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The coalescence of two black holes generates gravitational waves that carrydetailed information about the properties of those black holes and their binaryconfiguration. The final coalescence cycles are in the form of a {\itringdown}: a superposition of quasi-normal modes of the merged remnant blackhole. Each mode has an oscillation frequency and decay time that in generalrelativity is determined by the remnant's mass and spin. Measuring thefrequency and decay time of multiple modes makes it possible to measure theremnant's mass and spin, and to test the waves against the predictions ofgravity theories. In this {\it Letter}, we show that the relative amplitudes ofthese modes encodes information about a binary's {\it geometry}. Focusing onthe large mass-ratio limit, which provides a simple-to-use tool for effectivelyexploring parameter space, we demonstrate how a binary's geometry is encoded inthe relative amplitudes of these modes, and how to parameterize the modes inthis limit. Although more work is needed to assess how well this carries overto less extreme mass ratios, our results indicate that measuring multipleringdown modes from coalescence may aid in measuring important sourceproperties, such as the misalignment of its members' spins and orbit.

Learning about black hole binaries from their ringdown spectra