Two Tones From One Remnant: The 330 Mode, Its Separation From the Fundamental, and a Resolvability Criterion

Black-hole spectroscopy becomes a test, rather than a re-measurement, only when a second angular mode is resolved separately from the fundamental. This paper computes the frequency and damping of the l=m=3 mode for two representative remnants, evaluates the frequency separation against the natural linewidth set by the damping, and applies a Rayleigh-style criterion to state how much signal-to-noise ratio a genuine two-mode resolution requires. For the GW150914 remnant, the fundamental sits at 249.1 Hz and the 330 mode at 395.3 Hz, a ratio of 1.587 and a separation of 146.2 Hz. The mean natural linewidth of the two modes is 92.0 Hz, so the separation is 1.59 times the mean linewidth and the pair passes the Rayleigh-style criterion, though not by a wide margin. For the GW190521 remnant the 330 mode falls to 107.8 Hz with a separation of 39.7 Hz from its fundamental, in the part of the band where instrumental linewidth is no longer the only obstacle. The 330 mode of the GW150914 remnant lies at 395.3 Hz, separated from the fundamental by 146.2 Hz against a mean natural linewidth of 92.0 Hz, and resolving it alone at signal-to-noise ratio 8 requires a fundamental near 23. This study is computed from stated assumptions and published catalogue values; it is simulated, not measured.