Transcript Document

Data Analysis Techniques for
Gravitational Wave Observations
S. V. Dhurandhar
I U C AA
Pune, India
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
• 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
Source Strengths
Binary inspiral :
 23
h ~ 2.5  10
 M 


M
 sun 
5/3
1

  fa 
r

 

100
Mpc
100
Hz

 

2/3
Periodic:
h ~ 1.9 1025

 f 
I
 45

2 
10
gm
.
cm
500
Hz



2
1
 r    

  5 
10
kpc

 10 
Stochastic background:
~
h( f ) ~ 10 26
 f 


10
Hz


3 / 2
  GW ( f ) 
 1012 


1/ 2
Detector Sensitivity for the S2 run
*http://www.ligo.caltech.edu/~lazz/distribution/Data
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
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)
c( )   x (t ) q(t   ) dt
The matched filter :
Stationary ~noise:
h( f )
q( f ) 
Sh ( f )
~
Waveform: h
Noise: Sh (f)
Optimal filter in Gaussian noise:
Detection probability is maximised for a given false alarm rate
Matched filtering the inspiraling binary signal
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: 0 , 3 bank required
• For each template the maximised statistic is compared
with a threshold set by the false alarm rate. (SVD and Sathyaprakash)
Thresholding , false alarm & detection
Detection probability
Parameter Space
Parameter space for the mass range 1 – 30 solar masses
f a  40 Hz
Area : 8.5 sec2
Hexagonal tiling of the parameter space
LIGO I psd
Minimal match: 0 .97
Number of templates: ~ 104
Online speed: ~ 3 GFlops
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)
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
Relative size of templates in the 2 stages of hierarchy
Total gain factor
60 over the flat
search
Multi-detector search for GW signals
GEO: 0.6km
LIGO-LHO: 2km, 4km
VIRGO: 3km
TAMA: 0.3km
LIGO-LLO: 4km
AIGO: (?)km
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)
Spin
L
S2
S1
• 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 > .97)
 automatic search over several (extrinsic) parameters – no template bank
For searching single-spin binaries: 7 M < m1 < 12 M , 1 M < m2 < 3 M
Templates in just 3 parameters:
S1 , m1 and m2
76000 templates needed at .97 match (average) - LIGO I sensitivity
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
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):
2
spin-down
parameters
not included
 f 0   Tobs 
N patch  10 
 100 days
500
Hz

 

N op  3N p log2 N p  N patch ~ 1022
5
10
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
1013 patches in the sky
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,  = 40 yr; fmax = 200 Hz,  = 1000 yr
Stochastic Background
Cannot distinguish instrumental noise from signal with one detector
Cross-correlate the output of two detectors:
s1 (t )  h1 (t )  n1 (t )
s2 (t )  h2 (t )  n2 (t )
T
T
0
0
C   dt  dt' s1 (t ) s2 (t ' ) Q (t , t ' )
Q: filter
  C  T h2
  C
2
2

SNR 

 C
2
T 
  df P1 (| f |) P2 (| f |)
4 
(Allen & Romano)
(E. Flanagan)
Stochastic Background
Overlap reduction function g(f): Non-coincident & non-aligned detectors
SNR : functional of g(f), GW (f), P1(f), P2(f)
1 dGW
GW ( f ) 
crit d ln f
LIGO detector pair, Tobs = 4 months, PF = 5%, PD = 95%
Initial:
Advanced:
GW ~ 10-5 - 10-6
GW ~ 10-10 - 10-11
Unmodeled sources
Burst sources: Supernovae, Hypernovae, Binary mergers, Ring-downs of
binary blackholes
Excess power statistics: Sum the power in the time-frequency window
E  4 | sk |2 / Pk
Anderson, Brady, J.Creighton, Flanagan
k
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:
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   ai (Fi h (t  Di (q , )  i )  cross pol term)
Y: data from a single synthetic detector and P = || Y ||2
P = z  h and
2  z / E(h) and maximise 2
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 !
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)
Veto for inspirals (Allen et al)
Based on the fact that irrespective of the masses:
~
| h ( f ) |2  f  7 / 3
Divide the frequency domain into p subbands so that the signal has
equal power in each subband k and compute the c2 as :
c   (k 
2
k

p
)2
where k is the SNR in subband k (normalised templates)
Compare the value of c2 with a threshold for deciding detection
Better vetos:
follow the ambiguity function
Clustering of triggers for real events
Clustering of triggers for real events
• Condensing the `cloud of events’ – graph theory?
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
2
1
1

1
.
7

10
y
MWEG
R90%
A rate > than above means there is more than 90% probability
that one inspiral event will be observed with SNR > highest
SNR observed in S1 data. (gr-qc/0308069)
Setting upper-limits (contd.)
Upper limits can be set for other types of sources:
• Stochastic 
GW < 23 for S1 data L1- H2
• Continuous wave sources  h for a given source
Source: PSR J1939+2134 (fastest known rotating neutron star) located 3.6 kpc
from Earth - fGW ~ 1283.86 Hz
Best upper limit from S1 data (L1) ~ 10-22
Data Analysis as diagonistic tool
Detector characterisation:
• Understanding of instrumental couplings to GW
channel
• Calibration
• Line removal techniques – adaptive methods
LISA : ESA & NASA project
Space based detector for detecting low frequency GW
LISA sensitivity curve
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
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)
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