Data Analysis III: burst search

Burst sources of GWFiltering methods

Detection by a network of detectors

M.-A. Bizouard LAL-Orsay

Sources of GW burst … detectable by ITF or resonant detectors

• Massive star collapses

– Type II supernova

– Black hole formation

• Instabilities in newborn neutrons stars

• Mergers of couples of compact stars

• Black hole ring down

• Others ….

M82: starburst galaxy + Xray binary composed of black hole?11.106 light years from Earth

Massive star collapse

Type II supernova: collapse of the iron core of a massive (~10 solar masses) star -> formation of a neutron star

GW emission? depends on the asymetry (badly known)

Possible asymetry due to: - fast rotation - star companion

Recent predictions:

–collapse -> NS

–collapse -> Black Hole (too massive star.

direct of delayedformation)

galaxyyrSNkHzHzf

Mpch

/30/11100~

10@10~ 23

rate ? 10~

10@10~ 22kHzf

Mpch

Some waveforms

Type II supernovae: amplitude simulated: by Zwerger & Muller (A&A 97)by Dimmelmeir et al (A&A 02)

Collapse to Black Hole: Stark & Piran (PRL 95)

50 ms10 ms

Predictions are not robust huge variety of waveforms!

0.1 ms

Core collapse numerical simulations

• The maturity of modern collapse simulations is quite timely. • Lots of studies done over the last 3 decades. Especially core collapse SN.

Lots of physics have to be included:– Construction of accurate progenitor models (including realistic angular

momentum distributions)– Microphysics: realistic equations of state + neutrino transport + hyperon– 3D simulation to study non-axisymmetric effects– General Relativity– Magnetic field effects on angular momentum and rotation– …

• 2 D (axisymetric) simulation included microphysics but done in a Newtonian framework (Zwerger & Muller 97)

• 3D Newtonian (Bonazzola & Marck 93, Fryer &Warren 02) • 2 D GR including microphysics (Fryer & Heger 99 Dimmelmeier, Font &

Mueller, 02)• 3D including all aspects: not yet done.

Models core collapse simulation GW strength

Dimmelmeier et al. A&A 393 (02)

Sources @ 10 kpc

relativisticnewtonian

GW burst (core collapse) strength prediction

New born NS instabilities• SN remnant or accreting white dwarf collapse: very rapidly rotating NS

will develop non axi-symmetric dynamical instabilities (having angular dependence).

• The physics of NS is far from being simple. Many parameters play a role on the NS stability: equation of state for high density matter, matter superfluidity / superconducting, presence of proton, strange quarks, …)

• Pulsation normal modes: f and r modes are the most promising modes for GW emission ? – f-mode: fundamental (acoustic) pressure mode of the star (2-4 kHz). Rotation

change frequencies. GW emission damps f-modes within ~ a tenth of a second.– r-modes: inertial modes due to rotation (Coriolis force). Lots of work around

because have been thought to lead to high amplitude GW emission (GW radiation was thought to increase the amplitude of the mode)

– bar mode instabilities occur when rotational kinetic ratio exceeds β=0.27 – w-modes: pure GR effects (7 kHz) damped in a fraction of millisecond.

• GW Amplitude ? Last predictions are quite pessimistic. • Rate? Very unclear. The fraction of very rapidly NS after a SN is not

known (10-6 /yr/galaxy ?)Detection with first (and second) ITF generation :

hopeless!

Mergers of compact stars

unknown waveform !

known waveform:Damped sine and cosine !

Compact objects (NS, BH) merger

• Intermediate mass BH-BH merger: seems to be a promising source but incertitude on the population rate ….

• BH-BH merger rate: as high as 1 per year up to 100 Mpc?• Weak point: waveform not really known. Highly non linear regime.

