All-sky search for gravitational waves from neutron stars in binary systems

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All-sky search for gravitational waves from
neutron stars in binary systems

• strategy and algorithms
• H.J. Bulten

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analysis of PSS from binaries

• thesis work of Sipho van der Putten
• Sipho van der Putten, R. Ebeling (siesta)
• staff involved JFJ van den Brand, Th. Bauer,
  HJB, T.J. Ketel, S. Klous (grid)
• theory dept. G. Koekoek and J.W. van Holten

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motivation binary systems

• Virgo/Ligo better sensitivity at higher
  frequency (gt10 Hz)
• fixed quadrupole deformation
• most high-frequency neutron stars are in binary
  systems
• spin-up via gas transfer

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motivation

• Brady et al.
• PRD57,2101

binary
old
new?
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solitary neutron stars

• solitary neutron star Doppler shifts from earth
  movement
• Hierarchical search possible, T 1h (Rome group,
  e.g. Astona, Frasca, Palomba CQG 2005.)
• signal-to-noise

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solitary neutron stars

• alternative F-statistics approach (Ligo,
  Jaranowski et all PRD58, 063001)
• produce templates that remain in phase over the
  template search time
• parameters
• solitary neutron stars all-sky search
• many templates needed, e.g. Brady et al. PRD61,
  082001
• coherent all-sky search of length of 0.5days
  would take 10,000 Tflops (fmax1000 Hz)
• smaller spin-down, fmax200 Hz 5 days

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Binary Kepler orbitals

• ellipse
• We want to analyze
• orbital periods from 2 hours infinite
• masses companion star up to 15 solar masses
• eccentricities up to about 0.7
• frequency shifts up to 0.3, frequency changes
  df/dt up to 10-6 s-2
• 1 mHz shift in 1 second, at f1000Hz

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frequency shifts
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frequency derivative
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frequency shifts
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coherence

• phase signal
• signal should remain in-phase ,e.g. maximally 90
  deg. out of phase anywhere during observation
  time frequency within ½ bin - 1/(2Tobs)

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binary neutron stars

• how many extra parameters?
• e.g. orbital period gt2 hours, eccentricity
  lt0.6, mass companion lt15 solar masses,
  frequency lt1000 Hz
• coherent phase distance to neutron star within
  75 km w.r.t. template anywhere during the
  coherence time.
• all power coherent within 1 FFT-bin Tmax 30s
• FFT length 1 hour signal spreads over 4000
  bins.
• Tobs 1 hour
• detectable difference in orbital period 70 ms
• a factor of 100,000 in parameter space to scan
  all orbital periods between 2 and 4 hours in a
  blind search

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binary neutron stars

• additional parameters
• even with Tobs 1 hour, at least 100 billion
  times as many templates are required to keep the
  phase of the filter coherent for all
  possibilities within the boundaries
• T_orbit gt 2hour
• 0lt eccentricity lt 0.6
• all orientations of semi-major and semi-minor
  axes
• all starting phases in orbital
• up to 1000 Hz g.w. frequencies
• full parameter scan is not feasible.

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binary neutron stars

• different set of filters parameterize the phase
  as a function of time!
• assume that within Tobs, the frequency can be
  described by a second-order function of time
• third-order effects are assumed to be negligible.
  
• scan for presence of signal by calculating the
  correlation with the template

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Correlation

• Correlation is given by
• presence of signal defined by overlap with
  filter.
• data is not periodic make filter equal to zero
  for last N/2 samples and shift it maximally N/2
  samples to the right
• FFT interleave, to cover full dataset

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Filter search
data, split in overlapping periods
Filter zero-padded for half length check
correlations from t0 to t ½T (FFT1) check
correlations from t ½T to t1T (FFT2) check
correlations from t1T to t 1½T (FFT3) maximum
overlap amplitude and time known
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Filter search
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Example Filters
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parameter space

• phase should be given by filter
• coherent times up to about T500 seconds
• for times lt500 seconds, fourth-order corrections
  due to orbital movements are small
• quadratic change of frequency can be
  parameterized with about 120 parameters
• linear change of frequency

