1. The Search For Periodic Gravitational Waves
The Laser Interferometer Gravitational-Wave Observatory
Supported by the United States National Science Foundation
The Search For Periodic Gravitational Waves
Gregory Mendell, LIGO Hanford Observatory
on behalf of the LIGO Scientific Collaboration
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2. Gravitational Waves Gravitation = metric tensor: Weak Field Limit:
Gauge choice:
Quadrupole field:
At the detector:
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3. Laser Interferometer Gravitational-wave Detection
Gravitational-wave Strain:
free masses
LIGO’s arms: L = 4 km.
Sensitivity: L nominally around m. (Depending on frequency and duration of the signal this can be much larger or much smaller!)
suspended test masses
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4. Inside LIGO
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5. Calibration Measure open loop gain G, input unity gain to model
Extract gain of sensing function C=G/AD from model
Produce response function at time of the calibration, R=(1+G)/C
Now, to extrapolate for future times, monitor single calibration line in AS_Q error signal, plus any changes in gain beta, and form alphas
Can then produce R at any later time t, given alpha and beta at t
A photon calibrator, using radiation pressure, gives results consistent with the standard calibration.
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6. Noise Amp. Spectral Density:
[Sn(f)]1/2
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7. Astrophysical Sources
LMXBs
Pulsars
Black Holes
Astrophysical Sources
Need compact objects with large amounts of mass and high speed asymmetric motions to produce detectable gravity waves.
Stochastic Background
Supernovae
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Photos:

8. Neutron and Strange-quark Stars
NASA/CXC/SAO
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9. Continuous Periodic Gravitational-Wave Sources
“Mountain” on star
Free precession
wobble angle 
Wobbling star
Mountain mass & height: MR2
Low-mass x-ray
binary: balance GW torque with accretion torque .
Accreting star
Oscillating star
Unstable vibrations
with amplitude A
A. Vecchio on behalf of the LIGO Scientific Collaboration : GR17 – 22nd July, 2004
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10. The back of the envelope please…
Approximate form of the Quadrupole Formula
For fr = 200 Hz, M = 1.4 M, R = 106 cm, r = 1 kpc = 3  1021 cm:
Triaxial ellipsoid: h ~ (G/c4) MR24fr 2/r ~ 10–26 (for ellipticity  ~ 10–6)
Precession: h ~ (G/c4) sin (2)  MR2fr2/r ~ (for ellipticity  ~ 10-6, wobble angle  = /4)
LMXB Sco-X1: h ~ (balance GW torque with accretion torque)
R-modes: h ~ (G/c4) MA2R2 (16/9) fr2/r ~ 10–26 (for saturation amplitude A ~ 10–3)
Pulsar Glitch h ~ (G/c4) MA2R2fr2/r ~ 10 –32 ( for glitch amplitude A ~ 10–6)
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11. Estimate for population of objects spinning down due to gravitational waves
Blanford (1984) as cited by Thorne in 300 Years of Gravitation; see also LIGO Scientific Collaboration, gr-qc/ , submitted to Phys. Rev. D (2005); and LIGO Scientific Collaboration, S2 Maximum Likelihood Search, in prepartion.
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12. Periodic Signals GW approaching from direction n Solar System
Barycenter
T = arrival time at SSB
t = arrival time at detector
Relativistic corrections are included in the actual code.
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13. Phase and Frequency Modulation
Phase at SSB:
Phase at detector:
Frequency at detector:
df/dt :
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14. Figure: D. Sigg LIGO-P980007-00-D
Amplitude Modulation
Figure: D. Sigg LIGO-P D
Sensitivity is reduced by amplitude modulation. M is rotations about Euler angles to bring h^TT from the wave frame into the lab frame. Theta and phi are unsual polar coordinates, and Psi is the polarization or angle of rotation of the xy plane of the wave frame.
Polarization Angle 
Beam Pattern Response Functions:
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15. Maximum Likelihood
Likelihood of getting data x for model h for Gaussian Noise:
Chi-squared
Minimize Chi-squared = Maximize the Likelihood
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16. Coherent Matched Filtering
Jaranowski, Krolak, & Schutz gr-qc/ ; Schutz & Papa gr-qc/ ; Williams and Schutz gr-qc/ ; Berukoff and Papa LAL Documentation
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17. Time Domain Coherent Search Using Bayesian Analysis
Bayes’ Theorem:
Likelihood:
Posterior Probability:
Confidence
Interval:
Prior
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18. Incoherent Power Averaging
Break up data into M segments; FFT each segment.
Track the frequency.
StackSlide: add the power weighted by the noise inverse.
Hough: add 1 or 0 if power is above/below a cutoff.
PowerFlux: add power using weights that maximize SNR
Frequency
Time
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19. Incoherent Methods Short Fourier Transforms (SFTs): Periodogram:
Hough Number Count:
StackSlide Power:
Power Flux:
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20. Sensitivity of Periodic Search Techniques
False alarm & false dismissal rates determine SNR of detectable signal.
Coherent matched filtering tracks phase.
Incoherent power averaging tracks frequency only.
Optimal search needs 1023 templates per 1 Hz for 1 yr of data; Hierarchical approach needed.
These are for 1% false alarm & 10% false dismissal rates, an average sky position and source orientation. A hierarchical search employs a much large false alarm rate, then use coincidence and other vetoes to maximize sensitivity and narrow follow-up searches to the most promising candidates.
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21. Fake Pulsar vs. Fake Instrument Line
Power averaging reduces noise variance. Pulsar SNR increases with time.
Frequency demodulation broadens instrument line. SNR decreases with time.
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22. All Sky Loudest Events 1000 SFTs, Fake Pulsar & Noise:
DEC (radians)
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RA (radians)