• Use of EOB approach to go beyond the adiabatic PN approximation which breaks down before the last stable orbit [gr-qc/0211041]

3PN EOB templates …. but burst filters could be more robust to badly modeled merger waveforms than Wiener filtering

Equal mass BH reconstructed SNR fraction for inspiral, merger and

ringdown phases (LIGO sensitivity)(might be optimistic

[PRD 57, 4535 (1998)])

Black hole ring down• A BH that is distorted from its stationary Kerr configuration will radiate

GW that drive it back to the stationary state (hair loss theorem)

• A BH formed in core collapse will certainly be distorted: if large amounts of matter accrete onto it, it will continually driven into new states of distortion.

• Waveform: Quasi-normal mode: damped sine and cosine

• BH spectroscopy: deduce the BH parameters (mass M, spin a) from QNM parameters (f, τ).

• GW amplitude:

])1(63.01[123.0

Ma

Mf

)1(2

445.0

Maf

)()()( 15110

10.52/1

23

2/122

rMpc

fkHz

cMEh

(Q=πfτ)

)2sin(/ fte t

Detecting GW bursts: the problem

GW waveforms not accurately predicted

Standard technique of signal processing (match filtering) is prohibited

Need for robust methods

(sub-optimal)

What are the common characteristics of the burst sources?

• The burst signal duration is so short (~10 ms) that there is no modulation of the signal amplitude and phase due to Earth motion!

• The source is located in the sky. One detects the projection of the 2 polarizations and .

Needs at least 3 ITF (more preferably 4) to triangulate the position of burst source in the sky. Network analysis

• The typical frequency of the signal range from 100 Hz up to few kHz. Analysis performed on maximal sampling frequency: 20 kHz in Virgo

The false alarm rate is quite high if one wants to remain efficient to signals SNR < 10

SNR cut at 5 : fa rate : 41 / hour

SNR cut at 7: fa rate : 1.6 / year

h h

If Gaussian statistic …. But we are dominated by transient noise events!!!!

Network analysis is fundamental!

The techniques for the burst search

• Filters description:– Match filtering: for gaussian shaped and damped sine/cosine signals

– Sub-optimal methods for all other waveforms• Time domain filters:

• Time-frequency: power filtering and time-frequency transforms

• Filters comparison: Receiving Operator Characteristic curves (efficiency versus false alarm rate)

• Timing properties: estimation of the arrival time

• Whitening issue (mandatory for time domain filters)

• Usable (and optimal) if the signal is known a priori

• Principle: correlate data with a template (a copy of the expected signal)

• Signal to noise ratio:

• Signal to noise ratio can be seen as a scalar product :

• Templates placement: very simple in 1D, requires more work in 2D (see example with damped sine burst search)

0

~~

)(

*)( )( 4 dffS

ftfhh

h : detector output t : templateSh : detector noise spectral density

= <h|t>

Match filtering techniques

Signal and template : identical shapesTemplate : w = 1 ms. Signal : w = 1 msTemplate : normalized (t|t) =1 Signal : Intrinsic SNR = (h|h) = 10Filter max output : = (h|t) = 10

Signal and template : mismatched shapesTemplate : w = 1 ms. Signal : w = 5 msTemplate : normalized (t|t) =1 Signal : Intrinsic SNR = (h|h) = 10Filter max output : = (h|t) = 7

Match filtering exemple: peak correlator

Damped sine template placement

2D parameter space:• Q and f are uncorrelated• Q [2;16] and f [20;10000]

Minimal Match: ambiguity function between 2 close templates < MM

SNR loss < 1-MM

Templates are ellipses:

• No optimal method to cover a plane with varying ellipses

• Hexagonal tiling is optimal for circles on infinite plane. Then a geometrical transform

distort circles into ellipses.

