1. 1 Computing the F-Statistic for Continuous Gravitational Waves using a network of detectors Iraj Gholami, Reinhard Prix and Curt Cutler Max-Planck-Institut fuer Gravitationsphysik (Albert Einstein Institute) Potsdam, Germany 10 th GWDAW, 14-17 Dec 2005 Univ. of Texas at Brownsville (UTB)

2. 2 Derivation of F-Statistic for Multi-IFO with independent noises: Starting from the definition of the likelihood function having multiple data sources

3. 3 Defining the F-Statistic as the log of likelihood function: “X” indicates different detectors

4. 4

5. 5 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.

6. 6 rewriting the likelihood equation

7. 7 therefore Maximizing the likelihood function over the detector independent components

8. 8 important key point important key point; The formulation is the same as single detector, except, sum each element over different detectors we do sum each element over different detectors

9. 9 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

10. 10 Taking

11. 11 Now we can write the F-Statistic more explicitly we can re-write the F-Statistic in the form of If we define:

12. 12

13. 13 N: number of bins M: number of SFTs

14. 14

15. 15

16. 16 Our SFT to be:

17. 17

18. 18 Conclusion: