Adventures in LIGO Data Analysis

1. Adventures in LIGO Data Analysis Tiffany Summerscales Penn State University Dissertation Defense

2. Ultimate Goal: Gravitational Wave Astronomy • Gravitational waves = changes in the gravitational field that propagate like ripples through spacetime. • Fundamental prediction of the general theory of relativity. • Stretch and squeeze space transverse to direction of propagation. • Very weak, strain h = L / 2L Dissertation Defense

3. Ultimate Goal:Gravitational Wave Astronomy • Strongest gravitational waves produced by cataclysmic astronomical events. • Core-collapse supernovae • Neutron star & black hole inspirals • Big Bang • Strongest GWs are expected to have h ~ 10-21 Dissertation Defense

4. LIGO: The Detectors Dissertation Defense

5. LIGO: The Science 4 different searches for 4 different sources • Binary Inspirals • NS-NS, NS-BH and BH-BH inspirals • Stochastic Background • Background of grav waves from Big Bang or confusion limited sources • Pulsars • Search for known pulsars • Search for unknown pulsars / spinning NS  Einstein@Home • Bursts • Searches triggered by GRBs • Untriggered searches for any short duration signal • Signal = something that is not noise Dissertation Defense

6. PSU Burst Search Det Char Data Qual Data Cond ETG Vetoes Upper Limit Coincidence Waveform Recovery & Source Science Dissertation Defense

7. Thesis Projects • Remainder of Talk Outline • Brief description of 2 projects • Detector characterization & data quality • Data conditioning • Longer discussion of waveform recovery & source science project Det Char Data Qual Data Cond ETG Vetoes Upper Limit Coincidence Waveform Recovery & Source Science Dissertation Defense

8. Detector Characterization, Data Quality Det Char Data Qual Data Cond ETG Vetoes Upper Limit Coincidence Waveform Recovery & Source Science Dissertation Defense

9. Problem: Bilinear Couplings • Bilinear couplings = modulation of one noise source by another • Out of band noise sources may be converted to in-band noise sources decreasing sensitivity • Couplings appear only in high-order spectra which are computationally expensive • Model of couplings: • Each sample related to one k samples into the past and k samples into the future Dissertation Defense

10. Solution: Poisson Test • Impose a threshold • If data samples are independent, number of above threshold samples above threshold N in time interval T will follow a Poisson distribution • Correspondingly, intervals between threshold crossings t follow an exponential distribution • Apply 2 test to see if t follow an exponential distribution • Bin the t and find expected Ei and observed Oi in each bin • 2 should be equal to number of degrees of freedom  = NB-2 Dissertation Defense

11. Data Conditioning Det Char Data Qual Data Cond ETG Vetoes Upper Limit Coincidence Waveform Recovery & Source Science Dissertation Defense

12. Problem: LIGO data highly colored • All Event Trigger Generators (ETGs) find sections of data where the statistics differ from the noise • Need to identify and remove instrumental artifacts and correlations • Removal of correlations = whitening. (White data has the same power at all frequencies) Dissertation Defense

13. Solution: Data Conditioning Pipeline • Variety of filtering & other signal processing techniques used to remove artifacts and whiten data • Data broken up into frequency bands • Pipeline applied to data from science runs S2, S3, & S4 and conditioned data used in PSU BlockNormal analysis • Pipeline currently being applied to data from S5 Dissertation Defense

14. Waveform Recovery & Source Science Det Char Data Qual Data Cond ETG Vetoes Upper Limit Coincidence Waveform Recovery & Source Science Dissertation Defense

15. Motivation: Supernova Astronomy with Gravitational Waves • Problem 1: How do we recover a burst waveform? • Problem 2: When our models are incomplete, how do we associate the waveform with source physics • Example – The physics involved in core-collapse supernovae remains largely uncertain • Progenitor structure and rotation, equation of state • Simulations generally do not incorporate all known physics • General relativity, neutrinos, convective motion, non-axisymmetric motion Dissertation Defense

16. Problem 1: Waveform Recovery • Problem 1: How do we recover the waveform? (Deconvolution problem) • The detection process modifies the signal from its initial form hi • Detector response R includes projection onto the beam pattern as well as unequal response to various frequencies • Need a method for finding an h which is as close as possible to hi with knowledge only of d, R, and the noise covariance matrix N = E[nnT] Dissertation Defense

17. Maximum Entropy • Possible solution: maximum entropy – Bayesian approach to deconvolution used in radio astronomy, medical imaging, etc • Want to maximize • I is any additional information such as noise levels, detector responses, etc • The likelihood, assuming Gaussian noise is • Maximizing only the likelihood will cause fitting of noise Dissertation Defense

18. Maximum Entropy Cont. • Set the prior • S related to Shannon Information Entropy • Entropy is a unique measure of uncertainty associated with a set of propositions • Entropy related to the log of the number of ways quanta of energy can be distributed in time to form the waveform • Maximizing entropy = being as non-committal as possible about the signal within the constraints of what is known • Model m is the scale that relates entropy variations to signal amplitude Dissertation Defense

19. Maximum Entropy Cont. • Maximizing P(h|d,I) equivalent to minimizing •  is a Lagrange parameter that balances being faithful to the signal (minimizing 2) and avoiding overfitting (maximizing entropy) •  associated with constraint which can be formally established. In summary: half the data contain information abut the signal Dissertation Defense