Template number:

MMdQQdffsQfsQf ),(|),(),(

Example: N=698 damped sine templates for MM=97%

Time domain filters

• Time domain filters, rather signal independent:

one hypothesis about signal duration: several sliding windows (size N) in parallel for practical

implementation

Norm Filter (NF):

Mean Filter (MF):

ALF (linear fit of data (x=at+b) and slope and offset filter combination):

slope: offset:

1222

N

Nki

kii

NF

k xy

Nki

kii

MF

k xy N1

taxb22

ttxttxa

)1(2

baba

yk )12(

20122

NN

fa )1(

242

NNN

b

)121(

23

NN

yyykk

ALF

k

22

Time domain filters output

ROC for optimized filters

• Optimized filters = the window size is matched to the signal size• Signal = Gaussian peak (ω=1ms) (N=40 for MF and NF, N=140 for ALF)

MF and ALF performs identically in this ideal working caseOne always gain to combine slope (SF) and offset (OF) filtersNF performance are low (4 order of magnitude more f.a. @ same

efficiency)

NF

ALF

MF

ROC in a practical filter implementation• Filters are running with 10 sliding windows:

signal duration: 0.5, 0.75, 2, 1.25, 1.5, 2, 2.5 3.5 7.5 and 10 ms

• SNR = 5 (exhibit at most the filters’ difference)

• Gaussian peak (ω=1ms)

ALF performance is about the same!

MF efficiency decreases dramatically

compared to the use of 1 single signal

matched window !

The non matched windows increase

the number of false alarm for MF

f.a.rate @ efficiency = 50%

MF

ALFfilter ALF MF NFOptimal 10-7 10-7 >2.10-4

Practical 2.10-7 4.10-5 4.10-4

ROC in a practical implementation: signal robustness

• ROC for damped sine signal (f = 1 kHz, τ = 1 ms) and ZM (a2b2g1)

the filters performance may depend strongly on signal waveform.

(ALF: f.a. = 2.10-7 (Gaussian) 3.10-4 (Damped Sine) @ 50% efficiency)

MF : same behavior as for Gaussian because one of the window size matches one of the peaks of the damped sine signal.

ALF: best performance to detect ZM like waveforms

Damped sine ZM (a2b2g1)

Time frequency methods

• Time-frequency representation:– Spectrogram

– Wavelets

– Other transforms (Gabor, WignerVille, s-transform, …)

• Several methods used for detection

Power filters

• Several implementations in the literature: – Flanagan et al. (1998), Anderson et al (2000), Vicere 2002, Guidi et al

(2003), …

• The statistic is a measure of the rms noise in a given bandwidth (making different hypothesis on signal: duration, frequency spectrum, …)

– where sJ are the Fourier transformed data projected in a basis where the noise matrix <nI, nJ> is diagonal.

– where sJ are the coefficients of the data projected over a basis of N eigenfunctions of the noise matrix obtained using the Karhunen-Loeve decomposition. The data are previously filtered with a δ function (it is equivalent to divide by the power spectrum in frequency domain).

• Performance: does not seem (up to now) to exhibit as good efficiency as other filters for instance on SN waveforms (Zwerger & Mueller catalog).

N

JJ

JstQ1

2

2

)(

Examples of time-frequency transforms

Signal spectrum

ZM signal

“A trous” wavelet transformGabor transform

Spectrogram Pseudo Wigner Ville transform

Pure signal (Zwerger-Mueller SN). No noise added

Blurring by noise addition

Hypothesis: white gaussian noise

Spectrogram

Gabor transform

Pseudo Wigner Ville transform

Signal spectrum “A trous” dyadic wavelet

Noisy signal

Time frequency (s-transform)

dxiff

tftfS ee )(

22

||)(

2||

),(2

SNR = 10

Filters properties: timing accuracy

• Estimation of signal arrival time is important for:– working in a network of interferometers (coincidence analysis)– reconstructing the source location (maximal time delay between Hanford

and Virgo is 27 ms) – coincidence with ν detectors and optical telescopes

ν masses determination require a timing precision <1 ms

• Estimators construction:– Non trivial because it may depend both on the filters and the waveforms– Need to construct unbiased estimators or at least determine on simulation

their systematic bias and statistical errors

Study first the ideal case (Gaussian peak signal)one can obtain upper limit on the timing

resolutionMore realistic signals (ZM) results

Timing accuracy: Gaussian signal (ideal case)

• Definition: signal arrival time = time corresponding to the Gaussian peak maximum

• Time estimator for NF and MF:

