S. V. DHURANDHAR
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Transcript of S. V. DHURANDHAR
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October 2006 Data Analysis workshop in A&A Delhi University
S. V. DHURANDHAR
IUCAA
PUNE
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October 2006 Data Analysis workshop in A&A Delhi University
Lecture Set I: Gravitational Waves
• Gravitational Waves: Introduction
• General Relativity
• Differential Geometry: curvature, metric, tensors, manifolds – needed in GR & data analysis
• Wave equation, plane waves, transverse-traceless (TT) gauge, polarisations • Landau- Lipschitz quadrupole formula
• Detection of gravitational waves – experiment, sources, prospects
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October 2006 Data Analysis workshop in A&A Delhi University
Lecture Set II: Statistical theory of signal detection
• Time series data, sampling, uniform sampling, Fourier transform, DFT, data vector
• Random variables, noise, pdf, Gaussian distribution, multivariate Gaussian, vector random variable
• Characterisation of noise: coloured, stationary, Gaussian
• False alarm, false dismissal, Neyman-Pearson criterion
• Metric – Fischer information matrix, covariance matrix of errors
• Maximum likelihood detection, matched filter
• Parameter space, signal manifold, metric, template placement and curvature of signal manifold, ambiguity function
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October 2006 Data Analysis workshop in A&A Delhi University
p
F
, pWaves in
Sound Waves
Force field
Force field longitudinal Longitudinal Waves
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October 2006 Data Analysis workshop in A&A Delhi University
Electromagenetic Waves
Charge oscillates
E
kB
Test Charge
Force field transverse Transverse Waves
No medium required
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GRAVITATIONAL WAVES ???
MASSES OSCILLATE :
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NEWTON’S GRAVITY
Sun
Earth
mF
Field: Force on a test mass m:
Instantaneous propagation! Relativity ?
R
GM
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Inadequacy of Newtonian gravity
The Newtonian theory: Inverse square law
^
2
21 rr
mmGF
Gravity signals travel instantaneously
Inconsistent with SR
Need new theory of gravity
General Theory of Relativity
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Curved Space-time
CURVATURE GRAVITY
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EINSTEIN’S GRAVITY : General theory of Relativity
Gravitation: Manifestation of the curvature of spacetime
Gravitational waves: Waves in the
curvature of spacetime
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Dynamical variables in General Relativity
ijg
Waves in ,
Metric: Newtonian potential
Curvature: hijkR Gravitational field
ji : Tidal force
Dynamical variables:
ijg hijkR
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Curvature in 2 D
A
BC
The 3 angles of the triangle
do not add up to 180o
Signature of curvature !
2
1
RK
dSK
CBA
K > 0
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October 2006 Data Analysis workshop in A&A Delhi University
Negative curvature
0K
A
B
C
0
CBA
CBA
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4 – dimensional generalisations of metric and curvature
3,2,1,0,,,
22
222
lkjiRK
dxdxgds
dvGdudvFduEds
ijkl
jiij
Blackboard !!!
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Einstein’s Field Equations
ijijij Tc
GgRR
4
8
2
1
Geometry = Matter distribution
Spacetime grips mass, telling it how to move, and mass grips spacetime, telling it how to curve
— John Archibald Wheeler
Compare with Newton’s equation G42
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Schwarzschild Solution (1916)
Solution of Einstein’s Equations for a nonrotating uncharged point mass
2
2221222 )2
1()2
1(
c
GMm
drdrr
mdtc
r
mds
BlACKHOLE SOLUTION !
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Physics of gravitational waves
,
4
0
16
1
2
g h
h
Gh T
c
h h h
h h
Weak field :
Lorentz gauge :
Linearised Eintein’s Eq. :
Trace reverse:
Trace :
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Plane wave solutions: TT gauge
,
0
0 0
0 0
0
i k x
h
h A e
h A k
h h A
U A U
Source free:
Plane wave solutions:
Gauge conditions:
Transverse:
Traceless:
TT gauge
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GRAVITATIONAL WAVES
• Space-time warpage in the fabric of spacetime
travels with the speed c.
• Dynamic concentrations of matter
Decay in the orbit of the
binary pulsar PSR 1913+16
- Nobel Prize to
Hulse & Taylor 1993Gravitational Waves EXIST !
