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Transcript of 1 Computing the F-Statistic for Continuous Gravitational Waves using a network of detectors Iraj...
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Computing the F-Statistic for Continuous Gravitational Waves
using a network of detectors
Iraj Gholami, Iraj Gholami, Reinhard Prix and Curt CutlerReinhard Prix and Curt Cutler
Max-Planck-Institut fuer Gravitationsphysik(Albert Einstein Institute)
Potsdam, Germany
1010thth GWDAW, 14-17 Dec 2005 GWDAW, 14-17 Dec 2005Univ. of Texas at Brownsville (UTB)Univ. of Texas at Brownsville (UTB)
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Derivation of F-Statistic for Multi-IFO
with independent noiseswith independent noises::
Starting from the definition of the likelihood function
having multiple
data sources
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Defining the F-Statistic as the log of likelihood function:
“X” indicates different detectors
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Since the signal depends nonlinearly on the these 8 parameters, one can make a simple changes in the variables such that dependency of signal with the new variables is linear.
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rewriting the likelihood equation
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therefore
Maximizing the likelihood function over the detector independent components
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important key pointimportant key point;
The formulation is the same as single detector, except,
we do sum each element over different detectorssum each element over different detectors
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Because both the observation time and 1 day (time scale on which the a(t) and b(t)vary) are vastly larger than the period of the GWs, we can use the following estimation
Introducing the
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Taking
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Now we can write the F-Statistic more explicitly
we can re-write the F-Statistic in the form of
If we define:
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N: number of binsM: number of SFTs
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Our SFT to be:
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Conclusion: