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February 20, 2006 Dissertation Defense 1
Adventures in LIGO Data Analysis
Tiffany Summerscales
Penn State University
February 20, 2006 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
4~Rc
Gh
February 20, 2006 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
February 20, 2006 Dissertation Defense 5
LIGO: The Science
• 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
4 different searches for 4 different sources
February 20, 2006 Dissertation Defense 6
PSU Burst Search
Det CharData Qual
Data Cond VetoesETG
UpperLimit
WaveformRecovery
&Source Science
February 20, 2006 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 CharData Qual
Data Cond VetoesETG
UpperLimit
WaveformRecovery
&Source Science
February 20, 2006 Dissertation Defense 8
Detector Characterization, Data Quality
Det CharData Qual
Data Cond VetoesETG
UpperLimit
WaveformRecovery
&Source Science
February 20, 2006 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
)tan()tan1(
12
kjjjj nnnx
February 20, 2006 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
BN
i i
ii
E
EO
1
22
||
||
/
!
)/(),|( T
N
eN
TTNP
/)/exp()|( ttP
February 20, 2006 Dissertation Defense 11
Data Conditioning
Det CharData Qual
Data Cond VetoesETG
UpperLimit
WaveformRecovery
&Source Science
February 20, 2006 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)
February 20, 2006 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
February 20, 2006 Dissertation Defense 14
Waveform Recovery & Source Science
Det CharData Qual
Data Cond VetoesETG
UpperLimit
WaveformRecovery
&Source Science
February 20, 2006 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
February 20, 2006 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]
nRhd i
February 20, 2006 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
)|(),|(),|( IPIPIP hhddh
)],,,(exp[)]()(exp[),|( 2211
21 NdhRdRhNdRhhd TIP
February 20, 2006 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
)|(),|(),|( IPIPIP hhddh
)],(exp[)|( mhh SIP
i i
iiiii
iii m
hhmmhsS
2log2),(),( mh
2/122 )4( iii mh
February 20, 2006 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
),(2),,,(),,,|( 2 mhNdhRmNRdh SF
February 20, 2006 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
)2/exp(
)2/exp(),|(
2
2
dmhd
DP
)exp(
)exp()|(
SD
SP
hmh
February 20, 2006 Dissertation Defense 21
Maximum Entropy Performance, Strong Signal
• Maximum entropy recovers waveform with only a small amount of noise added
February 20, 2006 Dissertation Defense 22
Maximum Entropy Performance, Weaker Signal
• Weak feature recovery is possible• Maximum entropy an answer to the deconvolution
problem
February 20, 2006 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
0,)(1
L
jjj yxC 0,)(
||
1||
L
jjj yxC
February 20, 2006 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
|| grav
rot
E
E
12
0 1)(
A
rr
February 20, 2006 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
February 20, 2006 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)
February 20, 2006 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
February 20, 2006 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
February 20, 2006 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.
February 20, 2006 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.
February 20, 2006 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
February 20, 2006 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
February 20, 2006 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!