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30th January 2009 25th IAGRG meeting Kolkata
S. V. DHURANDHAR
IUCAA
PUNE
30th January 2009 25th IAGRG meeting Kolkata
General Relativity and other disciplines
• GR finds its home in astronomy & astrophysics
- The binary blackhole problem
- No exact two body solution: PN + Numerical Relativity
- Cosmology: observational data, COBE, WMAP, PLANCK
• Mathematics: Commutative algebra, differential
geometry in statistics
• Statistics: Hypothesis testing, statistical tests, maximum likelihood, etc. - Signal processing
• Gravitation has gone experimental: Gravitational waves
30th January 2009 25th IAGRG meeting Kolkata
Effect on a ring of test particles
01
222
ikhtc
Metric:
ikikik ecztheczthh )()(General Wave:
30th January 2009 25th IAGRG meeting Kolkata
But h is awfully small !
L ~ h L
Quadrupole formula :
21
1
2410
1001.0~
..4~
Mpc
R
cM
E
R
EK
c
Gh
nskinns
Change in arm-length:
30th January 2009 25th IAGRG meeting Kolkata
LIGO Louisiana 4 km armlength (US)
30th January 2009 25th IAGRG meeting Kolkata
We did it !
*http://www.ligocaltech.edu/~lazz/distribution/LSC_Data
30th January 2009 25th IAGRG meeting Kolkata
LISA: Space based detector for detecting low frequency GW
30th January 2009 25th IAGRG meeting Kolkata
Laser frequency noise
30th January 2009 25th IAGRG meeting Kolkata
30th January 2009 25th IAGRG meeting Kolkata
30th January 2009 25th IAGRG meeting Kolkata
into, Eastabrook, Armstrong; SVD, Nayak, Vinet, Pai
30th January 2009 25th IAGRG meeting Kolkata
30th January 2009 25th IAGRG meeting Kolkata
30th January 2009 25th IAGRG meeting Kolkata
30th January 2009 25th IAGRG meeting Kolkata
30th January 2009 25th IAGRG meeting Kolkata
Open Problems
30th January 2009 25th IAGRG meeting Kolkata
Inspiraling compact binaries
GW
• Broadband source best for interferometric detectors
• Waveform is well modeled by PN approximations – waveform obtained to 3.5 PN
• Numerical Relativity: great advances – merger
• Signal is way below the noise – data analysis: filtering
30th January 2009 25th IAGRG meeting Kolkata
Breakthrough in Numerical Relativity
Numerical Relativity
solves
• merger waveform
• 3.5% of total restmass energy as compared to 1.5 % in inspiral waveform!
Work in progress on stitching together waveformsIndia has the right talent/aptitude for NR
30th January 2009 25th IAGRG meeting Kolkata
Example of a filter
A sinusoidal signal is embedded in noise
0 0
( ) ( ) ( )
( ) cos 2
x t s t n t
s t A f t
Data:
Signal:
0 256f Hz
30th January 2009 25th IAGRG meeting Kolkata
Filtering the data
2( ) ( )
( ) | ( )|
iftx f x t e dt
C f x f
Best filter is the Fourier Transform:
Statistic:
Generalisation: Matched Filter
Provides parameters of the signal: eg. frequency
30th January 2009 25th IAGRG meeting Kolkata
Matched filtering the inspiraling binary signal
)(
)()(
)()()(
~~
fS
fhfq
dttqtxc
h
30th January 2009 25th IAGRG meeting Kolkata
Statistics and geometry
• Statistic: c()
• Where does one set the threshold? – False alarm probability – choose threshold such that this is small
• Detection probability: choose this high
• Signal depends on many parameters: masses, kinematical parameters: initial phase, time of arrival …
• Parameter space
• Parameter space can be viewed as a manifold with the
parameters as coordinates
• Metric: mismatch between signal and template
30th January 2009 25th IAGRG meeting Kolkata
Parameter space metric
Coordinates: i
gik
ik
Choose coordinates so that metric is simplest: Cartesian coordinates
For inspiral choose chirp times instead of masses: 03
Metric const: Uniform placement of templates
Number of templates = volume of parameter space / template size
30th January 2009 25th IAGRG meeting Kolkata
International Network of GW InterferometersInternational Network of GW Interferometers
LIGO-LLO: 4km
LIGO-LHO: 2km, 4kmGEO: 0.6km VIRGO: 3km
TAMA: 0.3km
AIGO: (?)km
1. Detection confidence 2. Source direction 3. Polarisation info1. Detection confidence 2. Source direction 3. Polarisation info
30th January 2009 25th IAGRG meeting Kolkata
eed for a (or two) detector(s) in Asia/Australia
• AIGO: Australian project
• INDIGO: Indian detector?
• Advantages: sky coverage, resolution of sources
30th January 2009 25th IAGRG meeting Kolkata
Optimum location of a detector
Courtesy A. Sengupta & S. Mitra
30th January 2009 25th IAGRG meeting Kolkata
ptimum location
30th January 2009 25th IAGRG meeting Kolkata
Network Sensitivity
IGO/INDIGO network doubles sensitivity
30th January 2009 25th IAGRG meeting Kolkata
Angular Resolution
An order of magnitude improvement
30th January 2009 25th IAGRG meeting Kolkata
Summary
Connection of GR to several fields:
Astrophysics, Mathematics, Statistics, Engineering …
Numerical Relativity
Gravitation going experimental
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