Data Analysis Techniques for

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Data Analysis Techniques for Gravitational
Wave Observations
S. V. Dhurandhar
I U C A A Pune, India
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Great strides taken by experimentalists in
improving sensitivity of GW detectors
Technology driven to its limits
Gravitational Wave Data Analysis Important
component of GW observation
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• Signals with parametrizable waveforms
• Deterministic
• Binary inspirals modelled on the Hulse-Taylor
  binary pulsar
• Continuous wave sources
• Stochastic
• Stochastic background
• Unmodeled sources
• Bursts and transients

h 10- 23 to 10-27
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Source Strengths
Binary inspiral
Periodic
Stochastic background
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Detector Sensitivity for the S2 run
http//www.ligo.caltech.edu/lazz/distribution/Da
ta
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Data Analysis Techniques

• Techniques depend on the type of
  source
• Binary Inspirals
• Matched filtering
• Continuous wave signals
• Fourier transforms after applying
  Doppler/spin-down corrections
• Stochastic background
• Optimally weighted cross-correlated data
  from independent detectors
• Unmodeled sources Bursts
• Time-frequency methods Excess power
  statistics

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Inspiraling compact binary
Waveform well modelled - PN approximations
(Damour, Blanchet, Iyer) - Resummation
techniques Pade, Effective one body
extend the validity of the PN formalism (Damour,
Iyer, Sathyaprakash, Buonanno, Jaranowski,
Schafer)
Waveform h Noise Sh (f)
The matched filter
Stationary noise
Optimal filter in Gaussian noise Detection
probability is maximised for a given false
alarm rate
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Matched filtering the inspiraling binary signal
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Detection Strategy
Signal depends on many parameters
Parameters Amplitude, ta , fa , m1 , m2 ,
spins
Strategy Maximum likelihood method

• Spinless case
• Amplitude Use normalised templates
• ta FFT
• Initial phase fa Quadratures only 2
  templates needed for 0 and p/2
• masses chirp times t0 ,
  t3 bank required
• For each template the maximised statistic is
  compared
• with a threshold set by the false alarm rate.
  (SVD and Sathyaprakash)

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Thresholding , false alarm detection
Detection probability
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Parameter Space
Parameter space for the mass range 1 30
solar masses
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Hexagonal tiling of the parameter space
LIGO I psd
Minimal match 0 .97 Number of templates
104 Online speed 3 GFlops
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Inspiral Search (contd)

• Reduced lower mass limit .2 Msun , fs 10
  Hz , then online speed 300 Gflops
• Hierarchical search required
• - 2 step search 2 banks - coarse fine
  (Mohanty SVD)
• Step I coarser bank fewer templates, low
  threshold - high false alarm rate
• Step II follow-up the false alarms by a fine
  search
• - Extended hierarchical search over ta
  and masses
• (Sengupta, SVD, Lazzarini)
  (Tanaka Tagoshi)

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Hierarchical search frees up CPU for searching
over more parameters
LIGO I psd - mass range 1 to 30 solar masses
92 power at fc 256 Hz
Factor of 4 in FFT cost
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Relative size of templates in the 2 stages of
hierarchy
Total gain factor 60 over the flat search
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Multi-detector search for GW signals
GEO 0.6km
VIRGO 3km
LIGO-LHO 2km, 4km
TAMA 0.3km
LIGO-LLO 4km
AIGO (?)km
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Inspiral search with a network of detectors

• Coincidence analysis
• event lists, windows in parameter space
  (S. Bose)
• Coherent search - phase information used
  (Pai, Bose, SVD) (S. Finn)
• Full data from all detectors necessary
  to carry out the data analysis
• A single network statistic constructed to
  be compared with a threshold
• Analytical maximisation over amplitude,
  initial phase, orientation of binary
  orbit
• FFT over the time-of-arrival
• direction search time-delay window
• Filter bank over the intrinsic
  parameters masses metric depends on extrinsic
  parameters
• Computational costs soar up in searching over
  time-delays ( x 103 for LIGO-VIRGO)

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Spin

• Orbital-plane precesses spin-orbit coupling
  - modulates the waveform (Blanchet, Damour, Iyer,
  Will, Wiseman, Jaranowski, Schafer)
• Too many parameters high computational cost
  (Apostolatus)
• Detection template families detection only
  (Buonnano, Chen, Vallisneri)
• - few physical parameters, model well the
  modulation (FF gt .97)
• - automatic search over several (extrinsic)
  parameters no template bank
• For searching single-spin binaries 7 M lt m1 lt
  12 M , 1 M lt m2 lt 3 M
• Templates in just 3 parameters S1 , m1 and
  m2
• 76000 templates needed at .97 match (average)
  - LIGO I sensitivity

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Periodic Sources
Target sources Slowly varying instantaneous
frequency eg.
Rapidly rotating neutron stars h 10-25 ,
10-26 Integration time months, years -
motion of detector phase modulates the signal
Doppler modulation depends on direction of
GW Df (n . v) f0/c 1 kHz wave gets
spread into a million Fourier bins in 1 year
observation time Intrinsic spin down
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Computational cost in searching for periodic
sources
Parameters f0, q, f, spin down
parameters Targeted search known pulsar
window in parameter space, heterodyne
All sky all frequency
search - A CHALLENGE f0 is also a
parameter Number of Doppler corrections
(patches in the sky)
spin-down parameters not included
Brady et al (1998)
Parameter space large typical Tobs 107
secs weak source Effective GW telescope size
2 AU, thus resolution l / D .2 arc sec
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Hierarchical Searches

