An Evidence Based Search For Neutron Star Ringdowns

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An Evidence Based Search For Neutron Star
Ringdowns

• James Clark

http//www.astro.gla.ac.uk/jclark
Supervisors Ik Siong Heng, Graham Woan
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Overview

• Objective Construct a (triggered) Bayesian
  search algorithm for neutron star ring-downs
• Neutron star ring-downs
• Bayesian model selection evidence
• Application analysis pipeline
• Preliminary sensitivity estimates
• Future work

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Neutron Star Ring-downs

• Possible GW emission from neutron stars via
  quasi-normal mode (QNM) oscillations. QNMs may
  be excited by (e.g.)
• Birth of neutron star in core-collapse supernova
• Soft gamma repeater (SGR) flares
• highly magnetised NS, B-field stresses induce
  crustal cracking excite QNMs, leading to GWs
  1
• Trigger GRB observations (e.g., SGR1806-20 GEO
  LHO data)
• Pulsar glitches
• Spin-down (and or de-coupling of crust/core,
  internal phase transition) induces crustal
  cracking due to relaxation of ellipticity
  starquake.
• Trigger pulsar timing data

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Neutron Star Ringdowns

• Following a starquake, fundamental oscillatory
  modes excited in the neutron star.
• Waveform damped sinusoid
• For a single detector, no polarisation
  information so model waveform as
• Current ringdown search does not include
  chi-squared cut at the end
• We develop a Bayesian approach to quantitatively
  establish consistency with ringdown waveform in
  follow up investigations
• Can also apply this as a search algorithm

For the f-mode 2
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Neutron Star Ring-downs
Fundamental polar mode f0 and tau eigenvalues
NS model equations of state 3
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Bayesian Model Selection

• Model selection is based on our state of belief
  in one model, relative to another. Use Bayesian
  posterior probability to measure this belief.
• Extend Bayes' Theorem to evaluate posterior for
  a given model, , given some data
  and background information or world view
• is the evidence for
  the model (likelihood, marginalised over some
  model parameters and weighted by the prior)
  

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Bayesian Model Selection

• For competing models , compute the
  odds ratio (ratio of posteriors probabilities)
• Odds ratio consists of 2 terms
• Prior odds express initial bias for one model
  over another
• Bayes' factor ratio of evidences for each model
  ? measure of relative likelihoods of each model.
  Penalises those models with more parameters /
  larger parameter space

Bayes factor
prior odds
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What are our models?Aim is to detect a known
waveform in a stretch of noisy interferometer
data with known properties ? Odds ratio
serves as a detection statistic for
ring-downs versus white noise
Operation Outline
What are our models? Aim is to detect a known
waveform in a stretch of noisy interferometer
data with known properties ? Odds ratio
serves as a detection statistic for
ring-downs versus white noise

• - probability
  that the data contains a ring-down waveform and
  white noise
• - probability
  that the data contains only white noise

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Signal model, likelihood priorsIn
practice, work in frequency domain to allow easy
identification of interesting frequency bands.
Also pre-marginalises over start time of
signal.The correct likelihood function for a
single datum , given an arbitrary signal
power Gaussian noise is a non-central
chi-squared distribution with non-centrality
parameter Here, the signal power is
modelled as having a Lorentzian lineshape and
parameterised by the peak amplitude of the
ringdown , the central frequency and the
decay time - assume uniform, independent
priors on these parameters.
Operation Outline

• Signal model, likelihood priors
• In practice, work in frequency domain to allow
  easy identification of interesting frequency
  bands. Also pre-marginalises over start time of
  signal.
• The correct likelihood function for a single
  datum , given an arbitrary signal power
  Gaussian noise is a non-central chi-squared
  distribution with non-centrality parameter
• Here, the signal power is modelled as having a
  Lorentzian lineshape and parameterised by the
  peak amplitude of the ringdown , the central
  frequency and the decay time - assume
  uniform, independent priors on these parameters.

