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We investigate the detectability of subdominant spin effects in merging black-hole binaries using current gravitational-wave data. Using a phenomenological model that separates the spin dynamics into precession (azimuthal motion) and nutation (polar motion), we present constraints on the resulting amplitudes and frequencies. We also explore current constraints on the spin morphologies, indicating if binaries are trapped near spin-orbit resonances. We dissect such weak effects from the signals using a sequential prior conditioning approach, where parameters are progressively re-sampled from their posterior distribution. This allows us to investigate whether the data contain additional information beyond what is already provided by quantities that are better measured, namely the masses and the effective spin. For the current catalog of events, we find no significant measurements of weak spin effects such as nutation and spin-orbit locking. We synthesize a source with a high nutational amplitude and show that near-future detections will allow us to place powerful constraints, hinting that we may be at the cusp of detecting spin nutations in gravitational-wave data. 

Binary black holes with misaligned spins will generically induce both precession and nutation of the orbital angular momentum 𝐋 about the total angular momentum 𝐉. These phenomena modulate the phase and amplitude of the gravitational waves emitted as the binary inspirals to merger. We introduce a “taxonomy” of binary black-hole spin precession that encompasses all the known phenomenology, then present five new phenomenological parameters that describe generic precession and constitute potential building blocks for future gravitational waveform models. These are the precession amplitude ⟨𝜃𝐿⟩, the precession frequency ⟨Ω𝐿⟩, the nutation amplitude Δ⁢𝜃𝐿, the nutation frequency 𝜔, and the precession-frequency variation Δ⁢Ω𝐿. We investigate the evolution of these five parameters during the inspiral and explore their statistical properties for sources with isotropic spins. In particular, we find that nutation of 𝐋 is most prominent for binaries with high spins (𝜒≳0.5) and moderate mass ratios (𝑞∼0.6). 

Abstract LISA, the Laser Interferometer Space Antenna, will usher in a new era in gravitational-wave astronomy. As the first anticipated space-based gravitational-wave detector, it will expand our view to the millihertz gravitational-wave sky, where a spectacular variety of interesting new sources abound: from millions of ultra-compact binaries in our Galaxy, to mergers of massive black holes at cosmological distances; from the early inspirals of stellar-mass black holes that will ultimately venture into the ground-based detectors’ view to the death spiral of compact objects into massive black holes, and many sources in between. Central to realising LISA’s discovery potential are waveform models, the theoretical and phenomenological predictions of the pattern of gravitational waves that these sources emit. This White Paper is presented on behalf of the Waveform Working Group for the LISA Consortium. It provides a review of the current state of waveform models for LISA sources, and describes the significant challenges that must yet be overcome.