Topics in Data Analysis from Gravitational Wave Interferometers, including a Cross Correlation Stati

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Topics in Data Analysis from Gravitational Wave
Interferometers, including a Cross Correlation
Statistic to Identify Co-incident bursts

• Brief Introduction to LIGO
• What is a Gravitational Wave?
• Primary Types of GWs
• Gravitational Wave Interferometers
• LIGO and its sister projects
• LISA
• GW Bursts
• Cross Correlation Statistic to Identify
  Co-incident Bursts
• Simulation of Time Delay Interferometry in LISA

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Gravitational Waves

• Static Gravitational fields are described in
  General Relativity as a curvature or warpage of
  space-time, changing the distance between
  space-time events
• Special Relativity requires that news about
  changes in the gravitational field cannot travel
  faster than the velocity of light (c)
• The news about the changing gravitational field
  propagates outward as gravitational radiation a
  wave of spacetime curvature.
• When a plane polarized Gravitational Wave passes
  through space, it stretches and squeezes space
  along mutually perpendicular axes which form a
  plane orthogonal to the direction of propagation
  of the GW.
• These strecthes and squeezes can be expressed as
  a strain in space .
• Plane polarized Gravitational Waves come in 2
  polarizations the Polarization and the x
  polarization.

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Types of extra-terrestrial GW emissions

• Bursts
• Collapse of a star into a Neutron star or Black
  Hole
• Fall of stars and small black holes into super
  massive black holes
• Asymmetric supernova explosions
• Chirps
• Coalescence of compact binaries
• Periodic Waves
• Rotating Neutron and Binary Star systems
• Stochastic Waves
• Primarily from the Big Bang

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Gravitational Wave Interferometers
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LIGO and its sister projects
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Laser Interferometer Space Antenna (LISA)

• Constellation of 3 spacecraft
• Able to search for low frequency gravitational
  waves owing to lack of seismic noise
• Primarily searches for low frequency periodic
  waves from compact binaries, neutron stars and
  black holes

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GW Bursts

• The waveform of a GW Burst depends primarily on
  the dynamics of the source and therefore, burst
  waveform templates are difficult to create and
  hence Matched filtering techniques cant be
  reliably employed.
• Classical Methodology adopted to detect Bursts by
  LDAS (LIGO Data Analysis System)
• LDAS contains algorithms like Slope, tfClusters
  and Power (also called DSOs or Event Trigger
  Generators) which identify peaks of excess power
  in sensitive frequency bands of the data-stream
• Upon identification, the algorithms fill up a
  meta-database with such candidate burst triggers
• Each burst trigger contains information about the
  central frequency, amplitude, start-time and
  duration of the corresponding burst.
• To identify co-incident bursts, we require the
  candidate burst triggers to have similar central
  frequencies, amplitude, duration and
  appropriately delayed start-times (which is 10 ms
  for the 2 LIGO observatories at Hanford and
  Livingston)

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Cross-correlation of coincident burst data

• After the search DSOs have identified data
  segments in which a burst is apparently present,
• And processing of the triggers identifies H2/L1
  pairs which are coincident in time (to the level
  of resolution of the DSOs, eg, 1/8 second for
  tfclusters, ie, not as good as the required ?10
  msec),
• And trigger level consistency cuts are made
  (overlapping frequency band, consistent
  amplitudes, etc)
• We still may have to reduce the coincident fake
  rate.
• SO, go back to the raw data and require
  consistency
• We seek a statistical measure which
• reduces false coincidences significantly while
• maintaining very high efficiency for even the
  faintest injected burst which triggers the DSOs
• And can provide a better estimate of the
  start-time coincidence
• We require this statistic to be robust even when
• the two IFOs have very different sensitivities
  as well as
• when there is a time delay of /- 10 ms between
  the injected signals in the two IFOs.

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Cross-correlation statistic

• Let
• X(t) DT seconds of data from H2
• Y(t) DT seconds of data from L1.
• CXY(f) Coherence function between X, Y.
• abs(CSD(X, Y)2)/(PSD(X)PSD(Y)
  )
• (CSD Cross Spectral Density, PSD Power
  Spectral density)
• Consider the statistic CCS Integral (CXY,
  fmin, fmax)
•  (or) since we are sampling the data at discrete
  time intervals, we use the following discrete
  analog of ()
• CCS S
  CXY(f)Df (between fmin and fmax)
•  
• In our analysis, we use the value of DT 1
  second.
• The statistic will depend upon the value of DT
  and this dependence needs to be explored.