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Applying Model Selection

• Noise model, likelihood priors
• If are the amplitude phase terms of
  an FFT (i.e., ), we
  can construct normalised variables
• In which case,
  follows a central distribution
• Equivalently, we can use the same likelihood as
  for with a prior on

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Analysis Pipeline
illustrative example spectrogram with ringdown

• Construct spectrogram centered on external
  trigger (e.g., pulsar glitch)
• Compute all possible likelihoods for
  pixels marginalise to get evidences in each
  time bin
• Assume no prior model bias and compute odds
  ratio
• Finally, identify events with

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Analysis Pipeline
illustrative example spectrogram with ringdown

• Construct spectrogram centered on external
  trigger (e.g., pulsar glitch)
• Compute all possible likelihoods for
  pixels marginalise to get evidences in each
  time bin
• Assume no prior model bias and compute odds
  ratio
• Finally, identify events with

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Analysis Pipeline

• Construct spectrogram centered on external
  trigger (e.g., pulsar glitch)
• Compute all possible likelihoods for
  pixels marginalise to get evidences in each
  time bin
• Assume no prior model bias and compute odds
  ratio
• Finally, identify events with

log odds from previous example
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Preliminary Sensitivity Estimates

• Evaluate search sensitivity through multiple
  signal injections of varying SNR into artificial
  white noise (performed in matlab in time-domain)
• Define signal-to-noise ratio for white noise
• Use the following parameter values to generate
  spectrograms

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Preliminary Sensitivity Estimates

• Choice of priors
• Assume parameters are independent so that

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Preliminary Sensitivity Estimates

• Choice of priors
• Assume parameters are independent so that

Note prior ranges FFT parameters highly
tune-able
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Preliminary Sensitivity Estimates
Characterise ( and choose threshold) using
receiver operating characteristic (ROC) curve
Calibrate false alarm probability for different
thresholds by examining results from h0 0
injections...
Note that all injections have same tau (0.2s) and
frequency (2000 Hz)
Decision theory by eye...
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Preliminary Sensitivity Estimates
Find that false alarm probability less than 1
for Log odds threshold 0
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Preliminary Sensitivity Estimates
Efficiency curve
Get 90 detection efficiency for SNR6.3
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Different Waveforms

• Algorithm appears reasonably sensitive to
  ring-downs what happens with other waveforms?
• Inject a sine-Gaussian with the following
  parameters into the 60s of the same white noise
  as before and compare with a ring-down

RD
SG
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Different Waveforms
Spectrogram containing RD SG
RD
SG
Still nothing obviously visible. Note frequency
range due to priors and PSD normalisation
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Different Waveforms
Output from odds algorithm
RD
SG
Confident detection of ring-down but surprisingly
strong detection of sine-Gaussian...
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Different Waveforms

• Explaining the response to different waveforms
• Notice that the noise model assigns very low
  evidence to anything with a high power so that
  glitches can potentially generate high odds,
  given non-zero ring-down evidence
• This effect will be present for anything that
  looks unlike white noise and doesn't have zero
  evidence for a ring-down
• Recall that our world view, , only allows 2
  possibilites, ring-down or noise.
• ? This is overly simplistic if we want to be able
  to discriminate ring-downs from, say,
  sine-Gaussians under these circumstances if the
  data does not resemble white noise, the only
  other possibility is a ring-down.

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Different Waveforms

• Recent idea
• Alter the 'null-detection' model to include the
  possibility of glitch-like signals. ,
  becomes
• In principle, straightforward case of calculating
  evidences for glitch signals and summing the
  results
• a very preliminary examination of evidences for
  sine-Gaussians and ring-downs for previous
  example encouraging...

the data set may consist purely of white noise
or a glitch waveform in addition to white noise
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Future Plans

• Short-term (pre-Christmas / GWDAW)
• Implement 'new world view' to handle (e.g.)
  sine-Gaussians and publish methodology! (started)
• Run code on GEO LIGO data from around
  SGR1806-20 need to know what happens with real
  data... (have data)
• Long-term
• Upper limits on SGR1806-20 based on posterior
  probabilities and/or search sensitivity
• Look at other sources (pulsar glitches, GRB
  ring-downs)
• Potentially have a framework for multi-detector
  analysis by comparing models in different data
  streams (speculative)

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References
1 J. A. de Freitas Pacheco, Astron. Astrophys.
336, 397 (1998), astro-ph/9805321 2 N.
Andersson and K. D. Kokkotas, Mon. Not. R.
Astron. Soc. 299, 1059 (1998), gr-qc/9711088 3
O. Benhar, V. Ferrari, and L. Gualtieri, Phys.
Rev. D 70, 124015 (2004), astro-ph/0407529
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Templates Evidence Behaviour
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Templates Evidence Behaviour
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Templates Evidence Behaviour