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S. V. DHURANDHAR

GRAVITATIONAL WAVES. 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

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S. V. DHURANDHAR

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  1. GRAVITATIONAL WAVES S. V. DHURANDHAR IUCAA PUNE Data Analysis workshop in A&A Delhi University

  2. 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 Data Analysis workshop in A&A Delhi University

  3. 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 Data Analysis workshop in A&A Delhi University

  4. Waves in , p Sound Waves Force field Force field longitudinal Longitudinal Waves Data Analysis workshop in A&A Delhi University

  5. Charge oscillates Test Charge E k B Force field transverse Transverse Waves No medium required Electromagenetic Waves Data Analysis workshop in A&A Delhi University

  6. GRAVITATIONAL WAVES ??? MASSES OSCILLATE : Data Analysis workshop in A&A Delhi University

  7. NEWTON’SGRAVITY Earth Sun Force on a test mass m: Field: Instantaneous propagation! Relativity ? Data Analysis workshop in A&A Delhi University

  8. 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 Data Analysis workshop in A&A Delhi University

  9. Curved Space-time CURVATURE GRAVITY Data Analysis workshop in A&A Delhi University

  10. Gravitational waves: Waves in the curvature of spacetime EINSTEIN’S GRAVITY : General theory of Relativity Gravitation: Manifestation of the curvature of spacetime Data Analysis workshop in A&A Delhi University

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

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

  13. A C B Negative curvature Data Analysis workshop in A&A Delhi University

  14. 4 – dimensional generalisations of metric and curvature Blackboard !!! Data Analysis workshop in A&A Delhi University

  15. 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 Data Analysis workshop in A&A Delhi University

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

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

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

  19. 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 1993 Gravitational Waves EXIST ! Data Analysis workshop in A&A Delhi University

  20. 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 GRAVITATIONAL WAVE ASTRONOMY PROBES OF THE UNIVERSE GW ASTRONOMY !! Data Analysis workshop in A&A Delhi University

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

  22. Detection: Effect on test particles Free test particles move along geodesics Curvature and geodesics: Sphere Hyperboloid Geodesics move closer Geodesics move away Data Analysis workshop in A&A Delhi University

  23. Gravitational wave and geodesics • Gravitational wave • curvature oscillates • The sign of the curvature keeps flipping • The length of the connecting vector oscillates Time Data Analysis workshop in A&A Delhi University

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

  25. mirror beamsplitter laser and injection optics mirror detector Detection principle on ground Michelson interferometer measuring changes in relative lengths of arms formed between “free” test mass Data Analysis workshop in A&A Delhi University

  26. PRINCIPLE OF DETECTION • Quadrupolar force field • Effect on a ring of test • particles: d L ~ h L Quadrupole formula : Data Analysis workshop in A&A Delhi University

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

  28. VIRGO 3 km armlength Pisa, Italy Data Analysis workshop in A&A Delhi University

  29. CURRENTDETECTORSTATUS • 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 Data Analysis workshop in A&A Delhi University

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

  31. Vacuum better than 10-6 torr 1.22 m aperture x 4000 m arms ~9.4 x 103 m3 (each site) ~109 Joule of stored energy 5m 10 kg test masses <1 nm rms surface error @ l =1.024 mm Q ~ 106 R>0.99999 @ l =1.024 mm Technology pushed to the limits 10 W @ l =1.024 mm ~1020g /s to split fringe to 10-10 dn/n~10-10 Seismic isolation of ground motion by > 10-6 Data Analysis workshop in A&A Delhi University

  32. The noise floor • Seismic noise at low frequencies • Thermal noise at mid frequencies • Shot noise at high frequencies – quantum nature of light Data Analysis workshop in A&A Delhi University

  33. LIGO DETECTORS • 2 x 4km armlength detectors + 1 x 2km detector Most sensitive detectors today ! h ~ few x 10-23 Data Analysis workshop in A&A Delhi University

  34. 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 Data Analysis workshop in A&A Delhi University

  35. Inspiraling compact binaries GW Data Analysis workshop in A&A Delhi University

  36. LISA 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 Data Analysis workshop in A&A Delhi University

  37. Data Analysis workshop in A&A Delhi University

  38. Data Analysis workshop in A&A Delhi University

  39. Data Analysis workshop in A&A Delhi University

  40. 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 Data Analysis workshop in A&A Delhi University

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