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We report on the methods and results of the first dedicated search for gravitational waves emitted during the inspiral of compact binaries with spinning component bodies. We analyze 788 hours of data collected during the third science run (S3) of the LIGO detectors. We searched for binary systems using a detection template family specially designed to capture the effects of the spin-induced precession of the orbital plane. We present details of the techniques developed to enable this search for spin-modulated gravitational waves, highlighting the differences between this and other recent searches for binaries with nonspinning components. The template bank we employed was found to yield high matches with our spin-modulated target waveform for binaries with masses in the asymmetric range 1.0M(circle dot) <m(1) <3.0M(circle dot) and 12.0M(circle dot) <m(2) <20.0M(circle dot) which is where we would expect the spin of the binary's components to have a significant effect. We find that our search of S3 LIGO data has good sensitivity to binaries in the Milky Way and to a small fraction of binaries in M31 and M33 with masses in the range 1.0M(circle dot) <m(1), m(2) <20.0M(circle dot). No gravitational wave signals were identified during this search. Assuming a binary population with spinning components and Gaussian distribution of masses representing a prototypical neutron star-black hole system with m(1) similar or equal to 1.35M(circle dot) and m(2) similar or equal to 5M(circle dot), we calculate the 90%-confidence upper limit on the rate of coalescence of these systems to be 15.9 yr(-1) L-10(-1), where L-10 is 10(10) times the blue light luminosity of the Sun.