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GRAVITATIONAL WAVE ASTRONOMY
PROBES OF THE UNIVERSE
GW ASTRONOMY !!
• Enormous differences between GW and EM
- Produced by bulk motions of matter
- Compact objects: Blackholes, neutron stars
• Not easily scattered: Hi fidelity info
• EM (f > Hz) while GW (f < Hz)
-Information orthogonal to EM - revolution
710 410
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Waves in the curvature of spacetime
Waves in curvature: K keeps flipping sign
tihijk eR )0(
In higher dimensions
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Detection: Effect on test particles
0K
Geodesics move closer Geodesics move away
Sphere Hyperboloid
Free test particles move along geodesics
0K
Curvature and geodesics:
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October 2006 Data Analysis workshop in A&A Delhi University
Gravitational wave and geodesics
Time
• Gravitational wave
curvature oscillates
• The sign of the curvature keeps flipping
• The length of the connecting vector oscillates ti
hijk eR )0(
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Effect on a ring of test particles
Polarisations
General wave: Linear combination of plus and cross
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Detection principle on ground
Michelson interferometer measuring changes in relative lengths of arms formed between “free” test mass
detector
mirror
laser and injection optics
beamsplitter
mirror
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PRINCIPLE OF DETECTION
• Quadrupolar force field
• Effect on a ring of test
particles:
L ~ h L
Quadrupole formula :
21
1
2410
1001.0~
Mpc
R
cM
E
c
Gh
nskin
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LIGO Louisiana 4 km armlength (US)
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VIRGO 3 km armlength Pisa, Italy
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CURRENT DETECTOR STATUS
• Several large scale laser interferometric detectors
constructed: armlength of 300 m to 4 km
- LIGO, VIRGO, GEO, TAMA, AIGO (?)
- LIGO, TAMA, GEO : already taking data
- VIRGO: engineering runs (science run – spring 2005 ?)
• Space based detector LISA – 5 million km
- Launch in 2015
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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 info
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Technology pushed to the limits
10 kg test masses<1 nm rms surface error @ =1.024 m
Q ~ 106
R>0.99999 @ =1.024 m
Seismic isolation of ground motion by > 10-6
10 W @ =1.024 m~1020 /s to split fringe to 10-10
/~10-10
Vacuum better than 10-6 torr1.22 m aperture x 4000 m arms~9.4 x 103 m3 (each site)~109 Joule of stored energy
5m
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The noise floor
• Seismic noise at low frequencies
• Thermal noise at mid frequencies
• Shot noise at high frequencies – quantum nature of light
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LIGO DETECTORS
• 2 x 4km armlength detectors + 1 x 2km detector
Most sensitive
detectors today !
h ~ few x 10-23
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GRAVITATIONAL WAVE SOURCES
• Inspiraling binaries:
Neutron stars (NS), Blackholes
h ~ 10-23 for 2 NS at 200 Mpc
• Rotating NS, Accreting NS – LMXBs
Sco X-1
• Supernovae
• Stochastic background – Early Universe
Parametric amplification
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Inspiraling compact binaries
GW
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October 2006 Data Analysis workshop in A&A Delhi University
LISA• A Collaborative ESA / NASA Mission to observe low-
frequency gravitational waves
• Cluster of 3 S/C in heliocentric orbit at 1 AU
• S/C contain lasers and free-flying test masses
• Equilateral triangle with 5 Million km arm-length
• Trailing the earth by 20°
• Equivalent to a Michelson interferometer– Thermal & seismic motions of mirror
masses and pendulums
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THE LISA PROJECT
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Orbit of LISA
Cluster rolls once per year around its centre
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LISA sensitivity curve
vibrationnoise
shot noise
armlength
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FUTURE DIRECTIONS
• LIGO, VIRGO, TAMA, GEO will operate continuously from 2004/5 for many years.
• Sensitivity upgrades: 1st generation - may well make first detections 2nd generation: > 2008 :increase band-width & sensitivity 3rd generation: > 2010 : being studied Each generation: Factor of 10 in amplitude
• LISA will open the low frequency window
10-4 Hz – 1
Hz – many detections + some with high SNR