings, Time above #’3
So, what would we see if we were to apply this to the simulations and playground triggers in S3 H1 data?
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13. Preliminary Results for H1 Injections + Playground Data
r2 threshold = 10
window
H1 Injections
Time above r2 threshold (ms)
This plot shows the distribution of injection and playground triggers for the time spent above threshold vs signal to noise.
* I also plot an approximate linear fit to the injections.
It it is clear here to see that the character of the time spent above my given looser threshold of r^2 = 10, shows a significant difference in lower values of SNR, where all the playground triggers lie.
If I was to use this fit as a possible threshold, then many of the triggers that passed the first step in the pipeline for a single ifo could be discarded.
Also note that the injections with the highest SNR have < 2 crossings, which says that I could possibly use a combination of time above threshold and threshold crossings.
These triggers passed the first 
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14. Preliminary Results for H1 Injections + Playground Data
r2 threshold = 10
H1 Injections
Time above r2 threshold (ms)
This plot shows the distribution of injection and playground triggers for the time spent above threshold vs signal to noise.
* I also plot an approximate linear fit to the injections.
It it is clear here to see that the character of the time spent above my given looser threshold of r^2 = 10, shows a significant difference in lower values of SNR, where all the playground triggers lie.
If I was to use this fit as a possible threshold, then many of the triggers that passed the first step in the pipeline for a single ifo could be discarded.
Also note that the injections with the highest SNR have < 2 crossings, which says that I could possibly use a combination of time above threshold and threshold crossings.
These triggers passed the first 
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15. Preliminary Results: H1 Injections + Playground Data
possibly vetoed
r2 threshold = 10
Crossings < 2
H1 Injections
Triggers
Time above r2 threshold (ms)
This plot shows the distribution of injection and playground triggers for the time spent above threshold vs signal to noise.
* I also plot an approximate linear fit to the injections.
It it is clear here to see that the character of the time spent above my given looser threshold of r^2 = 10, shows a significant difference in lower values of SNR, where all the playground triggers lie.
If I was to use this fit as a possible threshold, then many of the triggers that passed the first step in the pipeline for a single ifo could be discarded.
Also note that the injections with the highest SNR have < 2 crossings, which says that I could possibly use a combination of time above threshold and threshold crossings.
These triggers passed the first 
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Summary and Plans
Preliminary testing r2 thresholds set to 6, 8, & 10.
The test would be able to eliminate many of our loudest
false triggers --> reduces false alarm rate.
Additional tuning using S3 Hardware Injections and
testing on S3 playground data from L1,H2.
Will be incorporated into S3, S4 BNS search, and future
online analysis.
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