Testing coherent code for coalescing binaries network analysis

1. Simona Birindelli INFN Pisa, Università di Pisa Leone B. Bosi INFN Perugia, Università di Perugia Andrea Viceré INFN Firenze, Università di Urbino Testing coherent code for coalescing binaries network analysis

2. Determination of the source position With a network of three detectors we can determine the source position (modulo a reflection with respect to the three interferometers plane), using the coincident and coherent analysis methods. We have tested a combination of the two methods, which starts from a coincidence and tries to improve the determination using the coherent analysis, first using a “classic” maximization and successively with a fit of the network statistic. We have considered the usual network composed by Virgo and the two 4 Km LIGOs.

3. Coherent analysis method The basic idea is to “construct” an ideal detector equivalent to the network, to which each real detector coherently contributes with its sensitivity, position, orientation.. The most useful network statistic (see Phys.Rev D 64, 042004(2001) by Pai, Dhurandhar and Bose)is the Logarithm of the Likelyhood Ratio, and can be written as: where is a matrix with as many rows as the number of detectors in the network, depending on the interferometers locations, their relative sensitivities and source position is the usual Wiener correlator computed (for the two quadratures) by each experiment, and depends on stellar masses and location is the time delay of the I-th detector with respect to the network frame.

4. The work L.Bosi produced with Merlino coalescing binaries events using the LIGO-Virgo project 1b simulated data. The exact masses of the injections were used for the Wiener filter. The SNR threshold was set at SNR 4, thus resulting in a rate of several events per second. In order to make it possible to compute the logarithm of the likelyhood ratio for different directions in the sky(and therefore different shifts within the correlators) for each detector and for each events, the correlators were saved around the maxima.

5. First simple test: crude maximization of the LLR Choosing a pair of interferometers we have searched for double coincidences (events with both SNR > 6), then we have searched over the events of the third detector to find the one which maximizes the LLR (with masses and time delays compatible). More in details, for each double coincidence we have computed the LLR, and we have maximized it over the correlators (the two quadratures) of the third detector. So we have extrapolated the best guess for the signal arrival time at the third interferometer, and determined the source position with the same geometric method used for the coincident search. We have repeated the procedure with the other two possible combinations of interferometers (a couple, and the third), and chosen the reconstructions with maximum LLRs.

6. First test: accuracy of the reconstruction For each event we have computed the source position also with the triple coincidence method, using the time of the maximum of the correlator at each detector. The crude maximization of LLR did not seem to systematically improve the determination, probably because the maximization over one correlator only is not sufficient to optimize the position reconstruction

7. Perform a global maximization over all the correlators simultaneously: Starting from a triple coincidence, we have maximized the LLR around each maximum over the three correlators in order to refine the reconstruction. (Coherent maximization) Make a fit of the LLR behavior instead of simply take its maximum value: From its behavior near the maxima, we have supposed that the LLR could be fitted with a parabolic function of the three correlator indexes. (Coherent fit) Second test: global maximization and parabolic fit to the LLR

8. The reconstructed position is improved fitting the likelyhood respect to the coincidence one, and also respect to the LLR total maximization Comparison between the methods

9. The accuracy during a day is quite uniform for all the three methods NGC6744 reconstructed position - real position • Coincident dec mean = 0°46’ dec standard deviation = 6°40’ RA mean = -7°43’ RA std. dev = 52°35’ • Coherent fit dec mean = 0°04’ dec standard deviation = 2°45’ RA mean = 1°08’ RA std. dev = 4°20’ • Coherent max dec mean = 0°48’ dec standard deviation = 8°55’ RA mean = 1°10’ RA std. dev = 14°34’ M87 reconstructed position - real position • Coincident dec mean = 3°45’ dec standard deviation = 8°59’ RA mean = -3°31’ RA std. dev = 14°49’ • Coherent fit dec mean = 0°38’ dec standard deviation = 3°37’ RA mean = -0°02’ RA std. dev = 1°58’ • Coherent max dec mean = 0°09’ dec standard deviation = 7°20’ RA mean = 1°40’ RA std. dev = 9°21’

10. Conclusions and future work In this work we have done a comparison of coincident and coherent methods in the reconstruction of the source position, using a network of three interferometers. We have focused on signal emitted by binary newtron stars. The results obtained using LIGO-Virgo simulated data appear to be promising. The maximization of the network LLR followed by a fit of the LLR allowed to improve the source position The next step is to fully characterize the method, assessing its performance at different SNR levels, for a wider range of sky locations.