06/2006

Elba, Italy

Linqing Wen

MPI/AEI, Germany

Network Analysis of Gravitational Waves

Outline

Motivation

Principle

Network

Angular resolution

Detection/veto method

Waveform extraction

Conclusion

Network Analysis

Detection Confidence

Source Localization

Waveform Extraction

Test GR

10 ms

25 ms

30 ms

L1-AIGO : 40 ms

FAQ

Data analysts’ questions

How to combine data for best detection

Different noise level in different frequencies and in different detectors. Different directional sensitivity

How to distinguish between detector glitches and real GW events ? (veto/consistency check)

FAQ

Astronomers/astrophysicists’ questions

How well can we localize the source

How much information can we extract for the waveform?

(especially when we have no prior information on GW waveforms)

Principle

General Principle

Detector response is linear to two wave polarizations

assuming stationary Gaussian noise

constant antenna beam pattern

short duration data segments

For each frequency bin k:

Matrix A consists of time-delay info and directional sensitivity weighted by noise

For all data

* A: block diagonal.

Each frequency component can be treated independently

Tricks for Network Analysis
of Burst Sources

Consider two sets of unknown parameters:

source direction, wave polarizations at each frequency

Apply Fisher's information matrix

Principle for Network Analysis
(burst search)

Based on (1) Fisher's information matrix (2) Singular Value Decomposition method

-> determine best possible angular resolution

-> principle axes for waveform estimation

-> optimal detection statistic and null-stream for consistency check/veto

MLR principle

Maximum SNR principle

1. Network Angular Resolution
(Revisited)

Single-detector Angular Resolution

- Single GW detector is a good all sky monitor, but not a pointing instrument

Angular resolution (FWHM) ~100 degrees

Network Angular Resolution:
Previous Studies

Order of magnitude estimation from diffraction limit for single -f source

Previous studies

numerical study/Kip’s pvt comm. for 3-detector network

Guersel & Tinto 1989, Sylvestre 2003

Inspiral coherent search using Fisher’s information matrix/numerical result: Pai et al. 2001,

LISA as a 2-detector network: Fisher’s information matrix/MF/numerical result: Cutler 1998

Network Angular Resolution
(best case scenario)

We give explicit analytic expression for multi-detectors

Fisher's information matrix~ I(single) + I(time-delay)

Better directional resolution for

higher SNR, higher freq., more detectors

larger time-delay derivative against sky direction

Larger base-line

Network Angular Resolution
(best case)

Worse directional resolution if wave direction is along

line connecting 2-detectors

plane formed by 3-detectors

only one direction can be determined (better than single-detector sensitivity) if wave travels in the plane of a 3-detector network

Network Angular Resolution
(burst search)

when no information about waveform exist

angular resolution degraded by a projection factor

recall perfect case

if A is full-ranking, i.e., with no redundant degree of freedom, directional information cannot be determined

BH-BH merger waveform. f~ 500 Hz, Mean SNR (LHV) ~3, perfect case, mean omega~10 deg^2

1

1

perfect case, mean SNR ~3.3, mean~ 1 square deg

imperfect case degraded by a factor of 3

BH-BH (10-10 Msun) merger waveform, fc~ 500 Hz

T~ 7 ms, (Lazarus project). Detector noise = designed sensitivity

Chandra/HRC

FOV

Diffraction limit/SNR

Network Angular Resolution

Angular resolution ~ 0.2 square-degree can be reached for burst-search

At SNR~10, by L1-H1-V-G-T-A

Advanced detectors, with ~10 times better sensitivity, all networks are possible for the same sources

Direct astronomical follow-up observation of BH-BH coalescence is possible

for a large detector network and for advanced detectors.

GEO/TAMA improve the angular resolution

long baseline + complimentary antenna beam pattern

2. Detection Statistics, Null Stream Construction, Waveform Extraction
-Using Singular Value Decomposition Method

For all data

* A: block diagonal.

Each frequency component can be treated independently

Principle for Network Analysis
(burst search)

Our approach

perform singular value decomposition on A

For each frequency:

Principle for Network Analysis
(burst search)

Construct new data stream

projection of data onto basis with ranking based on directional sensitivity and noise level

d' is the MLR estimator for new wave parameter h' – diagonal elements of Fisher’s Info. Matrix

Reconstructed noise ~ N(0,1)

Signal Information

Ill-conditioned?

(unstable sol.

to h+, hx)

Null (no-signal)

region

treat non-null component d' as new data and apply analysis

MF or burst search related method

Similar to excess power method, we search for maximum SNR by windowing the axes

Detection Statistic
(burst search)

If add all non-zero component, recover standard treatment of Anderson (2001), Hughes & Flanagan (1998)

Identifying Null-stream
(burst search)

zero component of h' corresponds to null streams

there are “absolute” null-streams

See also Wen & Schutz, CQG (2005), Ajith et al (2006), Chatterji et al (2006) gr-qc

Waveform Extraction and regularize “ill-conditioned” Streams
(burst search)

For non-zero singular values only

Need treatments for ill-posed region

Recover Hughes & Flanagan (1998) estimator if no treatment is used

Which is equivalent to Guesel & Tinto (1998) for 3-detector case

Waveform Extraction and regularizing
ill-conditioned Streams
(burst search)

Regularize ill-posed region:

for nearly parallel antenna beam pattern (e.g., L1-H1)

See also Rakhmanov, M. (2006), Klimenko et al (2005)

or noiser detectors

Otherwise, unstable solution to h+, hx

Help improving angular resolution & detection statistic

All Roads Lead to Rome

LHG, SNR ~ 20, SNR(GEO)~2, src location in center

Standard MLR

“Absolute” Null-Stream, Prob(noise)

New Detection Statistic

Treated null-stream

LHGTV, SNR=25

Real GW Events

localization and consistency check

Wen L. and Schutz, B. 2005, CQG

Glitches

with “inconsistent” amplitudes h(H1) *= 0.5, h(G1) *=2

L1-H1-G1

L1-H1

LHG

LHV

Wen & Schutz (2006), to be submitted

Efficiency of Null-Streams

Localization of Null-Streams, Wen & Schutz (2006), TBS

Conclusion

We have studied angular resolution of the GW network

Analytic solution is given for both known and unknown waveforms

Apply to realistic network

We have proposed constructing new data stream based on singular value decomposition method

optimal detection statistic/null stream/waveform extraction/localization run in parallel

simple to implement

MLR -> a GW is detected only if both detection and null stream are consistent in sky direction

2004-12-17T23:31:02