• Time estimator for ALF:

double peak structure

• Systematic bias removed in the definition of the estimator

maxSNRe tt

2

2max

1max ttte

Timing accuracy: Gaussian signal – results

• The systematic bias and statistical precision on the signal arrival time have been evaluated using simulations (Gaussian white noise + signals calibrated according to the optimal SNR ρ)

• The statistical accuracy on the signal arrival time estimation depends on the optimal SNR ρ and the width ω of the signal

• Optimal filter results:

• NF filter:

• MF filter:

• ALF filter:

Linear dependency on ω for all filters except NF!All filters estimators have a statistical accuracy well below 1 ms for a standardpeak Gaussian signalALF shows good timing proprieties (not a priori obvious)

Timing accuracy: realistic case

• The good numbers obtained with Gaussian peaks must be moderated

• Realistic SN waveforms exhibit more complicated structure (several peaks). That implies:

– The systematic bias may depend on the signal waveform!

– One has to define a “robust” arrival time estimator

• An extended study has been performed with ALF filter (most efficient filter for ZM signal):

– Estimators: time of maximal SNR, time of the first bin above threshold, average time between 2 SNR peaks, …

No unbiased time estimator has been found

The minimal systematic bias is about 0.5 ms for a SNR=5

(average over the 78 ZM waveforms)

The time estimation precision may be much larger than 1 ms (especially for the type III ZM signals)

Whitening issue for time domain filters

• The ITF noise is colored.

• Time domain filters require to have a white noise (flat PSD) otherwise will trigger a lot because of the low frequencies.

• Need for whitening processing: simple linear filtering which “fits” the data.

Sensitivity to residual lines

• All the sub-optimal filters here require a data pre-whitening

• But how much are they sensitive to imperfect whitening?

Sensitivity to a single frequency component:

Measure with simulations the relative excess of false when varying the amplitude A and for f = 0.6 Hz (pendulum frequency)

f = 100, 200 and 400 Hz (power line harmonics)

Window size: MF, NF: N=50 ALF: N=170

Matched signal size: 2.5 ms for all the filters

)2sin( ftA

Sensitivity to residual lines

• Frequency dependence for MF:

MF averages frequencies higher than a cut off frequency:

Similar effect with ALF

• Specification defined using the spectrum flatness:

• Example: Amplitude line < 1%

HzNffc 4000

Hz6.0 Hz100

Hz200

Hz400

97.0

Network burst search

Burst detection algorithm also sensible to burst of noise as wellA single detector is not enough.

Coincidence with other detectors is mandatory to validate a detection.

Coincidence with other GW detectors: at least 3 for the signal reconstruction.

Coincidence with other messengers

Antenna Pattern poor directional information

Coïncidences in a network of GW detectors

3 6 interferometric detectors

Detection validationGW reconstruction

Geometrical acceptance

)()()( thFthFth xx

Fx and F+ depend on

• detector location

• Source position

• Polarisation angle Ψ

Sky map = Ψ averaged beam pattern functions

• LIGO maps similar by design

• Virgo and GEO more or less

similar due to geometrical proximity

• Virgo and LIGO maps are “orthogonal”

LIGO - Virgo antenna pattern

Coincidence

Definition of a time window depending on time delay between detectors

• The source location is not known: loose coincidence

has been determined on simulation (SNR dependence): (<0.3 ms for SNR>5)

for SNR>6

cDDnijij ji

ttt .

ijRMS

ijij ttt .max )42....10~( max msmst ij

ijRMSt

SNRmsmst ij

RMS

10

115.0~

Virgo-LIGO coincidence (loose)

Good sky global coverage:average efficiency of 67%

No more blind regions

Coincidence less likely:~ 20% for the 2 LIGO

~ 30% by adding VirgoLarge regions almost blind

Request on 1/3 Twofold coincidence

Loose coincidence in LIGO-Virgo network

• Best strategy: twofold coincidences (at least 2 among 3)

• Twofold coincidence is dominated by the 2 LIGO network (beam pattern matching)

