30 th January 200925 th IAGRG meeting Kolkata S. V. DHURANDHAR IUCAA PUNE.
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Transcript of 30 th January 200925 th IAGRG meeting Kolkata S. V. DHURANDHAR IUCAA PUNE.
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30th January 2009 25th IAGRG meeting Kolkata
S. V. DHURANDHAR
IUCAA
PUNE
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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
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30th January 2009 25th IAGRG meeting Kolkata
Effect on a ring of test particles
01
222
ikhtc
Metric:
ikikik ecztheczthh )()(General Wave:
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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:
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30th January 2009 25th IAGRG meeting Kolkata
LIGO Louisiana 4 km armlength (US)
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30th January 2009 25th IAGRG meeting Kolkata
We did it !
*http://www.ligocaltech.edu/~lazz/distribution/LSC_Data
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30th January 2009 25th IAGRG meeting Kolkata
LISA: Space based detector for detecting low frequency GW
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30th January 2009 25th IAGRG meeting Kolkata
Laser frequency noise
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30th January 2009 25th IAGRG meeting Kolkata
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30th January 2009 25th IAGRG meeting Kolkata
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30th January 2009 25th IAGRG meeting Kolkata
into, Eastabrook, Armstrong; SVD, Nayak, Vinet, Pai
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30th January 2009 25th IAGRG meeting Kolkata
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30th January 2009 25th IAGRG meeting Kolkata
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30th January 2009 25th IAGRG meeting Kolkata
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30th January 2009 25th IAGRG meeting Kolkata
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30th January 2009 25th IAGRG meeting Kolkata
Open Problems
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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
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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
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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
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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
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30th January 2009 25th IAGRG meeting Kolkata
Matched filtering the inspiraling binary signal
)(
)()(
)()()(
~~
fS
fhfq
dttqtxc
h
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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
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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
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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
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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
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30th January 2009 25th IAGRG meeting Kolkata
Optimum location of a detector
Courtesy A. Sengupta & S. Mitra
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30th January 2009 25th IAGRG meeting Kolkata
ptimum location
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30th January 2009 25th IAGRG meeting Kolkata
Network Sensitivity
IGO/INDIGO network doubles sensitivity
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30th January 2009 25th IAGRG meeting Kolkata
Angular Resolution
An order of magnitude improvement
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30th January 2009 25th IAGRG meeting Kolkata
Summary
Connection of GR to several fields:
Astrophysics, Mathematics, Statistics, Engineering …
Numerical Relativity
Gravitation going experimental