Time domain targeted pulsar search: Algorithm and Results

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Time domain targeted pulsar searchAlgorithm
and Results

• Réjean Dupuis Graham Woan
• IGR Glasgow

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Reasoning behind a time-domain analysis

• Searches targeted at known radio pulsars are not
  computationally expensive, so we can trade some
  efficiency for clarity and flexibility.
• We know the IFO sensitivity is non-stationary and
  that there are gaps and dropouts. This evolution
  is handled naturally in a time domain analysis.
• The data handling/transport problem can be
  efficiently reduced by heterodyning (mixing) the
  raw h(t) channel at near the expected pulsar
  signal frequency, and band-limiting the result to
  just a few Hz (gaining a factor of 1000 in
  compression).
• Pulsars with complex phase evolutions (especially
  the Crab) can be processed relatively simply.

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The signal a reminder

• We use the standard model for the detected strain
  signal from a non-precessing neutron star

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Data heterodyning

• Data are heterodyned in two stages
• A coarse complex heterodyne at a fixed frequency,
  to reduce the effective sample rate to 4 Hz and
  allow for easy data transportation.
• A fine complex heterodyne to take account of
  pulsar slowdown and Doppler shift, reducing its
  apparent frequency to 0 Hz, and reducing the data
  rate to (e.g.) 1 sample per 60 s.
• Accomplished with LAL routines LALCoarseHeterodyne
  ToPulsar and LALFineHeterodyneToPulsar, which
  include robust low-pass filtering routines to
  protect from strong out-of-band signals.
• Between these stages, the noise level is
  estimated from the variance of the data over each
  60 s period (assumed constant over this period).

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Heterodyne output

• Once fully heterodyned, the complex times series
  simply evolves with the IFO antenna pattern as
  the pulsar moves across the sky, and we fit a
  model to this signal of the form
• where a is the vector of the 4 unknown
  parameters.

1 day
Rey(t)
GPS seconds
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Model fitting 1

• The data from the heterodyne code, Bk, are
  modelled as Gaussian, with variances estimated
  from each nominal 60 s stretch. This is fair
  provided the central limit theorem holds and the
  data are stationary over the period.
• We take a Bayesian approach, and determine the
  joint posterior distribution of the probability
  of our unknown parameters, using uniform priors
  on over their
  accessible values, i.e.

posterior prior likelihood
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Model fitting 2

• The likelihood (that the data are consistent with
  a given set of model parameters) is proportional
  to exp(-?2 /2), whereThe sum is only over
  valid data, so dropouts and gaps are dealt with
  simply.
• Finally we marginalize over the uninteresting
  parameters to leave the posterior distribution
  for the probability of h0

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Upper limit definition

• The 90 confidence upper limit is set by the
  value h90 satisfying

h90
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Detection

• A detection would appear as a maximum
  significantly offset from zero. Note that an
  upper limit can still be defined(!)

Most probable h0
h90
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Validation 1

• Marginalised posterior pdf for h0, resulting from
  an end-to-end test using 24 h of stationary,
  fake, Gaussian noise.
• signal h0 0, f01234 Hz, RA dec 0, ? 0,
  ? ?/4 noise Sh 9 x 10-17 Hz-1/2 (?
  10-18 at 16384 samples/s)

• Note a naïve calculation would give a 1? upper
  limit of The apparent loss on sensitivity of
  12 is due to the precise definition of h0 and
  the attenuation from the mean beam for this
  (low dec) test source.

1? upper limit 3.1 x 10-22
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Validation 2

• Marginalized posterior pdf for h0, using 24 h of
  fake data and Gaussian noise.
• signal h0 6 x 10-21 , f01234 Hz, RA dec
  0, ? 0, ? ?/4 noise Sh 9 x 10-17
  Hz-1/2 (? 10-18 at 16384 samples/s)

• (The multiple peaks are an temporary artefact of
  the fake data generation method.)

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GEO E7 results 1

• Within a 4 Hz band around 1283 Hz (PSR
  J19392134), the noise is highly non-stationary,

• but the standard deviation weighted data appears
  Gaussian over 60 s (above). In fact we need to
  drop to 10 s to resolve stationarity in these
  data.

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GEO E7 results 2

• Assuming just 10 s stationarity, the noise is
  time-resolved and a consistent upper limit for
  PSR J19392134 can be determined

90 confidence upper limit h0 lt 4.5 x 10-20
Marginalised posterior probability
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Prospects for GEO S1

• S1 is more sensitive than, but possibly less
  stationary than, E7

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Morals and intentions

• Method works and can handle some truly horrific
  conditions.
• A good understanding of the noise is vital to the
  definition of a reliable upper limit.
• Monte Carlo runs will give a final check.
• Method still needs to be applied to LIGO data.