Transcript Document
S. V. DHURANDHAR IUCAA PUNE 30th January 2009 25th IAGRG meeting Kolkata General Relativity and other disciplines • GR finds its home in astronomy & astrophysics - The binary blackhole problem - No exact two body solution: PN + Numerical Relativity - Cosmology: observational data, COBE, WMAP, PLANCK • Mathematics: Commutative algebra, differential geometry in statistics • Statistics: Hypothesis testing, statistical tests, maximum likelihood, etc. - Signal processing • Gravitation has gone experimental: Gravitational waves 30th January 2009 25th IAGRG meeting Kolkata Effect on a ring of test particles Metric: 2 1 2 2 hik 0 c t General Wave: hik 30th January 2009 h (t z c ) eik h (t z c ) eik 25th IAGRG meeting Kolkata But h is awfully small ! Quadrupole formula : ns 4G K .E. h ~ 4 c R 1 E R 21 ~ 10 2 0 . 1 M c 100 Mpc kin ns Change in arm-length: dL ~ h L 30th January 2009 25th IAGRG meeting Kolkata LIGO Louisiana 4 km armlength (US) 30th January 2009 25th IAGRG meeting Kolkata We did it ! *http://www.ligocaltech.edu/~lazz/distribution/LSC_Data 30th January 2009 25th IAGRG meeting Kolkata LISA: Space based detector for detecting low frequency GW 30th January 2009 25th IAGRG meeting Kolkata Laser frequency noise 30th January 2009 25th IAGRG meeting Kolkata 30th January 2009 25th IAGRG meeting Kolkata 30th January 2009 25th IAGRG meeting Kolkata Tinto, Eastabrook, Armstrong; SVD, Nayak, Vinet, Pai 30th January 2009 25th IAGRG meeting Kolkata 30th January 2009 25th IAGRG meeting Kolkata 30th January 2009 25th IAGRG meeting Kolkata 30th January 2009 25th IAGRG meeting Kolkata 30th January 2009 25th IAGRG meeting Kolkata Open Problems 30th January 2009 25th IAGRG meeting Kolkata Inspiraling compact binaries GW • Broadband source best for interferometric detectors • Waveform is well modeled by PN approximations – waveform obtained to 3.5 PN • Numerical Relativity: great advances – merger • Signal is way below the noise – data analysis: filtering 30th January 2009 25th IAGRG meeting Kolkata Breakthrough in Numerical Relativity Numerical Relativity solves • merger waveform • 3.5% of total restmass energy as compared to 1.5 % in inspiral waveform! Work in progress on stitching together waveforms India has the right talent/aptitude for NR 30th January 2009 25th IAGRG meeting Kolkata Example of a filter A sinusoidal signal is embedded in noise Data: x(t ) s(t ) n(t ) Signal: s(t ) A0 cos 2 f 0 t f0 256 Hz 30th January 2009 25th IAGRG meeting Kolkata Filtering the data Best filter is the Fourier Transform: x( f ) x(t ) e 2 ift dt Statistic: C ( f ) | x( f )| Provides parameters of the signal: eg. frequency Generalisation: Matched Filter 30th January 2009 25th IAGRG meeting Kolkata Matched filtering the inspiraling binary signal c( ) x (t ) q (t ) dt ~ ~ q( f ) 30th January 2009 h (f) Sh ( f ) 25th IAGRG meeting Kolkata Statistics and geometry • Statistic: c() • Where does one set the threshold? – False alarm probability – choose threshold such that this is small • Detection probability: choose this high • Signal depends on many parameters: masses, kinematical parameters: initial phase, time of arrival … • Parameter space • Parameter space can be viewed as a manifold with the parameters as coordinates • Metric: mismatch between signal and template 30th January 2009 25th IAGRG meeting Kolkata Parameter space metric Coordinates: li gik Dli Dlk e Choose coordinates so that metric is simplest: Cartesian coordinates For inspiral choose chirp times instead of masses: 0 , 3 Metric const: Uniform placement of templates Number of templates = volume of parameter space / template size 30th January 2009 25th IAGRG meeting Kolkata International Network of GW Interferometers 1. Detection confidence 2. Source direction 3. Polarisation info GEO: 0.6km LIGO-LHO: 2km, 4km VIRGO: 3km TAMA: 0.3km LIGO-LLO: 4km AIGO: (?)km 30th January 2009 25th IAGRG meeting Kolkata Need for a (or two) detector(s) in Asia/Australia • AIGO: Australian project • INDIGO: Indian detector? • Advantages: sky coverage, resolution of sources 30th January 2009 25th IAGRG meeting Kolkata Optimum location of a detector Courtesy A. Sengupta & S. Mitra 30th January 2009 25th IAGRG meeting Kolkata Optimum location 30th January 2009 25th IAGRG meeting Kolkata Network Sensitivity AIGO/INDIGO network doubles sensitivity 30th January 2009 25th IAGRG meeting Kolkata Angular Resolution An order of magnitude improvement 30th January 2009 25th IAGRG meeting Kolkata Summary Connection of GR to several fields: Astrophysics, Mathematics, Statistics, Engineering … Numerical Relativity Gravitation going experimental 30th January 2009 25th IAGRG meeting Kolkata