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Evaluation of the CCS

• The idea behind this exercise is to determine the
  distribution of the CCS statistic before and
  after signal injection
• and thereby hope to find a value of the CCS
  statistic which can then be used as a test to
  identify coincident bursts.
• We would like this statistic to have a high
  efficiency of detection while maintaining a low
  fake rate.

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Determination of optimal values of fmin and fmax

• We are considering ZM waveforms at a distance of
  2 parsec (limit of sensitivity during E7)
• To find the optimal range of values of fmin and
  fmax, we plot CXY(f) for the case when there are
  no injected burst signals in H2, L1 and compare
  it with the case when we inject signals.

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CCS for different ZM waveforms
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Limits of integration

• From the above plots, it is clear that the region
  of interest lies between 250 Hz 1000 Hz.
• This is consistent with the fact that ZM
  supernovae have little power beyond 1000 Hz and
  the fact that LIGO has its peak sensitivity in
  this region.
• The plots also indicate that the CCS statistic
  would be of little use in detecting some weak
  waveforms (eg A1B1G5).
• Details
• The raw E7 data has been whitened and resampled
  to 4096 Hz
• The injected signals have been filtered through
  the calibrated transfer function (strain ?
  LSC-AS_Q counts), then whitened and resampled
  like the data.
• So far, we have been using 300-1000 Hz as our
  limits of integration.

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Procedure for evaluating CCS

• We take N (N 360 in our case) seconds of data
  from L1 and H2.
• We break the N second dataset into (N/DT)
  intervals of length DT each (DT 1 second in our
  case).
• We then estimate the distribution of the CCS
  statistic on the raw data by forming (N/ DT)2
  coincidences between them and computing the CCS
  statistic between the DT second intervals thus
  generated.
• We histogram the results to arrive at the
  distribution of the CCS statistic on the raw
  data.  
• Since the CCS statistic test will be used only on
  the data sections that trigger the DSOs, we
  perform the same analysis on the data sections
  between the times t and t DT where t
  corresponds to the time identified by the DSO as
  the start time of the burst which triggered the
  DSO.
• We then inject ZM waveform signals in the (N/DT)
  intervals of length DT.
• The distribution of the CCS statistic after
  signal injection is similarly studied.
• We then inject the ZM waveform signals with a
  time delay of 10 ms (H2/L1 light travel time)
  between them and estimate the CCS statistic by
  the above method.

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Summary of results
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Observations

• the distribution of the CCS statistic on the data
  sections identified by the DSOs as containing
  bursts is very similar to the distribution of the
  CCS statistic on random DT seconds of data from
  L1 and H2.
• The peak of the CCS statistic distribution when
  the signal between H2, L1 is delayed by 10 ms is
  occurs at a slightly lower bin than the peak of
  the distribution when there is no delay.
• However, we can still produce an efficient value
  of the CCS statistic which maintains high rates
  of efficiency while minimizing the fake rate.

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Cut on CCS. Efficiency vs fake rate reduction
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Some things to be done

• Estimate the CCS between 250-1000 Hz. We expect
  the results to be better than the results
  obtained above (using 300-1000 Hz).
• Explore the dependence of the CCS on DT. Can we
  estimate DT to ?10 msec or better?
• Explore other waveforms
• S1 data
• Automate, using LDAS (or DMT).

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Laser Frequency and Spacecraft motion noise in
LISA

• The dynamics of the LISA constellation is such
  that it is impossible to maintain equal arm
  lengths between LISA spacecraft.
• Laser frequency fluctuations are therefore not
  cancelled.
• The NdYAG Laser to be used in LISA offers a
  frequency stability of 10-13Hz1/2
• The GW sources for LISA cause fluctuations of the
  order of 10-20Hz1/2
• Similarly, random motions of the optical benches
  induce Doppler shifts (of similar order as the
  Laser Frequency Fluctuations).
• These noise sources must therefore be cancelled
  up to at least second order for effective
  performance of the LISA constellation.

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The LISA System

• Each vertex spacecraft contains two rigid optical
  benches (the benches are attached to each other
  by an optic fiber) shielding two (almost)
  inertial proof masses.
• Each optical bench has its own laser, which is
  used to both exchange signals with one of the
  distant spacecraft and also to exchange signals
  with the adjacent optical bench.
• Thus, there are six optical benches, six lasers,
  and a total of twelve Doppler time series
  observed.
• An outgoing light beam transmitted to a distant
  spacecraft is routed from the laser on the local
  optical bench using mirrors and beam splitters
  this beam does not interact with the local proof
  mass.
• Conversely, an incoming light beam from a distant
  spacecraft is bounced off the local proof mass
  before being reflected onto the photo-detector
  where it is mixed with light from the laser on
  that same optical bench.
• Beams between adjacent optical benches however do
  precisely the OPPOSITE.

