30th January 2009 25th IAGRG meeting Kolkata

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

01

222

ikhtc

Metric:

ikikik ecztheczthh )()(General Wave:

30th January 2009 25th IAGRG meeting Kolkata

But h is awfully small !

L ~ h L

Quadrupole formula :

21

1

2410

1001.0~

..4~

Mpc

R

cM

E

R

EK

c

Gh

nskinns

Change in arm-length:

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

into, 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 waveformsIndia 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

0 0

( ) ( ) ( )

( ) cos 2

x t s t n t

s t A f t

Data:

Signal:

0 256f Hz

30th January 2009 25th IAGRG meeting Kolkata

Filtering the data

2( ) ( )

( ) | ( )|

iftx f x t e dt

C f x f

Best filter is the Fourier Transform:

Statistic:

Generalisation: Matched Filter

Provides parameters of the signal: eg. frequency

30th January 2009 25th IAGRG meeting Kolkata

Matched filtering the inspiraling binary signal

)(

)()(

)()()(

~~

fS

fhfq

dttqtxc

h

30th January 2009 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: i

gik

ik

Choose coordinates so that metric is simplest: Cartesian coordinates

For inspiral choose chirp times instead of masses: 03

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 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 info1. Detection confidence 2. Source direction 3. Polarisation info

30th January 2009 25th IAGRG meeting Kolkata

eed 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

ptimum location

30th January 2009 25th IAGRG meeting Kolkata

Network Sensitivity

IGO/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