The Fisher matrix is an easily computed mathematical object that is widely used to predict the parameter-estimation performance of gravitational-wave observations from the local dependence of the waveforms on the source parameters. Unfortunately, Fisher-matrix analyses can often be incorrect, especially for complicated waveforms or relatively weak signals; until now there have been no straightforward criteria to decide when this is the case.
In this work I provide practical tools to diagnose the correctness of the Fisher-matrix formalism for specific observations. In particular, I discuss what happens when some source parameters influence waveforms only weakly; when it is necessary to take into account the prior probabilities of source parameters; and, most important, when gravitational-wave signals can be considered strong enough to believe Fisher-matrix results. I also extend the formalism to include terms of higher order in the local dependence of the waveforms on the source parameters.
When we talk about the expected accuracy for gravitational-wave observations we usually refer to the errors induced by instrument noise (say, as predicted by the Fisher matrix), but there is often an unspoken assumption that the theoretical errors due to inaccuracies in our models of the waveforms are negligible by comparison. However, the mergers of supermassive–black-hole binaries will appear in the LISA data with such contrast over noise that we wonder whether our waveform models are truly accurate enough.
In this work with Curt Cutler, we develop a computationally efficient method to estimate these theoretical errors, and we apply it to the LISA observations of supermassive–black-hole binaries. Using a somewhat simplified model of these waveforms, we conclude that for the strongest signals the theoretical errors will be larger than the noise-induced errors, and therefore additional modeling work may be needed to draw the maximum scientific payoff from the LISA observations. However, the expected theoretical errors in the determination of sky position (a crucial piece of information that will be used for optical searches of electromagnetic counterparts to the mergers) appear to be already under control.
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