• Alternate between coherent incoherent
  stages
• Hough transform (Schutz, Papa, Frasca)
  
• - short term Fourier Transforms
• - Look for patterns in peaks in the
  time-frequency plane which
• correspond to parameter values
• - histogram in parameter space do full time
  coherent search around the peak
• Stack and slide search (Brady T.
  Creighton)
• - Given fixed computing power look for an
  efficient search algorithm
• - Divide the data into N stacks, compute
  power spectra, slide and then sum
• Results gain 2-4 in sensitivity 20-60
  hierarchical , 99 confidence
• Classes of pulsars fmax 1
  kHz, t 40 yr fmax 200 Hz, t 1000 yr

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Stochastic Background
Cannot distinguish instrumental noise from
signal with one detector Cross-correlate the
output of two detectors
Q filter
(Allen Romano) (E. Flanagan)
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Stochastic Background
Overlap reduction function g(f)
Non-coincident non-aligned detectors
SNR functional of g(f), WGW (f), P1(f),
P2(f)
LIGO detector pair, Tobs 4 months, PF 5,
PD 95
Initial WGW 10-5 - 10-6
Advanced WGW 10-10 - 10-11
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Unmodeled sources
Burst sources Supernovae, Hypernovae, Binary
mergers, Ring-downs of
binary blackholes
Excess power statistics Sum the power in the
time-frequency window
Anderson, Brady, J.Creighton, Flanagan
E is distributed c2 if no signal and
noise Gaussian
non-central c2 if signal is present
Q How to distinguish non-gaussianity from the
signal? (statistic can detect
non-gaussianity) Network of detectors
autocorrelation v/s cross-correlation Slope
statistic
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Coherent detection of bursts with a network of
detectors
(J. Sylvestre)

• Linearly combine the data with time-delays and
  antenna pattern functions for a given source
  direction
• Polarisation plane Signal lies in the plane
  spanned by h (t) and hx (t)

Y data from a single synthetic detector and P
Y 2 P z h and r2 z /
E(h) and maximise r2 Only 2 parameters
needed in addition to source direction length
ratio, angle Direction to the source can be
found LHV network 1o 10 o
Source model required !
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Dealing with real data

• Algorithms, codes working - yielding sensible
  results
• Real detector noise is neither stationary nor
  Gaussian
• - algorithms have been developed for G S
  noise
• - need to adapt the algorithms to the real
  world
• Vetos
• - Excess noise level veto
• - Instrumental vetos
• For inspirals
• - time frequency veto (Bruce Allen et al)

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Veto for inspirals (Allen et al)
Better vetos follow the ambiguity function
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Clustering of triggers for real events
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Clustering of triggers for real events

• Condensing the cloud of events graph theory?

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Setting upper limits

• Although at this early stage no detection can be
  announced we can place upper limits for
  example on the inspiral event rate
• S1 data from the LIGO detectors gives

A rate gt than above means there is more than
90 probability that one inspiral event will be
observed with SNR gt highest SNR observed in S1
data. (gr-qc/0308069)

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Setting upper-limits (contd.)

• Upper limits can be set for other types of
  sources
• Stochastic - WGW
• Continuous wave sources - h for a given
  source

lt 23 for S1 data L1- H2
Source PSR J19392134 (fastest known rotating
neutron star) located 3.6 kpc
from Earth - fGW 1283.86 Hz
Best upper limit from S1 data (L1) 10-22
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Data Analysis as diagonistic tool

• Detector characterisation
• Understanding of instrumental couplings to
  GW channel
• Calibration
• Line removal techniques adaptive methods

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LISA ESA NASA project
Space based detector for detecting low frequency
GW
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LISA sensitivity curve
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Laser Interferometric Space Antenna (LISA)

• LISA is an unequal arm interferometer in a
  triangular configuration
• LISA will observe low frequency GW in the
  band-width of 10-5 Hz - 1 Hz.

Six Doppler data streams Unequal arms
Laser frequency noise
uncancelled
Suitably delayed data streams form data
combinations cancelling laser frequency noise
(Tinto, Estabrook, Armstrong) Polynomial
vectors in time-delay operators (SVD, Vinet,
Nayak, Pai)
Coherent detection
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LISA data analysis

• Polynomial vectors in 3 time-delay operators
• - Module of syzygies
• 4 generators a , b , g , z
• - linear combinations generate the module
• There are optimal combinations which perform
  better than the Michelson LISA curve
• The z combination can be used to switch off
  GW
• - calibration
• Current effort generalise to moving LISA,
  changing
• arm-lengths etc. (Tinto, SVD, Vinet, Nayak)

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Summary

• Data analysis important aspect of GW
  observation
• Different types of sources need different
  data analysis strategies
• Algorithms must be computationally efficient
  sophisticated analysis is required
• Algorithms, codes now being tested on real
  data
• LISA data analysis combining data streams
  for
• optimal performance