LIGO Supernova Astronomy with
Maximum Entropy
Tiffany Summerscales
Penn State University
January 12, 2006
Sources & Simulations
1
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
January 12, 2006
Sources & Simulations
2
Maximum Entropy
• Problem 1: How do we recover the waveform?
(deconvolution)
• The detection process modifies the signal from its initial form hi
d  Rhi  n
• Detector response R includes projection onto the beam pattern as
well as unequal response to various frequencies
• Possible solution: maximum entropy – Bayesian
approach to deconvolution used in radio astronomy,
medical imaging, etc.
• Want to maximize
P( h| d , I )  P(d | h, I ) P( h | I )
where I is any additional information such as noise levels, detector
responses, etc
January 12, 2006
Sources & Simulations
3
Maximum Entropy Cont.
P( h| d , I )  P(d | h, I ) P( h | I )
• The likelihood, assuming Gaussian noise is
P(d | h, I )  exp[ 21 ( Rh  d ) T N 1 ( Rh  d )]  exp[  21  2 ( R,h,d,N )]
» Maximizing only the likelihood will cause fitting of noise
• Set the prior
P( h | I )  exp[ S ( h, m)]
» 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 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
• Maximizing P( h| d , I ) equivalent to minimizing
F ( h| d,R,N,m)   2 ( R,h,d,N )  2 S ( h,m)
January 12, 2006
Sources & Simulations
4
Maximum Entropy Cont.
• Minimize
•
•
F ( h| d,R,N,m)   2 ( R,h,d,N )  2 S ( h,m)
 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 about the signal.
• Choosing m
•
•
•
•
•
•
The model m related to the strength 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
exp(S )
exp(   2 / 2)
P
(
h
|
m
)

P(d | h, m) 
2
Dh exp(S )
 Dd exp(   / 2)
Evaluate P(d|m) with m ranging over several orders of magnitude and pick
the m for which it is highest.
January 12, 2006

Sources & Simulations
5
Maximum Entropy Performance,
Strong Signal
• Maximum entropy recovers waveform with only a small amount
of noise added
January 12, 2006
Sources & Simulations
6
Maximum Entropy Performance,
Weaker Signal
• Weak feature recovery is possible
• Maximum entropy an answer to deconvolution
problem
January 12, 2006
Sources & Simulations
7
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
• Gives a qualitative indication of the source physics
January 12, 2006
Sources & Simulations
8
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 progenitors evolved incorporating magnetic effects and rotational
transport.
Progenitor rotation controlled with two parameters: fractional rotational energy  and
differential rotation scale A (the distance from the rotational axis where rotation rate drops to
half that at the center)

E rot
E grav
•
•
•
  r  2
 (r )   0 1    
  A 
1
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 the central core rotates more rapidly.
Transition from supranuclear to subnuclear bounce occurs for smaller value of 
January 12, 2006
Sources & Simulations
9
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
January 12, 2006
Sources & Simulations
10
Extracting Bounce Type
• Calculated maximum 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)
January 12, 2006
Sources & Simulations
11
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
January 12, 2006
Sources & Simulations
12
Extracting Rotational Information
• Plots above show cross correlations between reconstructed
signal and waveforms from models that differ only by fractional
rotational energy  (left) and differential rotation scale A (right)
• Reconstructed signal most closely resembles waveforms from
models with the same rotational parameters
January 12, 2006
Sources & Simulations
13
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.
• Can maximum entropy handle actual, very non-white
instrument noise?
Recovery of hardware injection waveforms
would answer these questions.
January 12, 2006
Sources & Simulations
14
Hardware Injections
• Attempted recovery of two hardware injections
performed during the fourth LIGO science run (S4)
» Present in all three interferometers
» Zwerger-Mϋller (ZM) waveform 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.
January 12, 2006
Sources & Simulations
15
Waveform Recovery
January 12, 2006
Sources & Simulations
16
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
January 12, 2006
Sources & Simulations
17
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 beta as
injected signal.
January 12, 2006
Sources & Simulations
18
Conclusions
• Problem 1: How do we reconstruct waveforms from data?
•
Maximum entropy - Bayesian approach to deconvolution, successfully
reconstructs signals
• 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
• Method successful even with realistic situations such as
hardware injections
• Maximum Entropy References:
Maisinger, Hobson & Lasenby (2004) MNRAS 347, 339
Narayan & Nityananda (1986) ARA&A 24, 127
MacKay (1992) Neural Comput 4, 415
January 12, 2006
Sources & Simulations
19