February 20, 2006Dissertation Defense 1 Adventures in LIGO Data Analysis Tiffany Summerscales Penn State University

February 20, 2006 Dissertation Defense 1

Adventures in LIGO Data Analysis

Tiffany Summerscales

Penn State University


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


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


LIGO: The Detectors


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


PSU Burst Search

Det CharData Qual

Data Cond VetoesETG

UpperLimit

WaveformRecovery

&Source Science


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


Detector Characterization, Data Quality

Det CharData Qual

Data Cond VetoesETG

UpperLimit

WaveformRecovery

&Source Science


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


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


Data Conditioning

Det CharData Qual

Data Cond VetoesETG

UpperLimit

WaveformRecovery

&Source Science


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)


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


Waveform Recovery & Source Science

Det CharData Qual

Data Cond VetoesETG

UpperLimit

WaveformRecovery

&Source Science


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


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


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


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



• 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



• 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


Maximum Entropy Performance, Strong Signal

• Maximum entropy recovers waveform with only a small amount of noise added


Maximum Entropy Performance, Weaker Signal

• Weak feature recovery is possible• Maximum entropy an answer to the deconvolution

problem


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


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


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


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)


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


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


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.


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.


Waveform Recovery, Strong Hardware Injection


Waveform Recovery, Weak Hardware Injection


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


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


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!

Documents

February 20, 2006Dissertation Defense 1 Adventures in LIGO Data Analysis Tiffany Summerscales Penn State University