S.Klimenko, August 2005, LSC, G050434-00-Z Constraint likelihood analysis with a network of GW...
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Transcript of S.Klimenko, August 2005, LSC, G050434-00-Z Constraint likelihood analysis with a network of GW...
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
S.Klimenko, August 2005, LSC, G050434-00-Z
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 -
S.Klimenko, August 2005, LSC, G050434-00-Z
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
S.Klimenko, August 2005, LSC, G050434-00-Z
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)
S.Klimenko, August 2005, LSC, G050434-00-Z
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
S.Klimenko, August 2005, LSC, G050434-00-Z
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
S.Klimenko, August 2005, LSC, G050434-00-Z
Network sensitivity factor
|| cr ggg
22
212 hhgSNR
H1-L1
+GEO
+VIRGO
+TAMA
need several detectors for more
uniform sky coverage
S.Klimenko, August 2005, LSC, G050434-00-Z
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)
S.Klimenko, August 2005, LSC, G050434-00-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
S.Klimenko, August 2005, LSC, G050434-00-Z
Comparison of different methods
Source at =120, =80,
SNR~13
H1-L1-GEO
hard
soft
standard
S.Klimenko, August 2005, LSC, G050434-00-Z
ROC
H1L1
GEO
<SNR>=5
<SNR>=8
sources uniformly distributed over the sky
sky average SNR = 7
H1-L1-GEO
S.Klimenko, August 2005, LSC, G050434-00-Z
Angular resolution for different methods
fraction of events in the cone|detected-injected|<16 degrees
H1L1
GEO
S.Klimenko, August 2005, LSC, G050434-00-Z
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
S.Klimenko, August 2005, LSC, G050434-00-Z
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
S.Klimenko, August 2005, LSC, G050434-00-Z
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)