20. Maximum Entropy Cont. • Choosing m • Pick a simple model where all elements mi = m • Model m related to the variance of the signal which is unknown • Using Bayes’ Theorem: P(m|d) P(d|m)P(m) • Assuming no prior preference, the best m maximizes P(d|m) • Bayes again: P(h|d,m)P(d|m) = P(d|h,m)P(h|m) • Integrate over h: P(d|m) = Dh P(d|h,m)P(h|m) where • Evaluate P(d|m) with m ranging over several orders of magnitude and pick the m for which it is highest Dissertation Defense

21. Maximum Entropy Performance, Strong Signal • Maximum entropy recovers waveform with only a small amount of noise added Dissertation Defense

22. Maximum Entropy Performance, Weaker Signal • Weak feature recovery is possible • Maximum entropy an answer to the deconvolution problem Dissertation Defense

23. Cross Correlation • Problem 2: When our models are incomplete, how do we associate the waveform with source physics? • Cross Correlation – select the model associated with the waveform having the greatest cross correlation with the recovered signal • For two normalized vectors x and y of length L, calculate C() for lags  between –L/2 and L/2 • Select the maximum C() • Gives a qualitative indication of the source physics Dissertation Defense

24. Waveforms: Ott et.al. (2004) • 2D core-collapse simulations restricted to the iron core • Realistic equation of state (EOS) and stellar progenitors with 11, 15, 20 and 25 M • General relativity and neutrinos neglected • Some models with progenitor evolution incorporating magnetic effects and rotational transport • Progenitor rotation controlled with two parameters: rotational parameter  and differential rotation scale A (the distance from the rotational axis where rotation rate drops to half that at the center) • Low  (zero to a few tenths of a percent): Progenitor rotates slowly. Bounce at supranuclear densities. Rapid core bounce and ringdown. • Higher : Progenitor rotates more rapidly. Collapse halted by centrifugal forces at subnuclear densities. Core bounces multiple times. • Small A: Greater amount of differential rotation so core center rotates more rapidly. Transition from supranuclear to subnuclear bounce occurs for smaller  Dissertation Defense

25. Simulated Detection • Select Ott et.al. waveform from model with 15M progenitor,  = 0.1% and A = 1000km • Scale waveform amplitude to correspond to a supernova occurring at various distances • Project onto LIGO Hanford 4-km and Livingston 4-km detector beam patterns with optimum sky location and orientation for Hanford • Convolve with detector responses and add white noise typical of amplitudes in most recent science run • Recover initial signal via maximum entropy and calculate cross correlations with all waveforms in catalog Dissertation Defense

26. Extracting Bounce Type • Calculated cross correlation between recovered signal and catalog of waveforms • Highest cross correlation between recovered signal and original waveform (solid line) • Plot at right shows highest cross correlations between recovered signal and a waveform of each type. • Recovered Signal has most in common with waveform of same bounce type (supranuclear bounce) Dissertation Defense

27. Extracting Mass • Plot at right shows cross correlation between reconstructed signal and waveforms from models with progenitors that differ only by mass • The reconstructed signal is most similar to the waveform with the same mass Dissertation Defense

28. Extracting Rotational Information • Plots above show cross correlations between reconstructed signal and waveforms from models that differ only by rotation parameter  (left) and differential rotation scale A (right) • Reconstructed signal most closely resembles waveforms from models with the same rotational parameters Dissertation Defense

29. Remaining Questions • Do we really know the instrument responses well enough to reconstruct signals using maximum entropy? • Maximum entropy assumes perfect knowledge of response function R • Can maximum entropy handle actual, very non-white instrument noise? Recovery of hardware injection waveforms would answer these questions. Dissertation Defense

30. Hardware Injections • Attempted recovery of two hardware injections performed during the fourth LIGO science run (S4) • Present in all three interferometers • Zwerger-Müller (ZM) waveforms with =0.89% and A=500km • Strongest (hrss = 8.0e-21) and weakest (hrss = 0.5e-21) of the injections performed • Recovery of both strong and weak waveforms successful. Dissertation Defense

31. Waveform Recovery, Strong Hardware Injection Dissertation Defense

32. Waveform Recovery, Weak Hardware Injection Dissertation Defense

33. Progenitor Parameter Estimation • Plot shows cross correlation between recovered waveform and waveforms that differ by degree of differential rotation A • Recovered waveform has most in common with waveform of same A as injected signal Dissertation Defense

34. Progenitor Parameter Estimation • Plot shows cross correlation between recovered waveform and waveforms that differ by rotation parameter  • Recovered waveform has most in common with waveform of same  as injected signal Dissertation Defense

35. Conclusions • Problem 1: How do we reconstruct waveforms from data? • Maximum entropy – Bayesian approach to deconvolution, successfully reconstructs signals • Method works even with imperfect knowledge of detector responses and highly colored noise • Problem 2: When our models are incomplete, how do we associate the waveform with source physics? • Cross correlation between reconstructed and catalog waveforms provides a qualitative comparison between waveforms associated with different models • Assigning confidences is still an open question • Have shown that recovered waveforms contain information about bounce type and progenitor mass and rotation • Gravitational wave astronomy is possible! Dissertation Defense