Gravitational waves from merging compact binaries: How accurately can one extract the binary’s parameters from the inspiral waveform?

Abstract

The most promising source of gravitational waves for the planned kilometer-size laser-interferometer detectors LIGO and VIRGO are merging compact binaries, i.e., neutron-star--neutron-star (NS-NS), neutron-star--black-hole (NS-BH), and black-hole--black-hole (BH-BH) binaries. We investigate how accurately the distance to the source and the masses and spins of the two bodies will be measured from the inspiral gravitational wave signals by the three-detector LIGO-VIRGO network using ``advanced detectors'' (those present a few years after initial operation). The large number of cycles in the observable waveform increases our sensitivity to those parameters that affect the inspiral rate, and thereby the evolution of the waveform's phase. These parameters are thus measured much more accurately than parameters which affect the waveform's polarization or amplitude. To lowest order in a post-Newtonian expansion, the evolution of the waveform's phase depends only on the combination scrM\ensuremath{\equiv}(${\mathit{M}}_{1}$${\mathit{M}}_{2}$${)}^{3/5}$(${\mathit{M}}_{1}$+${\mathit{M}}_{2}$${)}^{\mathrm{\ensuremath{-}}1/5}$ of the masses ${\mathit{M}}_{1}$ and ${\mathit{M}}_{2}$ of the two bodies, which is known as the ``chirp mass.'' To post-1-Newtonian order, the waveform's phase also depends sensitively on the binary's reduced mass \ensuremath{\mu}\ensuremath{\equiv}${\mathit{M}}_{1}$${\mathit{M}}_{2}$/(${\mathit{M}}_{1}$+${\mathit{M}}_{2}$) allowing, in principle, a measurement of both ${\mathit{M}}_{1}$ and ${\mathit{M}}_{2}$ with high accuracy.We show that the principal obstruction to measuring ${\mathit{M}}_{1}$ and ${\mathit{M}}_{2}$ is the post-1.5-Newtonian effect of the bodies' spins on the waveform's phase, which can mimic the effects that allow \ensuremath{\mu} to be determined. The chirp mass is measurable with an accuracy \ensuremath{\Delta}scrM/scrM\ensuremath{\approxeq}0.1%--1%. Although this is a remarkably small error bar, it is \ensuremath{\sim}10 times larger than previous estimates of \ensuremath{\Delta}scrM/scrM which neglected post-Newtonian effects. The reduced mass is measurable to \ensuremath{\sim}10%--15% for NS-NS and NS-BH binaries, and \ensuremath{\sim}50% for BH-BH binaries (assuming 10${\mathit{M}}_{\mathrm{\ensuremath{\bigodot}}}$ BH's). Measurements of the masses and spins are strongly correlated; there is a combination of \ensuremath{\mu} and the spin angular momenta that is measured to within \ensuremath{\sim}1%. Moreover, if both spins were somehow known to be small (\ensuremath{\lesssim}0.01${\mathit{M}}_{1}^{2}$ and \ensuremath{\lesssim}0.01${\mathit{M}}_{2}^{2}$, respectively), then \ensuremath{\mu} could be determined to within \ensuremath{\sim}1%. Finally, building on earlier work of Markovi\ifmmode \acute{c}\else \'{c}\fi{}, we derive an approximate, analytic expression for the accuracy \ensuremath{\Delta}D of mesurements of the distance D to the binary, for an arbitrary network of detectors. This expression is accurate to linear order in 1/\ensuremath{\rho}, where \ensuremath{\rho} is the signal-to-noise ratio. We also show that, contrary to previous expectations, contributions to \ensuremath{\Delta}D/D that are nonlinear in 1/\ensuremath{\rho} are significant, and we develop an approximation scheme for including the dominant of these nonlinear effects. Using a Monte Carlo simulation we estimate that distance measurement accuracies will be \ensuremath{\le}15% for \ensuremath{\sim}8% of the detected signals, and \ensuremath{\le}30% for \ensuremath{\sim}60% of the signals, for the LIGO-VIRGO three-detector network.