· Burst analysisFar less data needs to be considered when analysing burst sig-nals; only 2 seconds for each of 3 interferometers (0.1 MB) wereprocessed in the example below.

5.65 5.70 5.75 5.80

amplitude a × 10−19361 362

frequency f (Hz)0.92 0.96 1.00 1.04

phase φµ

788.95 789.00 789.05

center µ (ms)2.48 2.50 2.52 2.54 2.56

width σ (ms)●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

17.77h 17.75h 17.73h

−29

.2°

−29

°−

28.8

°

right ascension α

decl

inat

ion

δ

Posterior densities for the burst parameters, and a 95% pos-terior region for the sky location. True parameter valuesare indicated in red.

Due to the burst posterior’s different characteristics, more paral-lel chains need to be run in order to find the posterior’s globalmode; here 15 chains were used.

· OutlookWe are planning to extend our research to more complex sig-nal waveforms, like e.g. inspirals involving greater masses (blackholes), or taking into account the effects of spins the two com-panions may have.Eventually, the developed methodology is supposed to be in-stalled at the end of a ‘detection pipeline’ that monitors the in-terferometer measurements and triggers the analysis when thepresence of a signal is detected.

References[1] B. F. Schutz. Gravitational wave astronomy. Classical and Quantum Grav-

ity, 16(12A):A131–A156, December 1999.

[2] L. Blanchet. Post-Newtonian computation of binary inspiral waveforms.In I. Ciufolini, V. Gorini, U. Moschella, and P. Fre, editors, Gravitationalwaves: Proceedings of the Como school on gravitational waves in astro-physics. Institute of Physics Publishing, Bristol, 2001.

[3] L. S. Finn and D. F. Chernoff. Observing binary inspiral in gravitational ra-diation: One interferometer. Physical Review D, 47(6):2198–2219, March1993.

[4] C. Rover, R. Meyer, and N. Christensen. Bayesian inference on com-pact binary inspiral gravitational radiation signals in interferometric data.Submitted for publication; URL: http://arxiv.org/abs/gr-qc/0602067(preprint), February 2006.

[5] W. R. Gilks, S. Richardson, and D. J. Spiegelhalter. Markov chain MonteCarlo in practice. Chapman & Hall / CRC, Boca Raton, 1996.

· PriorsThe prior information about parameters is specified (in terms of‘proper’ priors) assuming for example that different orientationsare equally likely, potential events are evenly spread across timeand space, neutron star masses comply with predicted frequen-cies, and also taking into account the interferometers’ sensitivi-ties [4].

· MCMCMC Posterior simulationIntegration of the posterior distribution is solved using MCMCsimulation. The posterior distribution exhibits many local modesamongst which a ‘regular’ Metropolis-sampler would have trou-ble converging towards the global optimum. A Metropolis-Cou-pled MCMC (MCMCMC) scheme (also known as ‘parallel tem-pering’) was implemented. This algorithm runs several chainssampling from ‘tempered’ likelihoods p(z|ϑ)

1T with increasing

temperatures T in parallel, so only the first chain (T = 1) sam-ples from the actual likelihood. Additional steps then allow torandomly switch between chains, improving sampling and con-vergence [5].

· Chirp analysisIn the example below, data was generated to simulate a binaryneutron star inspiral event that is measured at three different in-terferometers. The amount of data was 23, 23 and 44 seconds forthe different sites, downsampled to 4–5 kHz (1.6 MB in total).

12.344 12.345 12.346

coalescence time tc (s)5 10 15

distance dL (Mpc)

1.4 1.6 1.8 2.0 2.2

individual masses m1, m2 (multiples of sun mass)

companion 1

companion 2

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●●

●

●

●

●

●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

● ●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

19h 18.5h

−60

°−

55°

right ascension α

decl

inat

ion

δ

Posterior densities for some of the inspiral parameters, anda 95% posterior region for the sky location. True parame-ter values are indicated in red.

The parallel tempering algorithm ran 6 parallel chains, generat-ing 80 draws per minute on a 3.2 GHz desktop PC.

The gravitational wave measurements are superimposed with in-terferometer-specific coloured noise. Combining data from sev-eral interferometers improves the signal-to-noise ratio, and alsomakes the estimation of certain parameters (e.g. sky location)possible.

frequency (Hz)

dete

ctor

noi

se (

spec

tral

den

sity

)

operational ranges

LIGO

VIRGO

30 40 100 200 500 100010

−100

10−9

0

Noise spectra of two different interferometers.

· ParametersThe measured detector response depends on the signal waveformas well as mutual orientations of detectors and source (sky loca-tion (δ, α) and polarisation angle (ψ)).A binary inspiral’s chirp waveform is determined by the massesof involved objects (m1, m2), coalescence time (tc), distance(dL), phase (φ0) and inclination (ι); not all of these are of pri-mary concern, the latter two as well as the polarisation are rathernuisance parameters.Burst waveforms can have a range of parametrisations; the sine-gaussian bursts we use are defined by their center (µ), width (σ),amplitude (a), frequency (f ) and phase (φµ).

· The coherent likelihoodThe individual likelihoods (for measurements from one interfer-ometer I) are computed based on Fourier transforms of data (z)and waveform (s(ϑ)), and the detector’s noise spectrum:

p(I)(z|ϑ) ∝ exp

(− 2

δt

iU∑i=iL

|data︷ ︸︸ ︷

z(I)(i×∆f)−waveform template︷ ︸︸ ︷s(I)(i×∆f , ϑ) |2

S(I)n (i×∆f)︸ ︷︷ ︸

noise spectrum

)

[3]. Waveform templates s(ϑ) are generated in the time domain,and then (numerically) Fourier-transformed for each likelihoodevaluation.Assuming that the noise is independent across different interfer-ometers, the coherent network likelihood is the product of theindividual detector likelihoods:

p(z|ϑ) =∏I

p(I)(z|ϑ).

· Gravitational wavesGeneral Relativity Theory predicts the existence of gravitation-al radiation—distortions in space-time that propagate throughspace at the speed of light. Around the world, laser interferom-eters are being built and put into operation in order to measurethe effect of gravitational waves. Direct measurement of grav-itational radiation would not only confirm the General Relativ-ity Theory, but also complement ‘traditional’ observations in theelectromagnetic spectrum [1].We propose a Bayesian framework to estimate event parametersfrom gravitational wave measurements.

· Measuring chirps & burstsAmong the event candidates that are expected to be detected firstare binary inspirals and bursts. Inspiralling binary star systemsemit sustained ‘chirp’ signals of increasing frequency and ampli-tude, whose evolution is predicted with great precision [2].

xlimits

Hanford

xlimits

Livingston

xlimits

Pisa

time (seconds)−0.20 −0.15 −0.10 −0.05 0.00

A binary system’s last moments before coalescence, mea-sured at different interferometer sites as ‘chirps’.

Burst signals are short-duration signals that are expected fromdifferent sources, e.g. from supernova events. Their waveformsare not exactly known. For now we use simple (e.g. sine-gaussi-an) waveforms.

xlimits

Hanford

xlimits

Livingston

xlimits

Pisa

time (seconds)−0.02 −0.01 0.00 0.01 0.02

A burst signal, arriving at different interferometers withdifferent time delays and amplitudes.

TheUniversity

of Auckland

Coherent Analysis of Gravitational Wave Chirps and BurstsChristian Rover Renate Meyer Nelson Christensen

Department of Statistics Department of Statistics Department of Physics & AstronomyThe University of Auckland The University of Auckland Carleton College, MN, USA