January 12, 2006Sources & Simulations 1 LIGO Supernova Astronomy with Maximum Entropy Tiffany Summerscales Penn State University

January 12, 2006Sources & Simulations 2 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, 2006Sources & Simulations 3 Maximum Entropy Problem 1: How do we recover the waveform? (deconvolution) The detection process modifies the signal from its initial form h i 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 where I is any additional information such as noise levels, detector responses, etc

January 12, 2006Sources & Simulations 4 The likelihood, assuming Gaussian noise is »Maximizing only the likelihood will cause fitting of noise 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 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 equivalent to minimizing Maximum Entropy Cont.

January 12, 2006Sources & Simulations 5 Maximum Entropy Cont. Minimize  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 Evaluate P(d|m) with m ranging over several orders of magnitude and pick the m for which it is highest.

January 12, 2006Sources & Simulations 6 Maximum Entropy Performance, Strong Signal Maximum entropy recovers waveform with only a small amount of noise added

January 12, 2006Sources & Simulations 7 Maximum Entropy Performance, Weaker Signal Weak feature recovery is possible Maximum entropy an answer to deconvolution problem

January 12, 2006Sources & Simulations 8 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, 2006Sources & Simulations 9 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) 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, 2006Sources & Simulations 10 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, 2006Sources & Simulations 11 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, 2006Sources & Simulations 12 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, 2006Sources & Simulations 13 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, 2006Sources & Simulations 14 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, 2006Sources & Simulations 15 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 (h rss = 8.0e-21) and weakest (h rss = 0.5e-21) of the injections performed Recovery of both strong and weak waveforms successful.

January 12, 2006Sources & Simulations 16 Waveform Recovery

January 12, 2006Sources & Simulations 17 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, 2006Sources & Simulations 18 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, 2006Sources & Simulations 19 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