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

October 2006Data 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 2006Data 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 2006Data Analysis workshop in A&A Delhi University, pWaves in Sound Waves Force field Force field longitudinalLongitudinal Waves

October 2006Data Analysis workshop in A&A Delhi University Electromagenetic Waves Charge oscillates E k B Test Charge Force field transverseTransverse Waves No medium required

October 2006Data Analysis workshop in A&A Delhi University GRAVITATIONAL WAVES ??? MASSES OSCILLATE :

October 2006Data Analysis workshop in A&A Delhi University NEWTON’S GRAVITY Sun Earth Field: Force on a test mass m: Instantaneous propagation!Relativity ?

October 2006Data Analysis workshop in A&A Delhi University Inadequacy of Newtonian gravity The Newtonian theory: Inverse square law Gravity signals travel instantaneously Inconsistent with SR Need new theory of gravity General Theory of Relativity

October 2006Data Analysis workshop in A&A Delhi University Curved Space-time CURVATUREGRAVITY

October 2006Data 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 2006Data Analysis workshop in A&A Delhi University Dynamical variables in General Relativity Waves in, Metric:Newtonian potential Curvature:Gravitational field : Tidal force Dynamical variables:

October 2006Data Analysis workshop in A&A Delhi University Curvature in 2 D A B C The 3 angles of the triangle do not add up to 180 o Signature of curvature ! K > 0

October 2006Data Analysis workshop in A&A Delhi University Negative curvature A B C

October 2006Data Analysis workshop in A&A Delhi University 4 – dimensional generalisations of metric and curvature Blackboard !!!

October 2006Data Analysis workshop in A&A Delhi University Einstein’s Field Equations 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

October 2006Data Analysis workshop in A&A Delhi University Schwarzschild Solution (1916) Solution of Einstein’s Equations for a nonrotating uncharged point mass BlACKHOLE SOLUTION !

October 2006Data Analysis workshop in A&A Delhi University Physics of gravitational waves Weak field : Lorentz gauge : Linearised Eintein’s Eq. : Trace reverse: Trace :

October 2006Data Analysis workshop in A&A Delhi University Plane wave solutions: TT gauge Source free: Plane wave solutions: Gauge conditions: Transverse: Traceless: TT gauge

October 2006Data 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 Nobel Prize to Hulse & Taylor 1993 Gravitational Waves EXIST !

October 2006Data 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

October 2006Data Analysis workshop in A&A Delhi University Waves in the curvature of spacetime Waves in curvature: K keeps flipping sign In higher dimensions

October 2006Data Analysis workshop in A&A Delhi University Detection: Effect on test particles Geodesics move closer Geodesics move away SphereHyperboloid Free test particles move along geodesics Curvature and geodesics:

October 2006Data 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

October 2006Data Analysis workshop in A&A Delhi University Effect on a ring of test particles Polarisations General wave: Linear combination of plus and cross

October 2006Data 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 2006Data 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 :

October 2006Data Analysis workshop in A&A Delhi University LIGO Louisiana 4 km armlength (US)

October 2006Data Analysis workshop in A&A Delhi University VIRGO 3 km armlength Pisa, Italy

October 2006Data 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 2006Data Analysis workshop in A&A Delhi University International Network of GW Interferometers International Network of GW Interferometers LIGO-LLO: 4km LIGO-LHO: 2km, 4km GEO: 0.6km VIRGO: 3km TAMA: 0.3km AIGO: (?)km 1. Detection confidence 2. Source direction 3. Polarisation info

October 2006Data Analysis workshop in A&A Delhi University Technology pushed to the limits 10 kg test masses <1 nm rms surface =1.024  m Q ~ 10 6 =1.024  m Seismic isolation of ground motion by > =1.024  m ~10 20  /s to split fringe to  / ~ Vacuum better than torr 1.22 m aperture x 4000 m arms ~9.4 x 10 3 m 3 (each site) ~10 9 Joule of stored energy 5m

October 2006Data 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 2006Data 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

October 2006Data Analysis workshop in A&A Delhi University GRAVITATIONAL WAVE SOURCES Inspiraling binaries: Neutron stars (NS), Blackholes h ~ for 2 NS at 200 Mpc Rotating NS, Accreting NS – LMXBs Sco X-1 Supernovae Stochastic background – Early Universe Parametric amplification

October 2006Data Analysis workshop in A&A Delhi University Inspiraling compact binaries GW

October 2006Data 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 2006Data Analysis workshop in A&A Delhi University

October 2006Data Analysis workshop in A&A Delhi University

October 2006Data Analysis workshop in A&A Delhi University

October 2006Data Analysis workshop in A&A Delhi University FUTURE DIRECTIONS LIGO, VIRGO, TAMA, GEO will operate continuously from 2004/5 for many years. Sensitivity upgrades: 1 st generation - may well make first detections 2 nd generation: > 2008 :increase band-width & sensitivity 3 rd generation: > 2010 : being studied Each generation: Factor of 10 in amplitude LISA will open the low frequency window Hz – 1 Hz – many detections + some with high SNR