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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: