Bi-orthogonal harmonics for the decomposition of gravitational radiation I: angular modes, completeness, and the introduction of adjoint-spheroidal harmonics

TL;DR

This paper introduces adjoint-spheroidal harmonics, a new class of functions that improve the representation and estimation of gravitational wave modes, especially for systems with angular momentum, by addressing mode-mixing issues.

Contribution

The paper develops and demonstrates the use of adjoint-spheroidal harmonics for better mode decomposition in gravitational wave analysis, overcoming limitations of spherical harmonics.

Findings

Spheroidal harmonics can represent arbitrary gravitational wave signals.

Adjoint-spheroidal harmonics possess a novel orthogonality property.

They reduce mode-mixing effects in mode estimation.

Abstract

The estimation of radiative modes is a central problem in gravitational wave theory, with essential applications in signal modeling and data analysis. This problem is complicated by most astrophysically relevant systems' not having modes that are analytically tractable. A ubiquitous workaround is to use not modes, but multipole moments defined by spin weighted spherical harmonics. However, spherical multipole moments are only related to the modes of systems without angular momentum. As a result, they can obscure the underlying physics of astrophysically relevant systems, such as binary black hole merger and ringdown. In such cases, spacetime angular momentum means that radiative modes are not spherical, but spheroidal in nature. Here, we work through various problems related to spheroidal harmonics. We show for the first time that spheroidal harmonics are not only capable of representing arbitrary gravitational wave signals, but that they also possess a kind of orthogonality not used before in general relativity theory. Along the way we present a new class of spin weighted harmonic functions dubbed ``adjoint-spheroidal" harmonics. These new functions may be used for the general estimation of spheroidal multipole moments via complete bi-orthogonal decomposition (in the angular domain). By construction, adjoint-spheroidal harmonics suppress mode-mixing effects known to plague spherical harmonic decomposition; as a result, they better approximate a system's true radiative modes. We discuss potential applications of these results. Lastly, we briefly comment on the challenges posed by the analogous problem with Teukolsky's radial functions