Limits on the sensitivity of spherical gravitational wave detectors and on the accuracy of reconstructed signals

A spherical geometry for a resonant-mass gravitational wave antenna offers significant improvements over traditional cylindrical antennas. However, completing a detector requires breaking the bare antenna's spherical symmetry by attaching multiple mechanical resonators and transducers. To fully assess the merits of such detectors, it is essential to be able to calculate the detector's sensitivity and the accuracy of the extractable signal information without relying on exact mathematical transducer or resonator symmetries to simplify the analysis, as has been done in previous work. Without making such assumptions, this paper generalizes the fundamental sensitivity limits, known for cylindrical detectors, that arise from the back-action noise present in any linear amplifier, and from thermal Brownian motion noise when detection bandwidth is limited. Optimal signal detection and estimation methods are derived by generalizing techniques used for one-dimensional detectors to the case of multiple interacting transducers. Formulas for the optimized signal-to-noise ratio are derived which generalize the connection between bandwidth and sensitivity known for one-dimensional detectors. A demand for isotropic sensitivity then gives requirements on transducer placement and matching. Comparing bandwidth anisotropies, the detector design proposed by Johnson and Merkowitz is found to be superior to an alternative proposal by Lobo and Serrano, and to be reasonably robust against asymmetries. In addition to sensitivity limits and optimal data analysis methods, limits are derived for the accuracy of reconstructed signal parameters such as direction, polarization, phase, and arrival time.