October 2006Data Analysis workshop in A&A Delhi University S. V. DHURANDHAR IUCAA PUNE.

October 2006 Data Analysis workshop in A&A Delhi University

S. V. DHURANDHAR

IUCAA

PUNE


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


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


p

F

, pWaves in

Sound Waves

Force field

Force field longitudinal Longitudinal Waves


Electromagenetic Waves

Charge oscillates

E

kB

Test Charge

Force field transverse Transverse Waves

No medium required


GRAVITATIONAL WAVES ???

MASSES OSCILLATE :


NEWTON’S GRAVITY

Sun

Earth

mF

Field: Force on a test mass m:

Instantaneous propagation! Relativity ?

R

GM


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


Curved Space-time

CURVATURE GRAVITY


EINSTEIN’S GRAVITY : General theory of Relativity

Gravitation: Manifestation of the curvature of spacetime

Gravitational waves: Waves in the

curvature of spacetime


Dynamical variables in General Relativity

ijg

Waves in ,

Metric: Newtonian potential

Curvature: hijkR Gravitational field

ji : Tidal force

Dynamical variables:

ijg hijkR


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


Negative curvature

0K

A

B

C

0

CBA

CBA


4 – dimensional generalisations of metric and curvature

3,2,1,0,,,

22

222

lkjiRK

dxdxgds

dvGdudvFduEds

ijkl

jiij

Blackboard !!!


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


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 !


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 :


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


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 !


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


Waves in the curvature of spacetime

Waves in curvature: K keeps flipping sign

tihijk eR )0(

In higher dimensions


Detection: Effect on test particles

0K

Geodesics move closer Geodesics move away

Sphere Hyperboloid

Free test particles move along geodesics

0K

Curvature and geodesics:


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(


Effect on a ring of test particles

Polarisations

General wave: Linear combination of plus and cross


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


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


LIGO Louisiana 4 km armlength (US)


VIRGO 3 km armlength Pisa, Italy


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


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


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


The noise floor

• Seismic noise at low frequencies

• Thermal noise at mid frequencies

• Shot noise at high frequencies – quantum nature of light


LIGO DETECTORS

• 2 x 4km armlength detectors + 1 x 2km detector

Most sensitive

detectors today !

h ~ few x 10-23


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


Inspiraling compact binaries

GW


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


THE LISA PROJECT


Orbit of LISA

Cluster rolls once per year around its centre


LISA sensitivity curve

vibrationnoise

shot noise

armlength


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

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October 2006Data Analysis workshop in A&A Delhi University S. V. DHURANDHAR IUCAA PUNE.

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