October 2006 Data Analysis workshop in A&A Delhi University
S. V. DHURANDHAR
IUCAA
PUNE
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
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
October 2006 Data Analysis workshop in A&A Delhi University
p
F
, pWaves in
Sound Waves
Force field
Force field longitudinal Longitudinal Waves
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
October 2006 Data Analysis workshop in A&A Delhi University
GRAVITATIONAL WAVES ???
MASSES OSCILLATE :
October 2006 Data Analysis workshop in A&A Delhi University
NEWTON’S GRAVITY
Sun
Earth
mF
Field: Force on a test mass m:
Instantaneous propagation! Relativity ?
R
GM
October 2006 Data Analysis workshop in A&A Delhi University
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
October 2006 Data Analysis workshop in A&A Delhi University
Curved Space-time
CURVATURE GRAVITY
October 2006 Data Analysis workshop in A&A Delhi University
EINSTEIN’S GRAVITY : General theory of Relativity
Gravitation: Manifestation of the curvature of spacetime
Gravitational waves: Waves in the
curvature of spacetime
October 2006 Data Analysis workshop in A&A Delhi University
Dynamical variables in General Relativity
ijg
Waves in ,
Metric: Newtonian potential
Curvature: hijkR Gravitational field
ji : Tidal force
Dynamical variables:
ijg hijkR
October 2006 Data Analysis workshop in A&A Delhi University
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
October 2006 Data Analysis workshop in A&A Delhi University
Negative curvature
0K
A
B
C
0
CBA
CBA
October 2006 Data Analysis workshop in A&A Delhi University
4 – dimensional generalisations of metric and curvature
3,2,1,0,,,
22
222
lkjiRK
dxdxgds
dvGdudvFduEds
ijkl
jiij
Blackboard !!!
October 2006 Data Analysis workshop in A&A Delhi University
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
October 2006 Data Analysis workshop in A&A Delhi University
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 !
October 2006 Data Analysis workshop in A&A Delhi University
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 :
October 2006 Data Analysis workshop in A&A Delhi University
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
October 2006 Data Analysis workshop in A&A Delhi University
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 !
October 2006 Data Analysis workshop in A&A Delhi University
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
October 2006 Data Analysis workshop in A&A Delhi University
Waves in the curvature of spacetime
Waves in curvature: K keeps flipping sign
tihijk eR )0(
In higher dimensions
October 2006 Data Analysis workshop in A&A Delhi University
Detection: Effect on test particles
0K
Geodesics move closer Geodesics move away
Sphere Hyperboloid
Free test particles move along geodesics
0K
Curvature and geodesics:
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(
October 2006 Data Analysis workshop in A&A Delhi University
Effect on a ring of test particles
Polarisations
General wave: Linear combination of plus and cross
October 2006 Data Analysis workshop in A&A Delhi University
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
October 2006 Data Analysis workshop in A&A Delhi University
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
October 2006 Data Analysis workshop in A&A Delhi University
LIGO Louisiana 4 km armlength (US)
October 2006 Data Analysis workshop in A&A Delhi University
VIRGO 3 km armlength Pisa, Italy
October 2006 Data Analysis workshop in A&A Delhi University
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
October 2006 Data Analysis workshop in A&A Delhi University
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
October 2006 Data Analysis workshop in A&A Delhi University
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
October 2006 Data Analysis workshop in A&A Delhi University
The noise floor
• Seismic noise at low frequencies
• Thermal noise at mid frequencies
• Shot noise at high frequencies – quantum nature of light
October 2006 Data Analysis workshop in A&A Delhi University
LIGO DETECTORS
• 2 x 4km armlength detectors + 1 x 2km detector
Most sensitive
detectors today !
h ~ few x 10-23
October 2006 Data Analysis workshop in A&A Delhi University
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
October 2006 Data Analysis workshop in A&A Delhi University
Inspiraling compact binaries
GW
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
October 2006 Data Analysis workshop in A&A Delhi University
THE LISA PROJECT
October 2006 Data Analysis workshop in A&A Delhi University
Orbit of LISA
Cluster rolls once per year around its centre
October 2006 Data Analysis workshop in A&A Delhi University
LISA sensitivity curve
vibrationnoise
shot noise
armlength
October 2006 Data Analysis workshop in A&A Delhi University
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