• Single detector less efficient than coincidences

• Threefold coincidences are rare

SNRopt=10

Full network loose coincidence

Twofold coincidence quite likely even at small false alarm rate

Threefold coincidences also possible

Larger coincidences much rarer

SNRopt=10

Coherent analysis

• Coherent statistics derived from a likelihood ratio (Pai, Bose & Dhurandhar (2001) method for coalescing binaries)

• General case: the source location is not known

bank of N templates to cover the full sky

Example: Gaussian peak signal:

N goes 1/w2

for w=1ms and MM=0.97 N ~ 5000 whatever the configuration

(up to 6 ITFs)

Virgo-LIGO coherent analysis results

Significant improvement in detection efficiency with respect to the coincidence case

Efficiency remains above 60% for SNRopt = 10 even at a false alarm rate of 1

per week (35% for a twofold coincidence)

Still no real hope to detect a weak signal (SNRopt = 5) in the 3 interferometer

Virgo-LIGO network

Full network coherent analysis

Clear enhancement of detection efficiencies by going from 3 to 6 ITFs

Almost certain detection for SNRopt =

10

Still more than 80% efficiency @ SNRopt = 7.5

Efficiency remains limited @ SNRopt

= 5 and below

GW burst and Gamma Ray Burst (GRB)

• GRB: short but very energetic pulses of gamma rays emitted at cosmological distances.

• Isotropic distribution on the sky

• Quite frequent: rate as high as 1 per day

• 2 populations: short and long (from 10 ms up to 100s)

• Origin: present consensus: associated with BH formation (hypernovae), compact binary inspirals, collapsars)

Good reason to look for coincidence with GW (bursts)!

LIGO GW burst-GRB coincidence search: they tried to detect GW burst in coincidence with recent detected GRB while taking data:

negative result up to now.

Few words about burst detection potentialas a conclusion

• Only SN in the Galaxy with the first generation of ITF– More efficient filter for SN waveforms (Zerger & Mueller catalog): ALF

– Average distance of detection: 22.5 kpc !

• Robustness to various waveforms: the best strategy is to use different filters in order to cover all kind of waveforms. There is no “universal” filter.

• Filters properties: the efficiency issue is one part of the problem.– Timing accuracy to reconstruct the arrival time

– Robustness of time domain filters to non perfect whitening processing (due to non stationarities in the data for instance)

• Network analysis:– Coherent analysis is more efficient (gain of factor at least 2 on efficiency)

but it is “heavier” to be implemented …

– Need to have a network composed of more than 3 interferometers …

References• Sources:

– Stellar collapse and GW: C. Fryer et al. astro-ph/0211609– Quasi-Normal Modes of Stars and Black Holes: K. Kokkotas et al, LLR www.

livingreviews.org/Articles/Volume2/1999-2kokkotas– Gravitational Waves from Gravitational Collapse, K. New, LLR www.

livingreviews.org/Articles/Volume6/2003-2new– E. Flanagan & S. Hughes, PRD 57, 4566 (1998)– N. Andersson, astro-ph/0211057– V. Ferrari et al., astro-ph/0310896, CQG 20, S841-S851 (2003)– S. Detweiler Proc. R. Soc. London, Ser. A, 352, 381-395, (1977)

• Detection in GW interferometers:– N. Arnaud et al. PRD 59, 082002 (1999), PRD 63, 042002 (2001), PRD 65, 033010 (2002),

PRD 67, 062004 (2003), PRD 67, 102003 (2003), gr-qc/0307101– A. Vicere PRD 66, 062002 (2002)– J. Sylvestre PRD 66, 102004 (2002)– E. Chassande-Mottin PRD 67, 102001 (2003)– W. Anderson et al.PRD 63, 042003 (2001)– E. Cuoco et al. PRD 64, 122002 (2001)– B. Abbot et al. (LIGO Collaboration) gr-qc/0312056

References

• Network:– N. arnaud et al. PRD 65, 042004 (2002), PRD 68, 102001 (2003)

– A. Pai et al. PRD 64, 042004 (2001)

– A. Searle et al. CQG 19, 1465 (2002)