23. All Sky Loudest Events 1000 SFTs, Fake Inst. Line & Noise:
DEC (radians)
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RA (radians)

24. Instrument lines in the S2 data.
LIGO Scientific Collaboration, gr-qc/ , submitted to Phys. Rev. D (2005).
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25. Frequentist Confidence Region
Matlab simulation for signal:
Estimate parameters by maximizing likelihood or minimizing chi-squared
Injected signal with estimated parameters into many synthetic sets of noise.
Re-estimate the parameters from each injection
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Figure courtesy Anah Mourant, Caltech SURF student, 2003.

26. Bayesian Confidence Region
Matlab simulation for signal:
Uniform Prior: x
Figures courtesy Anah Mourant, Caltech SURF student, 2003.
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27. Frequentist Confidence Region For A Source Population From The Loudest Event.
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28. S2 Time Domain Search Results
LIGO Scientific Collaboration & The Pulsar Group, Jodrell Bank Observatory, gr-qc/ , Phys. Rev. Lett. 94 (2005)
Best ho UL = 1.7 x Best ellipticity UL = 4.5 x 10-6 (I = 1045 gcm2)
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29. Detection of Fake HW Injected Signals
LIGO Scientific Collaboration & The Pulsar Group, Jodrell Bank Observatory, gr-qc/ , Phys. Rev. Lett. 94 (2005) ; LIGO Scientific Collaboration gr-qc/ , submitted to Phys. Rev. D (2005).
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30. S2 All-sky Hough Search Results
LIGO Scientific Collaboration, gr-qc/ , submitted to Phys. Rev. D (2005).
Best ho Upper Limit = 4.43 x10-23
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31. Many searches are underway. You can join via Einstein@home:
Like but for LIGO/GEO matched filter search for GWs from rotating compact stars.
Support for Windows, Mac OSX, and Linux clients
Our own clusters have thousands of CPUs.
has many times more computing power at low cost.
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32. Summary and Future Prospects
Initial LIGO is searching for highly deformed, rotating, highly compact stars in the solar neighborhood of the galaxy.
Four Science Runs Have Been Completed. The fifth Science Run is scheduled to begin next month and last over one year at or very near design sensitivity.
Advanced LIGO will search with ~10x the sensitivity of initial LIGO. It will be sensitive to more moderately deformed, rotating, highly compact stars throughout the galaxy.
Hierarchical searches, with coincidence and other vetoes, and follow-up targeted searches, are several ways to improve sensitivity beyond back-of-the envelope estimates presented in this talk.
Advanced LIGO using signal recycling can further improve sensitivity in a narrow frequency band.
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