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 4

LIGO: The Detectors

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 31

Waveform Recovery, Strong Hardware Injection

February 20, 2006 Dissertation Defense 32

Waveform Recovery, Weak Hardware Injection

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!