Observing binary inspiral in gravitational radiation: One interferometer

Abstract

Close binary systems of compact objects with less than ten minutes remaining before coalescence are readily identifiable sources of gravitational radiation for the United States Laser Interferometer Gravitational-Wave Observatory (LIGO) and the French-Italian VIRGO gravitational-wave observatory. As a start toward assessing the full capabilities of the LIGO-VIRGO detector network, we investigate the sensitivity of individual LIGO-VIRGO-like interferometers and the precision with which they can determine the characteristics of an inspiralling binary system. Since the two interferometers of the LIGO detector share nearly the same orientation, their joint sensitivity is similar to that of a single, more sensitive interferometer. We express our results for a single interferometer of both initial and advanced LIGO design, and also for the LIGO detector in the limit that its two interferometers share exactly the same orientation. We approximate the secular evolution of a binary system as driven exclusively by its leading-order quadrupole gravitational radiation. Observations of a binary in a single interferometer are described by four characteristic quantities: an amplitude $\mathcal{A}$, a chirp mass $\mathcal{M}$, a time $T$, and a phase $\ensuremath{\psi}$. We find the amplitude signal-to-noise ratio (SNR) $\ensuremath{\rho}$ of an observed binary system as a function of $\mathcal{A}$ and $\mathcal{M}$ for a particular orientation of the binary with respect to the interferometer, and also the distribution of SNR's for randomly oriented binaries at a constant distance. To assess the interferometer sensitivity, we calculate the rate at which sources are expected to be observed and the range to which they are observable. Assuming a conservative rate density for coalescing neutron-star binary systems of 8\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}8}$ ${\mathrm{yr}}^{\ensuremath{-}1}$ ${\mathrm{Mpc}}^{\ensuremath{-}3}$, we find that the advanced LIGO detector will observe approximately 69 ${\mathrm{yr}}^{\ensuremath{-}1}$ with an amplitude SNR greater than 8. Of these, approximately 7 ${\mathrm{yr}}^{\ensuremath{-}1}$ will be from binaries at distances greater than 950 Mpc. We give analytic and numerical results for the precision with which each of the characteristic quantities can be determined by interferometer observations. For neutron-star binaries, the fractional $1\ensuremath{\sigma}$ statistical error in the determination of $\mathcal{A}$ is equal to $\frac{1}{\ensuremath{\rho}}$. For $\ensuremath{\rho}>8$, the fractional 1\ensuremath{\sigma} error in the measurement of $\mathcal{M}$ in the advanced LIGO detectors is less than 2\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}5}$, a phenomenal precision. The characteristic time is related to the moment when coalescence occurs, and can be measured in the advanced detectors with a $1\ensuremath{\sigma}$ uncertainty of less than 3\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}4}$ s (assuming $\ensuremath{\rho}>8$). We also explore the sensitivity of these results to a tunable parameter in the interferometer design (the recycling frequency). The optimum choice of the parameter is dependent on the goal of the observations, e.g., maximizing the rate of detections or maximizing the precision of measurement. We determine the optimum parameter values for these two cases. The calculations leading to the SNR and the precision of measurement assume that the interferometer observations extend over only the last several minutes of binary inspiral, during which time the orbital frequency increases from approximately 5 Hz to 500 Hz. We examine the sensitivity of our results to the elapsed time of the observation and show that observations of longer duration lead to very little improvement in the SNR or the precision of measurement.