Precision as a Function of Loudness: A Fisher Budget for Ringdown Frequency and Damping Time

How loud must a ringdown be before its frequency is known to one per cent, and how much louder before its damping time is known to ten per cent? This paper answers with a Fisher-matrix budget for a damped sinusoid, evaluated for the fundamental mode of a GW150914-like remnant across a range of signal-to-noise ratios, and expresses the answers as thresholds that observing scenarios can be checked against. Damping time is expected on general grounds to be the harder measurement, and the budget's job is to replace that expectation with the actual factor by which it is harder at each loudness. At a ringdown signal-to-noise ratio of 8, representative of the loudest present events, the Fisher budget gives a fractional frequency uncertainty of 5.0 per cent and a fractional damping-time uncertainty of 15.5 per cent. The damping-time uncertainty exceeds the frequency uncertainty by a factor 3.1 at every tabulated loudness, so the difficulty ordering is a constant of the damped-sinusoid problem and not an artefact of a particular event. Reaching one per cent in frequency requires a ringdown signal-to-noise ratio of 40, while reaching ten per cent in damping time requires 12; both thresholds sit above the ringdown-only loudness of any single event recorded so far. At signal-to-noise ratio 8 the budget gives 5.0 per cent in frequency and 15.5 per cent in damping time, a difficulty ratio of 3.1, with thresholds of 40 for one per cent frequency precision and 12 for ten per cent damping precision. This study is computed from stated assumptions and published catalogue values; it is simulated, not measured.