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Phase parameters

• for coherent times up to 500 seconds, the
  frequency should be accurate within about 1mHz.
• phase description of data
• about 10 phases
• about 1 million values of f0
• about 500 values of alphadf/dt
• about 120 values of beta.
• however scan with FFT template
• in time direction can be determined
• templates can be re-used
• 600,000 templates reduce to about 5000

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shifting in time

• shifting a filter in time by a lag tau gives a
  filter with parameters
• you do not have to apply filters with with

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shifting in frequency

• frequency changes are smaller than 1 Hz within
  the set of filters
• produce filters in a small frequency band, a
  complete set for 1 fixed value of f(t0).
• reduction of a factor of
• Fourier-transform them
• heterodyne data, or alternatively compare the
  filter in frequency domain with the appropriate
  frequency band of the FFT of the data

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Scan

• Step in frequency if the filter has small
  frequency dependence, you have to step 1
  frequency bin. So a filter with a constant
  frequency is applied (Fmax/binwidth) times (e.g.
  1 million times for an FFT of 1000 second)
• if the filter has large linear or quadratic
  dependence, you can step with a stepsize
• total scans needed to analyze 0 - 1000 Hz, 1000
  seconds
• about 10,000 filters suffice.
• about 300 million correlations in total (300
  million FFTs)
• a few days of CPU-time on a single CPU, current
  desktop

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Hits

• a hit overlap is larger than pre-defined
  threshold
• PSD from FFT from complete set (needs to be
  optimized) sets noise threshold
• normalize data in frequency domain to have mean
  amplitude of in each bin

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Procedure tests

• we tested with white noise, 4096 samples per
  second, 1024 seconds FFT
• filters can pick signal with 20 times smaller
  amplitude (time domain) out of the noise (Total
  power signal is 800 times smaller than that of
  noise)
• overlap filter-signal is 1.0 if signal is equal
  to filternoise amplitude is reproduced
  correctly.
• frequency is reproduced correctly (filter gives
  only hits in the right frequency band)
• average overlap between filters is about 0.43 (at
  same frequency)

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First tests

• spectrum Gaussian-distributed noise with mean
  zero and amplitude
• one-sided PSD of
• signals 10 binary neutron stars
• frequency between 200 and 250 Hz
• random angles, deformations, etc
• maximum amplitude lt 10-23, total power of 10
  signals is 0.2 percent of the power in the noise
• FFT lenght 1024 seconds, 2048 samples/sec.
• 30 FFT sets (about 5 hours)

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Overlap of filters, only noise
maximum correlation for all filters applied
between 0 and 1000 Hz (81.5 million FFT
products, 4096 lags per filter)
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Overlap of filters with signal
maximum correlation with signal for all filters
applied between 0 and 1000 Hz (81.5 million FFT
products)
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signal-to-noise
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Power spectral density
PSD signalnoise
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PSD, signal only
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PSD, signal only
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Search results

• 30 FFTs, about 5h of data
• analyzed between 100 and 500 Hz
• 2405 different filters
• about 1.3 billion filter multiplications, 28731
  hits (10 pulsarsnoise)
• pulsars only 14972 hits

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Search results, all hits
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Search results
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Alternative cut on power
Cut 4 sigma on power
FFT number

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Alternative cut on power
Cut 4 sigma on power
7649 hits between 450 and 460 Hz
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highest PSD in data
FFT number

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PSD signal only

• signal highest PSD
• still data spread out over about 30 bins

FFT number

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Summary

• we propose an all-sky search for gravitational
  waves from neutron stars in binary systems
• a complete set of filters (complete to third
  order in frequency) is used to parameterize the
  signal.
• the correlation of the filters with the data
  yield
• time of overlap with better resolution than
  FFT-time
• amplitude and frequency of signal
• first and second derivative of the frequency as
  function of time

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Summary

• after first step, amplitude and frequency of the
  signal can be parameterized as a function of
  time.
• candidates can be followed from 1 FFT to the next
• Filters can be produced in a small frequency band
• compared to different frequency bands in the data
• stepsize in frequency determined by frequency
  dependence of filter
• amount of CPU time is manageable