S.Klimenko, August 2005, LSC, G050434-00-Z Constraint likelihood analysis with a network of GW detectors S.Klimenko University of Florida, in collaboration

S.Klimenko, August 2005, LSC, G050434-00-Z

Constraint likelihood analysis with a network of GW detectors

S.Klimenko

University of Florida,

in collaboration with

S.Mohanty,M.Rakhmanov,G.Mitselmakher

Likelihood analysis Two detector paradox Network sensitivity to GW polarizations Constraint likelihood Results & Summary

gr-qc/0508068


Likelihood analysis for bursts

Outlined at Flanagan, Hughes, PRD57 4577 (1998) Mohanty et al, CQG 21 S1831 (2004)

Likelihood for Gaussian noise with variance 2 k and GW

waveform u: xk[i] – detector output, Fk – antenna patterns

For unknown GW signal treat every sample of u as an independent variable find u[i] from variation of L

i k

kkkk

ixiixuL ][][2

1)( 22

2

x

x

ihhu

ihhu

~

2

~ ,

2kk

kkk

k

iFFA

iFFA

kkk AiuAiui ][~~][ detector response -


Solution for GW waveforms

Given by two linear equations:

Network response matrix MR

Network data vector

Network antenna patterns

Solution: -- estimator of GW

waveform

k k

kkAxX2

k k

kkr

k k

kc

AAg

Ag ,

~ ,

22

2

ccr

crs ggg

XgXgu ~

~

2

Sum over detectors

h

h

ggg

ggg

X

X

crc

ccr

)Re()Im(

)Im()Re(

)Im(

)Re(

detection statistic is obtained from the likelihood functional by substituting u with the solution


Two detector paradox

aligned detectors (unit noise variance for simplicity):

If separated LA has directional sensitivity (circle on the sky)

misaligned detectors (even infinitesimally):solution for GW waveform:

No directional sensitivity for two misaligned detectors!

2

21221

xxiiixixL

i

2211 , xx

power cross-correlation

21221141 2 xxxxxxLA

221121 xxxxLM

(Johnston, Mohanty)


Dominant polarization wave frame

such a wave frame where the network antenna pattern gc is positive and real

g – network sensitivity factor – network alignment factor

diagonal MR independent statistics for h1 & h2 -- GW polarizations in the DP frame

total signal

Only first component is detected if 0

MLRLhhgSNR 22 22

21

22

21 , hh

0

01

||0

0||g

gg

ggM

cr

crR

)()( ),( 221121 hLhLhhLMLR

-sum-square energies of GW polarizations


Network alignment factor

||

||

cr

cr

gg

gg

shows relative sensitivityto two GW components

For aligned network =0

blue<0.1

H1-L1

+GEO

+VIRGO

+TAMA

22

21 hhSNR

to be detected with the same SNR h2 should be

1/ times stronger then h1


Network sensitivity factor

|| cr ggg

22

212 hhgSNR

H1-L1

+GEO

+VIRGO

+TAMA

need several detectors for more

uniform sky coverage


constraint

In average for a population of bursts expect

Hard constraint ignore the second component

Soft constraint weight the second component

remove un-physical solutions produced by noise sacrifice small fraction of GW signals but enhance detection efficiency for the rest of sources

22

21 hh

)()( 2211 hLhLLhard

)( )( 2211 hLhLLsoft

another approach: penalized likelihood (Mohanty et al, LIGO-G050361-00-Z)


Two detector sky maps

Source at =120=80

Constraint likelihood gives directional

information in the case of two

detectors

H1-L1

H1-G1

Simulation•Equal detector sensitivity•white, G-noise•injections:-Lazarus BH-BH-2-polarizations


Comparison of different methods

Source at =120, =80,

SNR~13

H1-L1-GEO

hard

soft

standard


ROC

H1L1

GEO

<SNR>=5

<SNR>=8

sources uniformly distributed over the sky

sky average SNR = 7

H1-L1-GEO


Angular resolution for different methods

fraction of events in the cone|detected-injected|<16 degrees

H1L1

GEO


WaveBurst.net

SG20 injections into S4 data

Simulated LIGO noise

H1-H2-L1

WAT/DMT implementation (klimenko) First run on S4 data with SG20 injections (Yakushin)

Likelihood vs WaveBurst significance


Likelihood

Likelihood distribution of WaveBurst triggers for Simulated LIGO noise (black) SG20 injections into S4 data (blue, red)

WB significance >1.9

No selection on WB significance


Summary

Coherent network method for GW bursts searches Based on the likelihood analysis Obtain max likelihood ratio statistics by constrained variation of

likelihood functional

detection statistic uses both power and cross-correlation terms

give information about source location, waveform reconstruction

any number of detectors, arbitrary alignment

Plans Finish debugging of hierarchical search WaveBurst.net Study waveform and coordinate reconstruction on simulations Compare with r-statistics for H1-H2-L1 (useful cross-check) Analyze S 4 LIGO-GEO data get ready for S5 Implement non-hierarchical search (cpu consuming)

Documents

S.Klimenko, August 2005, LSC, G050434-00-Z Constraint likelihood analysis with a network of GW detectors S.Klimenko University of Florida, in collaboration