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The foundations are laid for the numerical computation of the actual worldline for a particle orbiting a black hole and emitting gravitational waves. The essential practicalities of this computation are here illustrated for a scalar particle of infinitesimal size and small but finite scalar charge. This particle deviates from a geodesic because it interacts with its own retarded field ψ ret . A recently introduced [1] Green's function G S precisely determines the singular part, ψ S , of the retarded field. This part exerts no force on the particle. The remainder of the field ψ R = ψ ret − ψ S is a vacuum solution of the field equation and is entirely responsible for the self-force. A particular, locally inertial coordinate system is used to determine an expansion of ψ S in the vicinity of the particle. For a particle in a circular orbit in the Schwarzschild geometry, the mode-sum decomposition of the difference between ψ ret and the dominant terms in the expansion of ψ S provide a mode-sum decomposition of an approximation for ψ R from which the self-force is obtained. When more terms are included in the expansion, the approximation for ψ R is increasingly differentiable, and the mode-sum for the self-force converges more rapidly.

We carry out two searches for periodic gravitational waves using the most sensitive few hours of data from the second LIGO science run. Both searches exploit fully coherent matched filtering and cover wide areas of parameter space, an innovation over previous analyses which requires considerable algorithm development and computational power. The first search is targeted at isolated, previously unknown neutron stars, covers the entire sky in the frequency band 160 -728.8 Hz, and assumes a frequency derivative of less than 4 10 ÿ10 Hz=s. The second search targets the accreting neutron star in the lowmass x-ray binary Scorpius X-1 and covers the frequency bands 464-484 Hz and 604-624 Hz as well as the two relevant binary orbit parameters. Because of the high computational cost of these searches we limit the analyses to the most sensitive 10 hours and 6 hours of data, respectively. Given the limited sensitivity and duration of the analyzed data set, we do not attempt deep follow-up studies. Rather we concentrate on demonstrating the data analysis method on a real data set and present our results as upper limits over large volumes of the parameter space. In order to achieve this, we look for coincidences in parameter space between the Livingston and Hanford 4-km interferometers. For isolated neutron stars our 95% confidence level upper limits on the gravitational wave strain amplitude range from 6:6 10 ÿ23 to 1 10 ÿ21 across the frequency band; for Scorpius X-1 they range from 1:7 10 ÿ22 to 1:3 10 ÿ21 across the two 20-Hz frequency bands. The upper limits presented in this paper are the first broadband wide parameter space upper limits on periodic gravitational waves from coherent search techniques. The methods developed here lay the foundations for upcoming hierarchical searches of more sensitive data which may detect astrophysical signals.

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The Laser Interferometer Gravitational-Wave Observatory (LIGO) has performed the fourth science run, S4, with significantly improved interferometer sensitivities with respect to previous runs. Using data acquired during this science run, we place a limit on the amplitude of a stochastic background of gravitational waves. For a frequency independent spectrum, the new Bayesian 90% upper limit is GW ; H 0 / 72 km s À1 Mpc À1 À Á Â Ã 2 < 6:5 ; 10 À5 . This is currently the most sensitive result in the frequency range 51Y150 Hz, with a factor of 13 improvement over the previous LIGO result. We discuss the complementarity of the new result with other constraints on a stochastic background of gravitational waves, and we investigate implications of the new result for different models of this background.