Detection, measurement, and gravitational radiation

The optimum design, construction, and use of the Laser Interferometer Gravitational Wave Observatory (LIGO), the French-Italian Gravitational Wave Observatory (VIRGO), or the Laser Gravitational Wave Observatory (LAGOS) gravitational radiation detectors depends upon accurate calculations of their sensitivity to different sources of radiation. Here I examine how to determine the sensitivity of these instruments to sources of gravitational radiation by considering the process by which data are analyzed in a noisy detector. The problem of detection (is a signal present in the output of the detector?) is separated from that of measurement (what are the parameters that characterize the signal in the detector output?). By constructing the probability that the detector output is consistent with the presence of a signal, I show how to quantify the uncertainty that the output contains a signal and is not simply noise. Proceeding further, I construct the probability distribution that the parametrization $\mu$ that characterizes the signal has a certain value. From the distribution and its mode I determine volumes $V\left(P\right)\mathrm{}$ in parameter space such that $\mu \in V\left(P\right)\mathrm{}$ with probability $P$ [owing to the random nature of the detector noise, the volumes $V\left(P\right)\mathrm{}$ are always different, even for identical signals in the detector output], thus quantifying the uncertainty in the estimation of the signal parametrization. These techniques are suitable for analyzing the output of a noisy detector. If we are designing a detector, or determining the suitability of an existing detector for observing a new source, then we do not have detector output to analyze but are interested in the "most likely" response of the detector to a signal. I exploit the techniques just described to determine the "most likely" volumes $V\left(P\right)\mathrm{}$ for detector output that would result in a parameter probability distribution with given mode. Finally, as an example, I apply these techniques to determine the anticipated sensitivity of the LIGO and LAGOS detectors to the gravitational radiation from a perturbed Kerr black hole.