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The estimation of a physical system's normal modes is a fundamental problemin physics. The quasi-normal modes of perturbed Kerr black holes, with theirrelated spheroidal harmonics, are key examples, and have diverse applicationsin gravitational wave theory and data analysis. Recently, it has been shownthat \textit{adjoint}-spheroidal harmonics and the related spheroidal multipolemoments may be used to estimate the radiative modes of arbitrary sources. Inthis paper, we investigate whether spheroidal multipole moments, relative totheir spherical harmonic counterparts, better approximate the underlying modesof binary black hole spacetimes. We begin with a brief introduction toadjoint-spheroidal harmonics. We then detail a rudimentary kind of spheroidalharmonic decomposition, as well as its generalization which simultaneouslyestimates pro- and retrograde moments. Example applications to numericalwaveforms from comparable and extreme mass-ratio binary black hole coalescencesare provided. We discuss the morphology of related spheroidal moments duringinspiral, merger, and ringdown. We conclude by discussing potentialapplications in gravitational wave theory and signal modeling.

Bi-orthogonal harmonics for the decomposition of gravitational radiation II: applications for extreme and comparable mass-ratio black hole binaries