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Notation

• Y31 is the fractional (or normalized by center
  frequency) Doppler series derived from reception
  at spacecraft 1 with transmission from spacecraft
  2. Similarly, Y21 is the Doppler time series
  derived from reception at spacecraft 1 with
  transmission at spacecraft 3.
• We also use a useful notation for delayed data
  streams Y31,23 Y31 (t-L2 - L3)
• Six more Doppler series result from Laser beams
  exchanged between adjacent optical benches these
  are similarly indexed as Zij
• The fractional frequency fluctuations of the
  laser on the optical bench on spacecraft 1 which
  exchanges signals with spacecraft 2 is labeled
  C1.
• The random velocity of this optical bench is
  labeled V1 while the random velocity of the proof
  mass associated with this bench is labeled v1.
• The shot noise contribution to the Doppler time
  series Yij is denoted by Yijshot , while the
  effect of a passing gravitational wave on the
  time series Yij is denoted by YijGW.

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Output at the Photodetectors

• Y21 C3,2 n2. V3,2 2n2.v1 - n2.V1 - C1
  Y21GW Y21shot
• Z21 C1 2n3.(v1 V1) C1
• Y31 C2,3 n3. V2,3 - 2n3.v1 n3.V1 - C1
  Y31GW Y31shot
• Z31 C1 - 2n2.(v1 V1) C1

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Noise Cancelling Combinations

• Work of Armstrong, Estabrook and Tinto (JPL)
• By taking appropriate combinations of the Doppler
  time series, we can cancel the Laser Frequency
  Fluctuations and Spacecraft motion effects up to
  second order
• In fact, complete cancellation of the Laser
  frequency noise is possible if we accurately knew
  the arm-lengths
• Tinto, Estabrook and Armstrongs analysis shows
  that these combinations are highly effective when
  the arm-lengths are known with realizable
  precision.
• Examples
• X Y32, 322 Y23,233 Y31,22 Y21,33 Y23,2
  Y32,3 Y21 Y31 (1/2) ( - Z21,2233
  Z21,33 Z21,22 Z21) (1/2) ( Z31,2233
  Z31,33 Z31,22 Z31)
• a Y21 Y31 Y13,2 Y12,3 Y32,12 Y23,13 -
  (1/2) (Z13,2 Z13,13 Z21 Z21,123 Z32,3
  Z32,12) (1/2) (Z23,2 Z23,13 Z31
  Z31,123 Z12,3 Z12,12)

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Details of the Simulation

• Doppler data received at each spacecraft has been
  preprocessed
• Distances between the spacecraft (L1, L2 and L3)
  are precisely known
• The simulation therefore deals with a system
  which consists of three almost but not precisely
  stationary spacecraft (ie each spacecraft is
  assumed to have a small random velocity), the
  spacecraft forming the vertices of a triangle
  with known sides.
• The Doppler data represented in this simulation
  is normalized by central frequency (300 THz
  corresponding to 1 mm wavelength laser light from
  the NdYAG Lasers).
• Noise spectra obtained from LISA Pre-Phase A
  report.
• The data for the simulation was generated and
  sampled at 2 Hz since LISA is maximally sensitive
  between 10-4 Hz 1 Hz.
• Since we require a frequency resolution of at
  least 10-4 Hz, the simulation was executed to
  obtain a week (604800 seconds) of LISA data and
  the power spectrum of the gathered data in the
  combinations described above was then estimated.
• The simulation in its current state accepts only
  elliptically polarized sinusoidal gravitational
  waves (LISA sensitivities have been traditionally
  given for sinusoidal waves).
• Simulation created in Matlab

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Results
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Results
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Results
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Results
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Results
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Results
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Results
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Results
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Conclusions

• The noise canceling combinations a and X
  successfully cancel the laser frequency
  fluctuation noise and spacecraft motion effects
  to acceptable levels while allowing us to detect
  the gravitational wave.
• The noise spectra obtained from the simulation
  are identical to the spectra obtained by
  Armstrong, Estabrook and Tinto through an
  analytic calculation of the appropriate transfer
  functions.
• Thus, the simulation quantitatively demonstrates
  that Time Delay Interferometry can be
  successfully implemented in LISA to recover the
  gravitational wave signal even when the system is
  swamped by laser frequency fluctuation noise and
  spacecraft motion effects. The gravitational wave
  sensitivity of LISA is then limited by
  acceleration noise (at low frequencies) and shot
  noise (at high frequencies).